--- created: 2024-04-17T19:38 updated: 2024-04-17T19:38 --- # Complex Geometry_ An Introduction - Daniel Huybrechts > [!WARNING] **Do not modify** this file > This file is automatically generated by scrybble and will be overwritten whenever this file in synchronized. > Treat it as a reference. ## Pages ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=11|Complex Geometry_ An Introduction - Daniel Huybrechts, page 11]] > = -§={z) = 1 and(1.3)The notation is motivated by the properties -§^{z) = -§={ •§g(z) = > the two tangent spaces can be given canonical bases (d/dx,d/dy) and (d/dr,d/ds), respectively. > the differential df(z) is given by the real Jacobian ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=12|Complex Geometry_ An Introduction - Daniel Huybrechts, page 12]] > Maximum principle. Let U C C be open and connected. If / : U —> > is holomorphic and non-constant, then |/| has no local maximum in U. If U is bounded and / can be extended to a continuous function / : U —> C, then |/| takes its maximal values on the boundary dU. Identity theorem. If /, g : U —> C are two holomorphic functions on a connected open subset U C C such that f(z) = g(z) for all z in a non-empty open subset V C U, then f = g. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=13|Complex Geometry_ An Introduction - Daniel Huybrechts, page 13]] > Liouville. > In particular, there is no biholomorphic map between C and a ball Be(0) with £ < oo. > polydiscs B£{w) = {z | \zi -Wi\ U > / : U —> C > Then / is said to be holomorphic if the Cauchy-Riemann equations (1.2) holds for all coordinates Zi = Xi + iyt, i.e. > By definition, a continuously differentiable) function / is holomorphic if the induced functions > ,...,z > z ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=14|Complex Geometry_ An Introduction - Daniel Huybrechts, page 14]] > Cauchy integral formula for functions of several variables and > The proposition can easily be applied to show that any continuous(l) func tion on an open subset U C Cnwith the property that the function is holo morphic with respect to any single coordinate is holomorphic itself (Osgood's Lemma, > the maximum principle, the identity theorem, and the Liouville theorem generalize easily to the higher dimensional situation. > the Riemann extension theorem holds true, > The Riemann mapping theorem definitely fails ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=15|Complex Geometry_ An Introduction - Daniel Huybrechts, page 15]] > The next result is only valid in dimension at least two. Proposition 1.1.4 (Hartogs' theorem) Suppose e — (ei,..., en) and e' > (e'j,..., e'n) are given such that for all i one has s\ < £». If n > 1 then ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=16|Complex Geometry_ An Introduction - Daniel Huybrechts, page 16]] > Of course, the theorem definitely fails for holomorphic functions of one variable. The informal reason for the theorem is that the singularities of / > given by the vanishing of a holomorphic function. But the zero set of > single holomorphic function would 'stick out' of the smaller disc. > Next we will prove the Weierstrass preparation theorem (WPT) > Definition 1.1.5 A Weierstrass polynomial is a polynomial in z\ of the form z{ + a1(w)zf~1+... + ad{w) where the coefficients ai(w) are holomorphic functions on some small disc in C""1vanishing at the origin. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=17|Complex Geometry_ An Introduction - Daniel Huybrechts, page 17]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=18|Complex Geometry_ An Introduction - Daniel Huybrechts, page 18]] > Proposition 1.1.7 (Riemann extension theorem) Let f be a holomor phic function on an open subset U C C". If g : U \ Z(f) —> C is holomorphic > locally bounded near Z(f), then g can uniquely be extended to a ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=19|Complex Geometry_ An Introduction - Daniel Huybrechts, page 19]] > Here, a thin subset is > subset which locally is contained in the zero set of a non-trivial holomorphic function. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=20|Complex Geometry_ An Introduction - Daniel Huybrechts, page 20]] > Proposition 1.1.10 (Inverse function theorem) Let f : U —> V be a holomorphic map between two open subsets U, V C Cn. If z G U is regular then there exist open subsets z € U' c U and f(z) £ V C V such that f induces a biholomorphic map f : U' —> V. > f : U —> Cnbe a holomorphic map, where m > n. Suppose ZQ € U is a point such that > U\ C C > ,U2 C C™ > g : U\ —> f/2 ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=22|Complex Geometry_ An Introduction - Daniel Huybrechts, page 22]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=23|Complex Geometry_ An Introduction - Daniel Huybrechts, page 23]] > The ring Oc" ,o is local and its maximal ideal m consists of all functions that vanish in 0. In other words, the set of units £>£„0consists of all functions / with /(0) ^ 0. > the WPT can be rephrased by saying that after an appropriate coordinate choice any function / G Oc»,o can be uniquely written as / = g • h, where h G 0c\o isa unitand g € OCn-i0[zx] is a Weierstrass polynomial. The WPT implies the following ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=24|Complex Geometry_ An Introduction - Daniel Huybrechts, page 24]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=25|Complex Geometry_ An Introduction - Daniel Huybrechts, page 25]] > The local UFD 0o\o is noetherian. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=27|Complex Geometry_ An Introduction - Daniel Huybrechts, page 27]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=28|Complex Geometry_ An Introduction - Daniel Huybrechts, page 28]] > Lemma 1.1.28 An analytic germ X is irreducible if and only if I(X) C Oc™ > is a prime ideal. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=30|Complex Geometry_ An Introduction - Daniel Huybrechts, page 30]] > that any prime ideal of height one in a UFD is principal. > More generally, any analytic germ of codimension one is the zero set of a single holomorphic function. In the theory of functions of one variable a meromorphic function / on an open subset U C C is a holomorphic function defined on the complement of > discrete set of points S C U such that / has poles of finite order in all points of S. Then one shows that locally around any point of 5 the function can be written as the quotient of two holomorphic functions. > A meromorphic function f on U is a function on the complement of a nowhere dense subset S C U with the ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=31|Complex Geometry_ An Introduction - Daniel Huybrechts, page 31]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=34|Complex Geometry_ An Introduction - Daniel Huybrechts, page 34]] > Definition 1.2.1 An endomorphism / : V —> V with I2= —id is called an almost complex structure on V. > Lemma 1.2.2 // / is an almost complex structure on a real vector space > by (a + ib) • v = a • v + b • ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=38|Complex Geometry_ An Introduction - Daniel Huybrechts, page 38]] > complex structure / on V associated to it which is denned as follows: For any 0 7^ v € V the vector I(v) € V is uniquely determined by the following three > with ( , ). ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=41|Complex Geometry_ An Introduction - Daniel Huybrechts, page 41]] > The Lefschetz operator comes along with its dual A. In order to define and to describe A we need to recall the Hodge *-operator on a real vector space. > vol = e\ A ... A e^. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=44|Complex Geometry_ An Introduction - Daniel Huybrechts, page 44]] > Corollary 1.2.27 Lei (V, ( , ),/) be an euclidian vector space with a com patible almost complex structure. The action of L, A, and H defines a natural sl(2)-representation on /\* V*. > of all 2 x 2-matrices of trace zero. A basis is given byX = (§J),7 = (1 0) > and B = (0 -?i)- A quick calculation shows that they satisfy [B, X] = 2X, [B,Y] = -2Y, and [X,Y] = B. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=45|Complex Geometry_ An Introduction - Daniel Huybrechts, page 45]] > An element a € f\ V* is called primitive if Aa. = 0. The linear subspace of all primitive elements a € f\ V* is denoted by PkC f\ V*. > There exists a direct sum decomposition of the form: ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=48|Complex Geometry_ An Introduction - Daniel Huybrechts, page 48]] > The Hodge-Riemann pairing is the bilinear form ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=49|Complex Geometry_ An Introduction - Daniel Huybrechts, page 49]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=52|Complex Geometry_ An Introduction - Daniel Huybrechts, page 52]] > By Ac(U) and Ap As before, the projection operators /\cU —> /\p'9U and A£{U) -> Ap'q(U) will be denoted by np'q ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=53|Complex Geometry_ An Introduction - Daniel Huybrechts, page 53]] > i) d = 8 + B. ii) d2= B2= 0 and 3d = -3d. > They satisfy the Leibniz rule, i.e. d{a Af3) = d(a) A /3 + (-l)p+qa A d{(3) ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=57|Complex Geometry_ An Introduction - Daniel Huybrechts, page 57]] > Let U C Cnbe an open subset and consider a Riemannian metric g on U. > The metric g is compatible with the natural (almost) complex structure on U > gx(v,w) = gx(I(v),I(w)) > w € AU1(U) n A2{U) denned by which is called the fundamental form of g. Moreover, h := g — iuj defines ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=62|Complex Geometry_ An Introduction - Daniel Huybrechts, page 62]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=63|Complex Geometry_ An Introduction - Daniel Huybrechts, page 63]] > A meromorphic function on a complex manifold X is a map ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=64|Complex Geometry_ An Introduction - Daniel Huybrechts, page 64]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=65|Complex Geometry_ An Introduction - Daniel Huybrechts, page 65]] > Thus, the function field K(X) contains a purely transcendental extension of C of degree a{X) < dim(X). Any other meromorphic function on X will be algebraic over this extension. In fact, due to a theorem of Remmert [97] one knows that the function field is a finite extension of this purely transcendental extension. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=66|Complex Geometry_ An Introduction - Daniel Huybrechts, page 66]] > The transition functions are just translations by vectors in Z2™. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=67|Complex Geometry_ An Introduction - Daniel Huybrechts, page 67]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=68|Complex Geometry_ An Introduction - Daniel Huybrechts, page 68]] > a complex torus X = Cn/ r , where the group operations are induced by the natural ones on Cn. Clearly, X is an abelian complex Lie group. > any connected compact complex Lie group is abelian > and in fact a torus > The action is free if for all 1 ^ g £ G and all x € X one has g • x ^ x. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=69|Complex Geometry_ An Introduction - Daniel Huybrechts, page 69]] > If G is discrete and the action on X is free and proper, > (The action is also called properly discontinuous in this > Any point x G X admits a neighbourhood x G UxC X such that g(Ux) n Ux= 0 for any 1 ^ g G G. > If x,y G X such that x ^ G-y, then there exist neighbourhoods x G f/xC X and y £Uy C X with f7xD #(£/,,) = 0 for any g € G. The first condition ensures that the quotient X/G admits holomorphic charts and the second that the quotient is Hausdorff. > If G does not act freely on a complex manifold X, then the quotient X/G may or may not be a manifold. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=70|Complex Geometry_ An Introduction - Daniel Huybrechts, page 70]] > A ball quotient is a quotient of D by any discrete group F C SU(1, n) acting freely on D. Usually one also assumes the quotient D/F to be compact. > Hopf manifolds. Let Z act on Cn\ {0} by (z\,..., zn) *—> (Xkzi,..., \kzn) for k G Z. For 0 < A < 1 the action is free and discrete. The quotient complex manifold X = (Cn\ {0})/Z is diffeomorphic to S1x S2""1 ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=71|Complex Geometry_ An Introduction - Daniel Huybrechts, page 71]] > Then Flag(V, fc1;..., kt) is the manifold of all flags WlcW2C ...cWeCV with dim(Wi) = kt > Flag(V, k) = Gik{V). > Flag(V, l,n) is the incidence variety {(£, H) \ £ C H} C P(V) x ¥(V*) of all pairs (£, H) consisting of a line £ c V contained in a hyperplane H C V. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=72|Complex Geometry_ An Introduction - Daniel Huybrechts, page 72]] > Some complex tori, but not all (see Exercise 3.3.6), are projective. Any Oy-she&f T on a complex submanifold Y G X can be considered as an O^-sheaf on X supported on Y. More precisely, one identifies J- with its direct image i*!F under the inclusion i : Y C X. > where Xy is the ideal sheaf of all holomorphic functions vanishing on Y. > An analytic subvariety of X is a closed subset Y C X such that for any point x € X there exists an open neighbourhood x € U C X such that Y D U is the zero set of finitely many holomorphic functions /i,... , /fc £ O(U). > we see that an analytic subvariety in a neighbour hood of a regular point is nothing but a complex submanifold. > the set of regular points Yleg= Y \ Y"Sing is a non-empty complex submanifold of X > For singular Y C X one defines Oy by this sequence. This amounts to giving Y the induced reduced structure. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=73|Complex Geometry_ An Introduction - Daniel Huybrechts, page 73]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=75|Complex Geometry_ An Introduction - Daniel Huybrechts, page 75]] > E(x) := n~1(x) > There exists an open cov ering X = [JUi and biholomorphic maps tpi : ?r~1([/i) = C/j x C commuting with the projections to [/, such that the induced map TT~1(X) = Cris C-linear. > vector bundle homomorphism from E to F is a holomorphic map tp : E —•> F with TTE = npoip such that the induced map f(x) : E(x) —> F(x) is linear ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=76|Complex Geometry_ An Introduction - Daniel Huybrechts, page 76]] > On the complement E \ s(X) one has a natural C*-action. The quotient F(E):=(E\s(X))/C* ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=77|Complex Geometry_ An Introduction - Daniel Huybrechts, page 77]] > Recall that in the category of differentiable vector bundles every short exact sequence splits > This is no longer true in the holomorphic setting, e.g. the Euler sequence in Section 2.4 does not split. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=78|Complex Geometry_ An Introduction - Daniel Huybrechts, page 78]] > of 0(1). > L ® L* is isomorphic to the trivial line bundle. This is best seen by using the cocycle description of L and the induced one for L*. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=79|Complex Geometry_ An Introduction - Daniel Huybrechts, page 79]] > Very roughly, Pic(X) has two parts. A discrete part, measured by its image in H2(X, Z), and a continuous part coming from the (possibly trivial) vector space H1(X, Ox)- ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=81|Complex Geometry_ An Introduction - Daniel Huybrechts, page 81]] > o/ sections, also called £?, is given by > Consider the holomorphic tangent bundle Tjr. The associated sheaf is sometimes denoted Qx- Furthermore, €>x admits an alternative de scription as the sheaf Der(Ox) of derivations. More precisely, Der(Ox) is the sheaf that associates to an open subset U C X the set of all C-linear maps D : OX(U) -> OX(U) satisfying the Leibniz rule D(f-g) = f-D(g) + D(f)-g. > local sections of 7~x are of the form ^2 a,i > where the a^z) are holomorphic, ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=83|Complex Geometry_ An Introduction - Daniel Huybrechts, page 83]] > However, the canonical ring R(X) = R{X, Kx) is expected to be finitely generated (abundance conjecture), at least if X is projective. In order to relate the Kodaira dimension to the algebraic dimension > we have to explain how to produce meromorphic functions from sections of line bundles. Let s\, S2 £ H°(X, > s\/s2 £ K(X) > ipi '• L\Ui — O\ji. Hence, V'iCsilf;) and ipi(s2\Ui) are holomorphic functions on Ui. The meromorphic function > Qo(R) is the subfield of the quotient field Q{R) that consists of elements of the form f/g with 0 ^ g, f £ Rmfor some m. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=85|Complex Geometry_ An Introduction - Daniel Huybrechts, page 85]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=86|Complex Geometry_ An Introduction - Daniel Huybrechts, page 86]] > hypersurfaces are always given as the zero locus of a global holomorphic section of a holomorphic line bundle. > A divisor D = ^ai[Yj] is called effective if a* > 0 for ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=87|Complex Geometry_ An Introduction - Daniel Huybrechts, page 87]] > the order ordyiX(/) of / in x with respect to Y is given by the equality / =fford(f> • h with h G O > Let / G K(X). Then the divisor associated to f is ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=88|Complex Geometry_ An Introduction - Daniel Huybrechts, page 88]] > Proposition 2.3.9 There exists a natural isomorphism ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=89|Complex Geometry_ An Introduction - Daniel Huybrechts, page 89]] > By definition O(D + D') is described by {(/, • f!) • (/, • /j)"1} = {&r > Hence, O(D + D') = O(D) ® O(D'). > thus O(-D) ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=90|Complex Geometry_ An Introduction - Daniel Huybrechts, page 90]] > then O(f*D) ^ f*O{D). ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=91|Complex Geometry_ An Introduction - Daniel Huybrechts, page 91]] > Two divisors D,D' G Div(X) are called linearly equiva lent, D ~ D', if D — D' is a principal divisor. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=92|Complex Geometry_ An Introduction - Daniel Huybrechts, page 92]] > Let 0 / s G H°(X,L). Then the line bundle O(Z{s)) is isomorphic to L. > For any effective divisor D G T)iv(X) there exists a section 0 / s £ H°(X,O(D)) with Z(s) = D. > The image of the natural map Div(X) —* Pic(X) is genera ted by those line bundles L G Pic(X) with H°(X,L) ^ 0. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=93|Complex Geometry_ An Introduction - Daniel Huybrechts, page 93]] > More generally, for any effective divisor D = £^aj[Yi] we obtain a short exact sequence > where OD is the quotient of Ox by all holomorphic functions vanishing of order at least <2j along Yj. > Let L be a holomorphic line bundle on X. A meromorphic section of L is a collection of meromorphic functions fi £ K-x{Ui) °n open subsets Ui of an open covering X — \JUi such that fi = ipij • fj, where ^ > the cocycle defined by trivializations tpi : L\ui= Ou^- > a projective manifold any line bundle has a non-trivial meromorphic section, but for a general complex manifold this is not true. > A holomorphic function s £ H°(X, Ox) on a complex manifold X defines ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=94|Complex Geometry_ An Introduction - Daniel Huybrechts, page 94]] > when X is compact, there exist only very few (or none at all) non-constant holomorphic functions. Replacing holomorphic functions by holomorphic sec tions s & H°(X,L) of a holomorphic line bundle L gives us more flexibil > s0,..., SJV G H°(X, L) of a holomorphic line bundle L only define a holomor phic map to some projective space P^ and, moreover, that this map might not be everywhere well-defined on X. > Let L be a holomorphic line bundle on a complex manifold X. A point x e X is a base point of L if s(x) = 0 for all s e H°(X, L). The base locus Bs(L) is the set of all base points of L. > Bs(L) = Z(SQ) n ... n Z(SN) is an analytic subvariety. > Let L be a holomorphic line bundle on a complex man ifold ifold X and suppose that so,..., SJV € H°(X, L) is a basis. Then Instead of taking a basis SQ, • • •, SJV one could just take any collection of sections, nei ther necessarily linearly independent nor generating H°(X,L). This yields a holomorphic map on the complement of the base locus > The map cp^ as in the proposition is said to be associated to the complete linear system H°(X,L), whereas a subspace of H°(X, L) is simply called a linear system of L. Sometimes, one denotes the complete linear system F(H°(X,L)) also by \L\. One says that L is globally generated by the sections So, •. ., SJV if Bs(L, SQ, ..., sjv) = 0. > A line bundle L on a complex manifold is called ample if for some k > 0 and some linear system in H°(X, Lk) the associated map

an embedding. By definition, a compact complex manifold is projective if and only if it admits an ample line bundle. > The induced map the Veronese embedding. > where (IQ, ..., in) runs through all multi-indices with Y^j=o h=d- > Let us consider the case X = P1and d = 2. Then the Veronese map defines an isomorphism of P1with the hypersurface Z(x$X2 — x\) of P2via (z0: zi) H-> 02: zozi : z\). ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=96|Complex Geometry_ An Introduction - Daniel Huybrechts, page 96]] > The Segre map is the induced map ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=97|Complex Geometry_ An Introduction - Daniel Huybrechts, page 97]] > Proposition 2.3.30 If D is a principal divisor on a compact curve then deg(D) = 0. > Lemma 2.3.31 Let f G K{X) be a meromorphic function on a curve X. Then the induced map f : X \ P(f) —> C extends naturally to a holomorphic ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=98|Complex Geometry_ An Introduction - Daniel Huybrechts, page 98]] > the map X ^Pic(X), xi *~O(x-x0), which depends on a chosen point XQ G X. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=100|Complex Geometry_ An Introduction - Daniel Huybrechts, page 100]] > This way we associate to any homogeneous polynomial s of degree k a global holomorphic section of O(k), ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=101|Complex Geometry_ An Introduction - Daniel Huybrechts, page 101]] > The canonical bundle Kpn is isomorphic to O(—n — 1). ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=102|Complex Geometry_ An Introduction - Daniel Huybrechts, page 102]] > det(F) = det(£')det(Gr) > Proposition 2.4.4 (Euler sequence) On Iexact sequence of holomorphic vector bundlesthere exists a natural short > It suffices to show that the kernel of the dual map ©™=00(-l) -* O is canonically isomorphic to J2pn. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=103|Complex Geometry_ An Introduction - Daniel Huybrechts, page 103]] > Remark 2.4-5 The dual of the Euler sequence twisted by 0(1) takes the form > V* is naturally identified with the space of homogeneous linear forms on V and, therefore, V* ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=104|Complex Geometry_ An Introduction - Daniel Huybrechts, page 104]] > Corollary 2.4.6 One has kod(Pn) = -oo. Proof. > This This yields in particular an example of a complex compact manifold with kod(X) < a(X). Other examples are provided by projective complex tori. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=105|Complex Geometry_ An Introduction - Daniel Huybrechts, page 105]] > Corollary 2.4.9 //X cP " is a smooth hypersurface of degree d, i.e. defined by a section s £ H°(Pn, O(d)), then Kx^O(d-n-l)\ > Show that the canonical bundle Kx of a complete intersection X = Z(/i) n ... O Z(/fe) C Pnis isomorphic to O(£,deg(/i) - n - l))|x ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=106|Complex Geometry_ An Introduction - Daniel Huybrechts, page 106]] > Show that Enis isomorphic to the hypersurface Z(x$y\ — 2:12/2) C P1x P2, where > Describe the tangent, cotangent, and canonical bundle of Pnx Pm ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=109|Complex Geometry_ An Introduction - Daniel Huybrechts, page 109]] > Note that blowing-up along a smooth divisor Y C X does not change X, i.e. in this case X = X. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=111|Complex Geometry_ An Introduction - Daniel Huybrechts, page 111]] > X#Pn ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=112|Complex Geometry_ An Introduction - Daniel Huybrechts, page 112]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=113|Complex Geometry_ An Introduction - Daniel Huybrechts, page 113]] > Definition 2.6.1 An almost complex manifold is a differentiable manifold X together with a vector bundle endomorphism / : TX >• TX, with I2= -id. > Proposition 2.6.2 Any complex manifold X admits a natural almost complex structure. > Not every real manifold of even dimension admits an almost complex structure. The easiest example is provided by the four-dimensional sphere. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=114|Complex Geometry_ An Introduction - Daniel Huybrechts, page 114]] > If X is a complex manifold, then Tl'°X is naturally isomorphic > to the holomorphic tangent bundle Tx > Definition 2.6.5 The bundles Tl'°X and T°'lX are called the holomorphic respectively the antiholomorphic tangent bundle of the (almost) complex ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=115|Complex Geometry_ An Introduction - Daniel Huybrechts, page 115]] > f : X —> Y > f* : Ap'q(Y) —» Ap'q(X). ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=116|Complex Geometry_ An Introduction - Daniel Huybrechts, page 116]] > f : X —> Y > Tx —> f*Ty, > f*fly —> J?x, > H°{Y, Q^) -^ H°{X, f2 ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=117|Complex Geometry_ An Introduction - Daniel Huybrechts, page 117]] > Definition 2.6.16 An almost complex structure / on X is called integrable if the condition i) or, equivalently, ii) in Proposition 2.6.15 is satisfied. > Proposition 2.6.17 An almost complex structure I is integrable if and only if the Lie bracket of vector fields preserves T^ , i.e. [Tx' ,TX' } C T ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=118|Complex Geometry_ An Introduction - Daniel Huybrechts, page 118]] > Theorem 2.6.19 (Newlander—Nierenberg) Any integrable almost com plex structure is induced by a complex structure. > This is the analogue of the coho mology group Hq(X, (2X):used to defines the Hodge numbers of a compact complex manifold, in the context of almost complex structures. > Then the (p, q)-Dolbeault cohomology is the vector space ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=119|Complex Geometry_ An Introduction - Daniel Huybrechts, page 119]] > We conclude this section by the vector bundle analogue of Theorem 2.6.19. It is in some sense a linearized version of the Newlander-Nierenberg theorem, ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=120|Complex Geometry_ An Introduction - Daniel Huybrechts, page 120]] > complex structure. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=121|Complex Geometry_ An Introduction - Daniel Huybrechts, page 121]] > Theorem 2.6.26 allows to study all holomorphic structures on a given complex vector bundle E by means of the set of 5_E-operators on E. Be aware that one still has to divide out by the gauge group, i.e. diffeomorphisms of E which are linear on the fibres and which cover the identity on X. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=122|Complex Geometry_ An Introduction - Daniel Huybrechts, page 122]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=123|Complex Geometry_ An Introduction - Daniel Huybrechts, page 123]] > a Kahler metric, which by definition coincides locally with the standard hermitian structure on C" up to terms of order at least two. > X if for any point x G X the scalar product gxon TXX is compatible with the almost complex structure Ix. The induced real (1, l)-form 10 := g(I( ), ( )) is > Locally the fundamental form to is of the form where for any x £ X the matrix (hij(x)) is a positive definite hermitian > the hermitian structure g is uniquely de termined by the almost complex structure / and the fundamental form w. Indeed, g( , )=w( ,/()). > Lefschetz operator ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=124|Complex Geometry_ An Introduction - Daniel Huybrechts, page 124]] > dual Lefschetz > Corollary 3.1.2 Let (X,g) be an hermitian manifold. Then there exists > direct sum decomposition of vector bundles > forms. > A*(X) —+ Ap'q(X), > adjoint opera tor d* is defined as > Laplace operator is > A = d*d + dd*. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=125|Complex Geometry_ An Introduction - Daniel Huybrechts, page 125]] > d* := — * o 8 o * and 8* = — * o d o > Hodge ^-operator maps Ap'q(X) to An-q'n~p{X). Thus, > Ad:=d*d + dd* and A8:= 8*5 + 88*. > Ad,A8:AP'q(X) *A > Definition 3.1.6 A Kahler structure (or Kahler metric) is an hermitian structure g for which the fundamental form to is closed, i.e. du> = 0. In this case, the fundamental to form is called the Kahler form. > Hermitian structures exist on any complex manifold (cf. Exercise 3.1.1), but, as we will see shortly, Kahler structures do not always exist. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=126|Complex Geometry_ An Introduction - Daniel Huybrechts, page 126]] > Let tu be a dosed real (1, l)-form on a complex manifold X. If OJ is positive definite, i.e. u> is locally of the form to = | ^hijdzi A dSj such that (hij(x)) is a positive definite hermitian matrix for any x £ X, then there exists a Kahler metric on X such that u is the associated fundamental form.U Thus, the set of closed positive real (1, l)-forms to G A1'1(X) is the set of all Kahler forms. Corollary 3.1.8 The set of all Kahler forms on a compact complex manifold X is an open convex cone in the linear space {u> G A1'1(X)nA2(X) \ dto = 0}. > Examples 3.1.9 i) The Fubini-Study metric is a canonical Kahler metric on ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=128|Complex Geometry_ An Introduction - Daniel Huybrechts, page 128]] > Any complex curve admits a Kahler structure. In fact, any hermitian metric is Kahler, as a two-form on a complex curve is always closed. For the existence of hermitian structures see Exercise 3.1.1. > On the unit disc Dn ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=129|Complex Geometry_ An Introduction - Daniel Huybrechts, page 129]] > Corollary 3.1.11 Any projective manifold is Kahler. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=132|Complex Geometry_ An Introduction - Daniel Huybrechts, page 132]] > Show that any complex manifold admits an hermitian structure. > In particular, b2(X) > 1. Deduce from this that S is the only sphere that admits a Kahler structure. > A closed two-form ui on M is a symplectic structure (or form) if UJ is everywhere non-degenerate, i.e uinis a volume form. Show that any Kahler manifold (X, g) possesses a natural symplectic volume ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=134|Complex Geometry_ An Introduction - Daniel Huybrechts, page 134]] > Definition 3.2.1 Let (X,g) be a compact hermitian manifold. Then one de fines an hermitian product on AQ(X) by > Thus, each Ap'q(X) is an infinite-dimensional vector space endowed with ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=139|Complex Geometry_ An Introduction - Daniel Huybrechts, page 139]] > Definition 3.2.14 Let X be a compact Kahler manifold. The Kahler class associated to a Kahler structure on X is the cohomology class [u>] G i/1'1(X) of its Kahler form. The Kahler > is the set of all Kahler classes associated to any Kahler structure on X. > Let (X, g) be a Kahler manifold. Show that the Kahler form us is harmonic. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=140|Complex Geometry_ An Introduction - Daniel Huybrechts, page 140]] > Show that holomorphic forms, i.e. elements of H°(X, f2p), on a compact Kahler manifold X are harmonic with respect to any Kahler metric. > Show that Hp'q(¥n) = 0 except for p = q < n. In the latter case, the space is one-dimensional. Use ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=142|Complex Geometry_ An Introduction - Daniel Huybrechts, page 142]] > Proposition 3.3.2 (Lefschetz theorem on (1, l)-classes) LetX be a com pact Kahler manifold. Then Pic(X) —> H1'1(X, Z) is surjective. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=143|Complex Geometry_ An Introduction - Daniel Huybrechts, page 143]] > The Lefschetz theorem above thus says that the natural inclusion NS(X) C Hl'l{X, Z) is an equality. If X is projective, yet another description of the Neron-Severi group can be given. Then, NS(X) is the quotient of Pic(X) by the subgroup of numerically trivial line bundles. A line bundle L is called numerically trivial if L is of degree zero on any curve C C X. If X is not projective it may very well happen that there are no curves in X, but yet NS(X) ^ 0. Definition 3.3.4 Let X be a compact complex manifold. Then the rank of the image Pic(X) —> H2(X,R) is called the Picard number p(X). Thus, if X is in addition Kahler the Picard number satisfies p(X) = rk{H1'1(X,Z))=rk(NS(X)). > to the continuous part of the Picard group, i.e. the kernel of Pic(X) —> H2(X, Z). > Definition 3.3.5 Let X be a complex manifold. Its Jacobian Pic°(X) is the kernel of the map Pic(X) -> H2(X, Z). Using the exponential sequence the Jacobian can always be described as the quotient Hl{X, O)/H1(X, Z), but only for compact Kahler manifolds one should expect this to be anything nice. Note that for compact manifolds the natural map H1(X, Z) —> HX(X, O) is really injective. > Corollary 3.3.6 If X is a compact Kahler manifold, then Pic°(X) is in ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=144|Complex Geometry_ An Introduction - Daniel Huybrechts, page 144]] > Pic(X) is fibred by tori of complex dimension b\(X) > PicQ(X) There is another complex torus naturally associated with any compact Kahler manifold, the Albanese torus. > HX{X,1) -> H°(X,f2x)* given by [7] H-> (a H J a), > Since X is Kahler, the image forms a lattice. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=145|Complex Geometry_ An Introduction - Daniel Huybrechts, page 145]] > i) If X is of dimension one, i.e. X is a curve, then Alb(X) = Pic°(X). > Furthermore, the Albanese map alb : X —> Alb(X) coincides with the Abel-Jacobi map X —> Pic(X), x i—> O{x — XQ) defined in Section 2.3. > Let X be a complex torus given as V/F with F a lattice inside the com plex vector space V. Hence, there exist natural isomorphisms TQX = V and H°(X,nx) = V*. > Fi(X,Z) ^ F and the embedding #i(X,Z) > H°(X,f2x)* is the given inclusion J1c V. With XQ = 0 the Albanese map X -> Alb(X) = V/r is the identity. Without representing a complex torus X as a quotient Vy.T, the Albanese map enables us to write X, canonically up to the choice of a base point XQ, as X = H°(X, (2x)*/Hi(X, Z). ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=146|Complex Geometry_ An Introduction - Daniel Huybrechts, page 146]] > Proposition 3.3.13 (Hard Lefschetz theorem) Let (X,g) be a compact Kahler manifold of dimension n. Then for k < n > Another way to phrase Proposition 3.3.13 is to say that any Kahler class on X yields a natural st(2)-representation on H*(X,W). ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=147|Complex Geometry_ An Introduction - Daniel Huybrechts, page 147]] > The Hodge numbers hp'q(X) of a compact Kahler manifold X of dimension n are visualized by the Hodge diamond: > ~T\ > t Hodge Serre > conjugation Serre duality (Corollary 3.2.12) shows that the Hodge diamond is invariant under rotation by IT. Complex conjugation Hp'q(X) = Hq>p(X) provides an other symmetry: Thus, the Hodge diamond is also invariant under reflection in > a reflection in the horizontal line passing through hn'° and h°'n ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=148|Complex Geometry_ An Introduction - Daniel Huybrechts, page 148]] > Corollary 3.3.16 (Hodge index theorem) Let X be a compact Kahler surface, then the intersection pairing > has index {2h2>°(X) + l,/iu( * ) - !)• Restricted to H1'1^) it is of index > The corollary can be used to exclude that certain compact > ferentiable manifolds of dimension four admit a Kahler structure > Moreover, since one can show that any compact complex surface with b\(X) even admits a Kahler structure (this ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=149|Complex Geometry_ An Introduction - Daniel Huybrechts, page 149]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=150|Complex Geometry_ An Introduction - Daniel Huybrechts, page 150]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=151|Complex Geometry_ An Introduction - Daniel Huybrechts, page 151]] > Definition 3.3.20 A class in HC'C{X,Q) := HC'C(X) n H*(X,Q) is called analytic if it is contained in the Q-vector space generated by all fundamental classes [Z] G HC>C(X,Z). > Hodge Conjecture. Let X be a projective complex manifold. Then any > It is known that the Hodge conjecture is false when for mulated over Z, i.e. not every class in HC'C(X, Z) is contained in the group generated by fundamental classes. > The Lefschetz theorem on (1, l)-classes provides evidence for the Hodge conjecture. First of all, one verifies that the image of O(D) under Pic(X) > H2(X,W) is in fact [D]. This > will see that the Kodaira vanishing theorem ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=152|Complex Geometry_ An Introduction - Daniel Huybrechts, page 152]] > Show that any complex line bundle on Pncan be endowed with a unique holomorphic structure. Find an example of a compact (Kahler, projective) manifold and a complex line bundle that does not admit a holomorphic complex structure. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=153|Complex Geometry_ An Introduction - Daniel Huybrechts, page 153]] > Kahler manifolds are special instances of symplectic manifolds. > So, one might wonder whether every symplectic manifold is in fact Kahler. Even in dimension four, this does not hold. The first example of a symplectic fourfold which is not a Kahler surface was constructed by Thurston. > one had tried to prove that irreducible fourfolds are always complex. This turned out to be false due to counterexamples of Gompf and Mrowka. > Of course, one has tried to prove the Hodge conjecture in special cases. But even for abelian varieties, i.e. projective complex tori, the question is still open. > Picard and Albanese torus are instances of a whole series of complex tori associated to any compact Kahler manifolds, so called intermediate Jacobians. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=154|Complex Geometry_ An Introduction - Daniel Huybrechts, page 154]] > Any dga (A = © A1, d) gives rise to a complex of k-vector spaces ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=155|Complex Geometry_ An Introduction - Daniel Huybrechts, page 155]] > Definition 3.A.8 A dga (A,d) is called connected if the inclusion k C .4° induces an isomorphism k = H°(A, d). It is called simply connected if in addition H1(A,d) =0 > a differentiable manifold M is connected if and only if its de Rham algebra (A*(M), d) is connected. If M is simply connected, then also (A*(M),d) is simply connected, but the converse does not hold. Any differentiable manifold with finite fundamental group is a counterexample. > Definition 3. A.10 A dga (M = ® M.1, d) is called minimal if the following conditions are satisfied: i) M° = k. > M.+:= (Bi>o-M* is free; i- e- there exist homogeneous elements xi,X2,... of positive degree d\,d2,. •., such that M.+= S+(xi,22,...) (the > by x\,X2, • •., cf. Section 3.B). > The Xj's in ii) can be chosen such that d\ < d condition ii) says that any a £ A4+is a lin ear combination of terms of the form xl± • x22• ... and that all relations are ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=156|Complex Geometry_ An Introduction - Daniel Huybrechts, page 156]] > Definition 3.A.15 The minimal model of a dga (A, dj{) is a minimal dga together with a dga-quasi-isomorphism / : (A4, dj^) —> (A, dj\). ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=158|Complex Geometry_ An Introduction - Daniel Huybrechts, page 158]] > Definition 3.A.20 A dga [A, d^) is called formal if (A, dj) is equivalent to a dga (B, ds) with rfg = 0. Clearly, (A, djC) is formal if and only if [A, djCj is equivalent to its coho mology dga (H*(A, dA),d = 0). Definition 3.A.21 A differentiable manifold M is formal if its de Rham al gebra (A*(M),d) is a formal dga. A general differentiable manifold is not formal, but some interesting ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=160|Complex Geometry_ An Introduction - Daniel Huybrechts, page 160]] > Merkulov has recently shown in [90] a weak version (forgetting the multiplicative structure) of formality for any symplectic manifold whose cohomology satisfies the Hard Lefschetz theorem. > There are certainly manifolds that are formal without being Kahler, but sometimes the formality property of Kahler manifolds can indeed be used to exclude a given compact complex manifold from being Kahler. In > Massey triple-products. Let M be a differentiable manifold and let a G HP(M,R), (3 G Hq(M,R), and 7 G Hr(M,R) be cohomology classes satisfying 0 = aj3 G Hp+q(M,R) and 0 = /?7 G Hq+r(M, R). Thus, if a, /3,7 represent a, (3, and 7, respectively, then a A f3~ = df and ft A 7 = dg for certain forms / G Ap+q~1(M) and Definition 3.A.31 The Massey triple-product ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=161|Complex Geometry_ An Introduction - Daniel Huybrechts, page 161]] > where G is the Green operator, i.e. G is the inverse of the Laplace operator A > It turns out that the de Rham complex of a compact Riemannian manifold is endowed with the structure of an A ^ -algebra ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=162|Complex Geometry_ An Introduction - Daniel Huybrechts, page 162]] > It is easy to find examples of manifolds which are not formal. The following is a real version of the complex Iwasawa manifold of dimension > Thus, there are non-trivial Massey triple products on M and, hence, M is not formal. > Note that the necessary condition bi(X) = 0(2) for a complex manifold to be Kahler, is satisfied in this example. Thus, only the finer information about the non formality shows that X is non-Kahler. Non-formality of X does not only show that the complex manifold X is not Kahler, but that there is no complex structure on the underlying differentiable manifold that is Kahler. > If M is a compact manifold, its cohomology H*(M,M.) in general reflects only a small part of its topology. Surprisingly, the de Rham complex A*{M) encodes much more of it, in fact the real homotopy of M is determined by it. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=163|Complex Geometry_ An Introduction - Daniel Huybrechts, page 163]] > Show that all spheres are formal. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=164|Complex Geometry_ An Introduction - Daniel Huybrechts, page 164]] > The two basic objects needed for a reasonable field theory in physics are a Hilbert space and a Hamiltonian. In In a geometric context, the first proposal for the Hilbert space would be the space of L2-functions on the given Riemannian manifold with the Laplacian as the Hamiltonian. The symmetries of the theory are then encoded by linear operators on the Hilbert space that commute with the Hamiltonian, i.e. the Laplacian. > soon passes on to the space of all differential forms. The space of differential forms can still be completed to a Hilbert space by introducing L2-forms, but > Every element can be decomposed into its even and its odd part, where even and odd is meant with respect to the usual degree of a differential form. > of even degree, e.g. functions, are bosonic and odd forms, e.g. one-forms, are fermionic. > troduces a natural super Lie algebra of symmetry operators, e.g. certain dif ferential operators for differential forms which commute with the Laplacian. > the size of the Lie algebra in physicists jargon is measured by the amount of super symmetry. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=166|Complex Geometry_ An Introduction - Daniel Huybrechts, page 166]] > The space of complex differential forms A^(M) be comes a super vector space by writing it as > one might also work with the odd operators <5i := d + d* and Q2:= i(d — d*). > Let Q be the super Lie subalgebra of End(A^(M)) generated by d, d*,A. Then g is of dimension three with a one-dimensional even part go spanned by A. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=167|Complex Geometry_ An Introduction - Daniel Huybrechts, page 167]] > One might then try to generate a super Lie algebra by the operators A,d,8,d*,8*, but this, in general, gives something messy. They are not 'closed', i.e. the set of generators > in general, span the vector space of the super Lie algebra they generate, > Even worse, the super Lie algebra might be of infinite dimension. > The Kahler condition has two effects. Firstly, due to the Kahler identitities the super Lie algebra spanned by A,d,8,d* ,8* is closed. Its even part is one-dimensional and the odd part is of dimension four. The Lie algebra generators form in fact a basis of the underlying vector space. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=168|Complex Geometry_ An Introduction - Daniel Huybrechts, page 168]] > For so called hyperkahler manifolds one obtains N = (4,4) su persymmetry algebras. Roughly, a hyperkahler metric is a Kahler metric which is Kahler with respect to a whole two-dimensional sphere worth of complex structures. > All the induced Lefschetz and differential operators are used to form this bigger algebra. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=169|Complex Geometry_ An Introduction - Daniel Huybrechts, page 169]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=170|Complex Geometry_ An Introduction - Daniel Huybrechts, page 170]] > By Q(/c) one denotes the unique one-dimensional weight —Ik Hodge structure given by z i—> z~kz~k > Let V be a rational vector space such that VR is endowed with an almost complex structure. Then, /\ V has a natural weight k Hodge structure given by the bidegree decomposition /\cV = (§)Vp'q(see Section 1.2). > Definition 3.C.4 Let if be a rational Hodge structure of weight k. The in duced Hodge filtration ... Fl+1Hc C FlHc C ... C He is given by > Thus, from the Hodge filtration one recovers the Hodge stucture itself. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=171|Complex Geometry_ An Introduction - Daniel Huybrechts, page 171]] > The tensor product H ® Q(fc) is called the /cth Taie twisf of iJ and is abbreviated by H{k). > Definition 3.C.5 Let H be a Hodge structure of weight k. A polarization > H is a bilinear form ( , ) :ffxif >-Q satisfying the two conditions: i) (p(z)a, p(z)(/3)) = (zz)k(a,0) andii) ( ,p(i) ) is symmetric and positive definite. > Lemma 3.C.6 Let ( , ) be a polarization of a weight k Hodge structure H. > The pairing ( , ) is symmetric if k is even and alternating otherwise. > With respect to the C-linear extension of the pairing the direct sum decomposition He = ®p>q{Hp'q® Hq'p) is orthogonal. > On the real part of Hp'q© Hq'pthe pairing ip~q( , ) is positive definite. > 7 Let ( , ) be a polarization of a weight one Hodge structure H. Then ( , ) is alternating and can be considered as an element in H1'0® H0'1C /\ HQ. Since the polarization is rational, this two-form is in fact contained in the Q-vector space /\ H*. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=172|Complex Geometry_ An Introduction - Daniel Huybrechts, page 172]] > Proposition 3.CIO There is a natural bijection between the set of isomor phism classes of integral Hodge structures of weight one and the set of iso morphism classes of complex tori. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=174|Complex Geometry_ An Introduction - Daniel Huybrechts, page 174]] > The Riemannian metric gives rise to an hermitian metric on the > tangent bundle. > 4.1.1 An hermitian structure h on E —> M is an hermitian scalar product hxon each fibre E(x) which depends differentiably on x. The pair (E, h) is called an hermitian vector bundle. > Let L be a (holomorphic) line bundle and let s\,...,Sk be global (holomorphic) sections generating L everywhere, i.e. at every point at least one of them is non-trivial. Then one defines an hermitian structure on Lby > where t is a point in the fibre L{x) and ip is a local trivialization of L around the point x. > one sometimes says that h is given by ( ^ jsij2)"1. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=175|Complex Geometry_ An Introduction - Daniel Huybrechts, page 175]] > bundle E can be decomposed as E = F © F1- and F1- is canonically (as a complex vector bundle) isomorphic to the quotient E/F. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=176|Complex Geometry_ An Introduction - Daniel Huybrechts, page 176]] > Proposition 4.1.4 Every complex vector bundle admits an hermitian metric. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=178|Complex Geometry_ An Introduction - Daniel Huybrechts, page 178]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=179|Complex Geometry_ An Introduction - Daniel Huybrechts, page 179]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=180|Complex Geometry_ An Introduction - Daniel Huybrechts, page 180]] > Let L be a holomorphic line bundle of degree d > 2g(C) — 2 on a compact curve C. Show that HX{C,L) = 0. Here, for our purpose we define the genus g(C) of C by the formula deg(Kx) = 2g(C) - 2. In other words, H1(C,Kc ® L) = 0 for any holomorphic line bundle L with deg(L) > 0. In this form, it will later be generalized to the Kodaira vanishing theorem for arbitrary compact Kahler manifolds. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=181|Complex Geometry_ An Introduction - Daniel Huybrechts, page 181]] > A°(E) is just the sheaf of sections of E. Sections of E cannot be differentiated canonically, i.e. the exterior differential is in general not defined > A substitute for the exterior differential is provided by a connection on E, which is not canonical, but always available. > Definition 4.2.1 A connection on a vector bundle E is a C-linear sheaf ho momorphism V : A°(E) —> A1(E) which satisfies the Leibniz rule > for any local function f on M and any local section s of E. Definition 4.2.2 A section sof a vector bundle E is called parallel (or flat or constant) with respect to a connection V on E if V(s) = 0. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=182|Complex Geometry_ An Introduction - Daniel Huybrechts, page 182]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=183|Complex Geometry_ An Introduction - Daniel Huybrechts, page 183]] > The difference between the direct sum Vi © V2 of the two induced connec tions and the connection V on E we started with is measured by the second fundamental form. Let E\ be a subbundle of a vector bundle E and > that a connection on the latter is given. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=184|Complex Geometry_ An Introduction - Daniel Huybrechts, page 184]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=185|Complex Geometry_ An Introduction - Daniel Huybrechts, page 185]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=186|Complex Geometry_ An Introduction - Daniel Huybrechts, page 186]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=187|Complex Geometry_ An Introduction - Daniel Huybrechts, page 187]] > Definition 4.2.17 Let E be a holomorphic vector bundle on a complex mani fold X. A holomorphic connection on E is a C-linear map (of sheaves) D > E -> QX®E with for any local holomorphic function / on X and any local holomorphic section s oiE. > if / is a holomorphic function, then d(f) is a holomorphic section of /\ ' X, i.e. a section of fix (use Bd(f) = —dd(f)). > Writing a holomorphic connection D locally as d + A shows that D also induces a C-linear map D : A°(E) —> Alr°(E) which satisfies D(f • s) = 9(/)®s + /-Z)(s). Thus, D looks like the (l,0)-part of an ordinary connection and, indeed, V := D + 8 defines an ordinary connection on E. However, the (l,0)-part of an arbitrary connection need not be a holo morphic connection in general. It might send holomorphic sections of E to those of A1'°(E) that are not holomorphic, i.e. not contained in Qx ® E. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=188|Complex Geometry_ An Introduction - Daniel Huybrechts, page 188]] > The Atiyah class > of the holomorphic vector bundle E is given by the Cech cocycle > A holomorphic vector bundle E admits a holomorphic connection if and only if its Atiyah class A(E) G H1(X,f2x <8> End(.E)) is trivial. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=190|Complex Geometry_ An Introduction - Daniel Huybrechts, page 190]] > However, in general a connection V need not satisfy V2= 0, i.e. V is not a differential. The obstruction for a connection to define a differential is measured by its curvature. > The curvature Fy of a connection V on a vector bundle E is the composition > Usually, the curvature Fy will be considered as a global section of ,42(End(£)), i.e. FvG ^2(M,End(£1)). ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=191|Complex Geometry_ An Introduction - Daniel Huybrechts, page 191]] > i) Let us compute the curvature of a connection on the trivial bundle M x Cr. If V is the trivial connection, i.e. V = d, then Fv = 0. > Fv= d(A) +A/\A. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=194|Complex Geometry_ An Introduction - Daniel Huybrechts, page 194]] > Let (X,g) be an hermitian manifold. Then the tangent bundle is na turally endowed with an hermitian structure. The curvature of the Chern connection on Tx is called the curvature of the hermitian manifold (X,g). ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=208|Complex Geometry_ An Introduction - Daniel Huybrechts, page 208]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=209|Complex Geometry_ An Introduction - Daniel Huybrechts, page 209]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=214|Complex Geometry_ An Introduction - Daniel Huybrechts, page 214]] > Levi-Civita Connection > we will clarify the relation between the Chern connection on the holomorphic tangent bundle of an hermitian manifold and the Levi-Civita connection on the underlying Riemannian manifold. > manifold is Kahler. > A connection on M by definition is a connection on the real tangent bundle TM, i.e. an M-linear map D : A°(TM) —> A1{TM) satisfying the Leibniz rule > For any two vector fields u and v we denote by Duv the one-form Dv with values in TM applied to the vector field u. > the Leibniz rule reads Du(f -v) = f-Du(v) + (df)(u)-v. A connection is metric if dg(u,v) = g(Du,v) + g(u,Dv). In other words, D is metric if and only if g is parallel, i.e. D(g) = 0, where D is the induced connection on T*M T*M > the Lie bracket is an M-linear skew-symmetric map [ , ] : A°(TM) x A°(TM) -> A°(TM) which locally for u = ]T\ ai -g^- and v = j ^ k -§^r is defined by > [/ • u, v] = f • [u, v] — df(v) > The torsion of a connection D is given by > for any two vector fields u and v. > To is skew-symmetric, i.e. To ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=215|Complex Geometry_ An Introduction - Daniel Huybrechts, page 215]] > as an element of A2(TM). > V). A connection V is called torsion free if TD= 0. > our connection is of the form D — d + A. Here, A is a one-form with values in End(TM). > we will write A(u) G .4°(End(TM)) for the endomorphism that is obtained by applying the one-form part of A to the vector field u. > the canonical isomorphism A1(TM) = _4°(End(TM)). > If D = d + A then TD(U, V) — A(u) • v - A(v) • u. > Classically, one expresses the connection matrix A in terms of the Christof fel symbols F^ as > In particular, D is torsion free if and only if Fjj — F^ for all i,j,k. > Theorem 4.A.3 Let (M,g) be a Riemannian manifold. Then there exists a unique torsion free metric connection on M; the Levi-Civita connection. > E.g. the exterior differential can be expressed in terms of torsion free connections. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=216|Complex Geometry_ An Introduction - Daniel Huybrechts, page 216]] > A form a £ Ak(M) is parallel if D(a) — 0. > Let D be a torsion free connection on M. Any D-parallel form is closed. > the complex vector bundles TX'°X and (TX,I) are identified via the > £ any hermitian connection V on TX'°X induces a metric connection D on the Riemannian manifold (X,g). ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=217|Complex Geometry_ An Introduction - Daniel Huybrechts, page 217]] > the torsion Ty £ A2(X) of an hermitian connection V. > Tv(u,v) = ^~l{Vui{v) - Vvt(u)) - [u,v] > Duv — Dvu — [u,v] — TD(U,V). > Let V fee a torsion free hermitian connection on the her mitian bundle (Tll0X,gc). > Then V is the Chern connection on the holomorphic bundle T\ endowed with the hermitian structure gc > The induced connection D on the underlying Riemannian manifold is the Levi-Civita connection. > The hermitian manifold (X,g) is Kahler. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=218|Complex Geometry_ An Introduction - Daniel Huybrechts, page 218]] > From this slightly lengthy discussion the reader should only keep in mind that the following four conditions are equivalent: > The complex structure is parallel with respect to the Levi-Civita con nection. > (X,g) is Kahler. > Levi-Civita connection D and Chern connection V are identified by £. > The Chern connection is torsion free. > Let us now turn to the curvature tensor of a Riemannian manifold (M, g). ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=219|Complex Geometry_ An Introduction - Daniel Huybrechts, page 219]] > In Riemannian geometry one also considers the Ricci tensor r(u, v) := tv(w t—> R(w, u)v) = tr( w I »- R(w, v, u) ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=221|Complex Geometry_ An Introduction - Daniel Huybrechts, page 221]] > one can define the parallel transport of tangent vectors along a path in M. > Let 7 : [0,1] —> M be a path connecting two points x := 7(0) and y := 7(1). The pull-back connection 7*D on j*(TM) is necessarily flat over the one-dimensional base [0,1] and 7*(TM) can therefore be trivialized by flat sections. In this way, one obtains an isomorphism, the parallel transport along the path 7: > In other words, for any v G TXM there exists a unique vector field v(t) with v(t) G Tl(t)M, v(0) = v, and such that v(t) is a flat section of 7*(TM). Then > In particular, if 7 is a closed path, i.e. 7(0) = 7(1) = x, then P1£ 0(TxM,gx)^0(m). ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=222|Complex Geometry_ An Introduction - Daniel Huybrechts, page 222]] > Theorem 4.A.15 (de Rham) If (M,g) is a simply connected complete (e.g. compact) Riemannian manifold then there exists a decomposition (M, g) = (Mi,gi) x ... x (Mfe, gk) with irreducible factors (Mi,gi). > Secondly, many groups can occur as holonomy groups of symmetric spaces, > If symmetric spaces are excluded then, surprisingly, a finite list of remaining holonomy groups can be given. > assume that (M,g) is ir reducible and not locally symmetric. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=223|Complex Geometry_ An Introduction - Daniel Huybrechts, page 223]] > (Recall that the Levi-Civita connection induces connections on all tensor bundles, e.g. on End(TM), so that we can speak about parallel tensor fields.) ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=224|Complex Geometry_ An Introduction - Daniel Huybrechts, page 224]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=225|Complex Geometry_ An Introduction - Daniel Huybrechts, page 225]] > If (X, g) is an hermitian manifold, w := g(I( ), ( )) is its fundamental form. By definition (X, g) is Kahler if and only if to is closed, i.e. dio = 0. > For the time being, we let E be an arbitrary holomorphic vector bundle with an arbitrary hermitian metric h. Recall that the curvature Fy ofthe Chern connection on (E,h) is of type (1,1), i.e. Fve A1'1(X,End(E)).The fundamental form u> induces an element of the same type w • id^; €A1'1{X, End(E)). These two are related to each other by the Hermite-Einsteincondition: Definition 4.B.I An hermitian structure h on a holomorphic vector bundle E is called Hermite-Einstein if i • A^Fy = A • ids for some constant scalar A G K. Here, Auis the contraction by w. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=226|Complex Geometry_ An Introduction - Daniel Huybrechts, page 226]] > Definition 4.B.3 The slope of a vector bundle E with respect to the Kahler form u> is defined by ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=227|Complex Geometry_ An Introduction - Daniel Huybrechts, page 227]] > Lemma 4.B.4 Any holomorphic line bundle L on a compact Kdhler manifold X admits an Hermite-Einstein structure. > Remark 4-B.5 Clearly, the first Chern class c\(L) € H2(X,M.) can uniquely be represented by an harmonic form. The ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=228|Complex Geometry_ An Introduction - Daniel Huybrechts, page 228]] > often called the Bogomolov-Liibke inequality, > When does a holomorphic bundle that satisfies the above inequality reallyadmit a Hermite-Einstein metric? This is a difficult question, but a completeanswer is known due to the spectacular results of Donaldson, Uhlenbeck, andYau. It turns out that the question whether E admits an Hermite-Einsteinmetric can be answered by studying the algebraic geometry of E. In particular,one has to introduce the concept of stability. Definition 4.B.8 A holomorphic vector bundle Eonacompact Kahler mani fold X is stable if and only if ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=229|Complex Geometry_ An Introduction - Daniel Huybrechts, page 229]] > A holomorphic vector bundle E is polystable if E = 0 Ei with Ei stable vector bundles all of the same slope /-i(E) = fi(Ei). > A holomorphic vector bundle E on a compact Kahler manifold X admits an Hermite-Einstein met ric if and only if E is polystable. > On curves, Hermite-Einstein metrics are intimately related to unitary representation of the fundamental group of the curve. > This kind of result is nowadays known as Kobayashi-Hitchin correspondence > An hermitian manifold (X,g) is called Kahler-Einstein ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=230|Complex Geometry_ An Introduction - Daniel Huybrechts, page 230]] > Kahler-Einstein. > Explicitly, this means that the curvature of the Levi-Civita connection > for some constant scalar factor A. > A Riemannian metric jon a differentiate manifold M is Einstein if its Ricci tensor r(M, g) satisfies > for some constant scalar factor A. If g is a Kahler metric on the complex manifold X = (M,/), then the Ricci curvature Ric(X, g) is denned by Ric(-u,t>) = r(I(u),v) ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=231|Complex Geometry_ An Introduction - Daniel Huybrechts, page 231]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=232|Complex Geometry_ An Introduction - Daniel Huybrechts, page 232]] > It turns out that the Fubini-Study metric, the only Kahler metric on Pnthat has been introduced, is indeed Kahler-Einstein. > i.e. A = n + > det(7i»n) = 0(n + 1) > X = Cn/F is trivial. The Chern connection for any constant Kahler structure on X is flat. Thus, the Kahler-Einstein condition is satisfied with the choice of the scalar A = 0. Complex tori are trivial examples of Ricci-flat manifolds. Any other ex > is much harder to come by. > The standard Kahler structure LO = (i/2)dd(l- \\z\\2) > This way, one obtains negative Kahler-Einstein structures on all ball quo tients. > The inequality is usually called the Miyaoka-Yau inequality. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=233|Complex Geometry_ An Introduction - Daniel Huybrechts, page 233]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=235|Complex Geometry_ An Introduction - Daniel Huybrechts, page 235]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=236|Complex Geometry_ An Introduction - Daniel Huybrechts, page 236]] > If ci(JQ is negative, i.e. —ci(X) can be represented by a Kahler form, the question is completely settled by the following theorem, due to Aubin and Yau. Theorem 4.B.24 (Aubin, Yau) Let X be a compact Kahler manifold > that c\(X) is negative. Then X admits a unique Kahler-Einstein metric up to scalar factors. > for ci(X) positive the situation is, for the time being, not fully understood. One knows that in this case a Kahler-Einstein metric need not exist. E.g. the Fubini-Study metric on P2is Kahler-Einstein, but the blow-up of P2in two points for which K\ is still ample does not admit any Kahler-Einstein metric. In order to ensure the existence of a Kahler-Einstein metric, a certain stability condition on X has to be added ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=237|Complex Geometry_ An Introduction - Daniel Huybrechts, page 237]] > Give an algebraic argument for the stability of the tangent bundle of Pn ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=238|Complex Geometry_ An Introduction - Daniel Huybrechts, page 238]] > Hirzebruch-Riemann-Roch formula. This formula allows to com pute the dimension of the space of global sections of a given vector bundle in terms of its Chern classes. In fact, the higher cohomology groups enter this formula as correction terms. > projectivity of a Kahler manifold is encoded by the position of its Kahler cone within the natural weight-two Hodge structure. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=239|Complex Geometry_ An Introduction - Daniel Huybrechts, page 239]] > h°(X, L) of an ample line bundle L de termines the dimension of the projective spaces in which X can be embedded > far reaching generalizations of the Hirzebruch-Riemann-Roch formula, most notably the Grothendieck-Riemann-Roch formula and the Atiyah-Singer in dex theorem, > (Hirzebruch—Riemann—Roch) > What is meant by the integral on the right hand side, of course, is the evaluation of the top degree component ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=240|Complex Geometry_ An Introduction - Daniel Huybrechts, page 240]] > Examples 5.1.2 i) Line bundles on a curve. > The special case L = Oc yields x(C, Oc) =2and, therefore, deg(Kc) = 2(^1(C, Oc) — 1). > g(C) = (1/2) deg(Kc) + 1 = hl(C,Oc) = h°(C,Kc), > Let us first consider the case of a tri vial line bundle. > Noether's formula: > (X, Ox) = > If L is any line bundle on a compact complex surface X. > X{X, L) = ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=241|Complex Geometry_ An Introduction - Daniel Huybrechts, page 241]] > Hirzebruch Xy-9enus lsthe polynomial > y = 0: Then Xy=o — x(^i ^x > the arithmetic genus of X. > xy=i = sgn(X) > y = — 1: > is the Euler number of X. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=242|Complex Geometry_ An Introduction - Daniel Huybrechts, page 242]] > For y = 0 > (X, Ox) = Jxtd(X). > For y = 1 it yields the so called Hirzebruch signature theorem > where L(X) is the L-genus which in terms of the Chern roots is just > For y = — > (the Gauss-Bonnet formula), ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=243|Complex Geometry_ An Introduction - Daniel Huybrechts, page 243]] > Let D : F(E) ->• F(F) be an elliptic differential operator between vector bundles E E and F on a compact oriented differentiate manifold M. Then Then the analytic index index(Z?) := dim Ker(£>) — dim Coker(D) and and the topological index j(D) satisfy mdex(D) = -y(D). > frequently happens that one wants to compute x(^^) °f a coherent sheaf J-, which is not locally free, e.g. J- = Xz the ideal sheaf of a submanifold Z C X. If X is projective, then there exists a locally free resolution 0 —> En—>...—> E\ —> J- —> 0. Applying the above formula to > sheaves Ei and using the additivity of the Euler-Poincare characteristic and the Chern character, one immediately obtains a Hirzebruch-Riemann-Roch formula for J-. > the direct image sheaves Ktf*E of a locally free sheaf E are in general not locally free anymore. As an example one might consider the structure sheaf Oy of a smooth hypersurface Y C X. The structure sheaf sequence 0 —> O{—Y) —> O —> Oy —> ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=244|Complex Geometry_ An Introduction - Daniel Huybrechts, page 244]] > Thus, any of the above men tioned results holds true for arbitrary complex manifolds and coherent sheaves. > The Hilbert polynomial ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=245|Complex Geometry_ An Introduction - Daniel Huybrechts, page 245]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=246|Complex Geometry_ An Introduction - Daniel Huybrechts, page 246]] > Assuming a certain positivity of L, the higher coho mology groups of L can be controlled due to the so called Kodaira(-Nakano Akizuki) vanishing theorem. In conjunction with the Hirzebruch-Riemann Roch formula (5.2) this often yields effective (and topological) bounds for the dimension of the space of global holomorphic sections H°(X, L) of a holomor phic line or vector bundle. > important applications like the Weak Lefschetz theorem and Serre's theorem. > A line bundle L is called positive if its first Chern class ci(L) € H2(X,M) can be represented by a closed positive real (l,l)-form. Note that a compact complex manifold X that admits a positive line bun dle L is automatically Kahler. > Since any closed real (1, l)-form representing c\(L) is the curva ture of a Chern connection > a line bundle L is positive if and only if it admits an hermitian structure such that the curvature of the induced Chern connection is positive > The algebraic inclined reader might replace 'positive' by 'ample'. The equivalence of both concepts will be proved in the next section. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=248|Complex Geometry_ An Introduction - Daniel Huybrechts, page 248]] > Consider 0(1) on Pn, which is positive due to Example 4.3.12 > Hq{Pn, flp® 0(m)) = 0 for p + q > n and m > 0. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=249|Complex Geometry_ An Introduction - Daniel Huybrechts, page 249]] > Proposition 5.2.6 (Weak Lefschetz theorem) LetX be a compact Kabler manifold of dimension n and let Y C X be a smooth hypersurface such that the induced line bundle O{Y) is positive. Then the canonical restriction map > is bijective for k < n — 2 and injective for k < n — > where the latter is the dual of the normal bundle sequence. > Twisting the first one with Qpxand taking the p-th exterior product of the second one yields short exact sequences of the form: ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=250|Complex Geometry_ An Introduction - Daniel Huybrechts, page 250]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=251|Complex Geometry_ An Introduction - Daniel Huybrechts, page 251]] > As an application of Serre's vanishing theorem we will prove the following classification result for vector bundles on P1 ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=252|Complex Geometry_ An Introduction - Daniel Huybrechts, page 252]] > Already an explicit classification of rank two vector bundles on the projective plane P2is impossible, > Let (E, h) be an hermitian holomorphic vector bundle on a compact Kahler manifold X. Suppose that the curvature Fv of the Chern connection is trivial, i.e. the Chern connection is flat. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=253|Complex Geometry_ An Introduction - Daniel Huybrechts, page 253]] > a{X) = dim(X) = n, i.e. X is Moishezon. > Kodaira vanishing does not hold for arbitrary smooth projective varieties in positive characteristic. > Another far-reaching generalization of the Kodaira vanishing theorem is the Kawamata-Viehweg vanishing which predicts the same sort of vanishing but this time for line bundles which are not quite positive, but only big and nef. The result is often used in Mori theory. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=254|Complex Geometry_ An Introduction - Daniel Huybrechts, page 254]] > Not every compact complex manifold is Kahler and not every compact Kahler manifold is projective. Of course, one would like to have a criterion that decides whether a Kahler manifold is projective. Such a criterion is provided by the Kodaira embedding theorem > suffices to be able to describe the Kahler cone inside H2(X, K) in order to decide whether an ample line bundle exists. The analogous question, which complex manifolds are in fact Kahler, is essentially open. In particular, it is not known whether being Kahler is a purely topological property. > In other words, Then ip^ is injective if and only if for two two arbitrary distinct points x\ ^ x2£ X there exists a section s G H°(X,L) with s(xi) = 0 and s{x2) ^ 0. One says that <£>L (or L) separates points. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=255|Complex Geometry_ An Introduction - Daniel Huybrechts, page 255]] > the differential dtpLiX: TXX —* TV^PNis injective. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=258|Complex Geometry_ An Introduction - Daniel Huybrechts, page 258]] > The projectivity of a compact Kahler manifold can now be read off the position of the Kahler cone Kx C H2(X, R) relative to the integral lattice Im(#2(X,Z) cH2(X,R)). Corollary 5.3.3 A compact Kahler manifold X is •protective if and only if JCxnH2(XZ)^ The Kahler cone Kxis by definition the cone of all Kahler classes on X and hence contained in Jf1'1(X). ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=259|Complex Geometry_ An Introduction - Daniel Huybrechts, page 259]] > Any compact complex curve is projective. Indeed, any curve is Kahler and since H^^X) = H2{X,C) the open subset KxC H2{X,R) contains integral classes. > Every compact Kahler manifold X with H°'2(X) = 0 is projective. > The reader may notice that we actually only used the easy direction of the Kodaira embedding theorem, namely that an ample line bundle is positive and thus satisfies the assumption of the Kodaira vanishing theorem. > The corollary also shows that the Neron—Severi group NS(.X') = H^'l(X, Z) of a projective manifold is indeed spanned by the fundamental classes of di visors. See ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=260|Complex Geometry_ An Introduction - Daniel Huybrechts, page 260]] > In particular, the blow-up X of a projective manifold X is again projective.) > that any line bundle L of degree deg(L) > 1g on a compact curve C of genus g is very ample, i.e. the linear system associated with L embeds C. > The Kodaira embedding theorem is complemented by a theorem of Chow saying that any complex submanifold X C P^ can be described as the zero set of finitely many homogeneous polynomials. > that a compact complex manifold X is projective if and only if X is Kahler and Moishezon ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=261|Complex Geometry_ An Introduction - Daniel Huybrechts, page 261]] > The set of all complex structures on a given differen tiable manifold comes itself with a natural differentiable and in fact complex structure. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=262|Complex Geometry_ An Introduction - Daniel Huybrechts, page 262]] > Recall that two complex manifolds (M, /) and (M', /') are isomor phic if there exists a diffeomorphism F : M —> M' such that dF o I = I' o dF. Thus, the set of diffeomorphism classes of complex structures / on a fixed differentiable manifold M is the quotient of the set > of all complex structures by the action of the diffeomorphism group > We start out with the set > of all almost complex structures on M. > the space Aac(M) is a nice space, i.e. (after completion) it is an infinite dimensional manifold. This, in general, is no longer true for the subspace AC(M) C Aac(M) of integrable almost complex structures. > Recall that an almost complex structure I is uniquely determined by a de composition of the tangent bundle TCM = T1-0© T0'1with / = i • id on Tlfi and / = -i • id on T0-1. In fact, giving T0-1c TCM is enough, for T1-0= T > If I(t) is a continuous family of almost complex structures with /(0) = /, > has a continuous family of such decompositions TcM = Tt' ®Tt' ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=263|Complex Geometry_ An Introduction - Daniel Huybrechts, page 263]] > Let us now consider the power series expansion of a given cf>. Then Then CJ>Q = 0 and the higher order coefficients 0,>o will be regarded as sections of f\ ' (giT1'0 > In particular, the higher order coefficients i correspond to elements in A°'1(Tx)- The integrability condition for I(t) can be expressed as ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=264|Complex Geometry_ An Introduction - Daniel Huybrechts, page 264]] > 6.1.1 Using the local description, one easily checks that d[a:f3] > [Ba,0\ ± [a, 8(3]. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=265|Complex Geometry_ An Introduction - Daniel Huybrechts, page 265]] > Let us now consider the power series expansion of the Maurer-Cartan equation, i.e. we replace cf> in (6.2) by (f> = J2^11 > This yields a recursive system of equations: 0 =501 > The Maurer Cartan equation is, in particular, saying that the first-order deformation of the complex structure / is described by a enclosed (0, l)-form i] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=266|Complex Geometry_ An Introduction - Daniel Huybrechts, page 266]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=267|Complex Geometry_ An Introduction - Daniel Huybrechts, page 267]] > Proposition 6.1.5 Let X be a complex manifold. There is a natural bijection between all first-order deformations of X and elements of H1(X,Tx). > In other words, the Kodaira-Spencer class determines the first-order de formations of X and any class in H1(X,Tx) occurs as a Kodaira-Spencer class. The principal task in deformation theory is to integrate given first-order deformations v 6 H1(X,Tx), i.e. to find a one-parameter family It such that its Kodaira-Spencer class is v. In general this is not possible. Obstructions may occur at any order. Thus, it frequently happens that we find 4>\t + ... + 4>kt > but a k+i does not exist. > That there in general exist obstructions can already be guessed at order two: Once a lift 2 such > d(f>2 = —[4>iii\- But it may very well happen that the form [<£i,<£i] is > 9-exact, no matter how \ is chosen. Using Remark 6.1.1 we can associate to any first order deformation v € Hl(X,Tx) a class \v,v] € H2(X,TX). Corollary 6.1.6 A first-order deformation v £ H1(X,Tx) cannot be inte grated if[v,v] € H2(X,Tx) does not vanish. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=268|Complex Geometry_ An Introduction - Daniel Huybrechts, page 268]] > Moreover, the isomorphism 77 induces canonical isomorphisms ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=271|Complex Geometry_ An Introduction - Daniel Huybrechts, page 271]] > due to Tian and Todorov. > Let X be a Calabi-Yau manifold and letv G H1(X,Tx). Then there exists a formal power series (pit + fat2+ ... with (pi G A°'1(Tx) satisfying the Maurer-Cartan equations > with [0i ] = v and such that ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=272|Complex Geometry_ An Introduction - Daniel Huybrechts, page 272]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=273|Complex Geometry_ An Introduction - Daniel Huybrechts, page 273]] > a formal solution converges if the coefficients 4>k are d^-x-exact. > as the mere existence of a formal solution ensures that all possible obstructions are trivial. > It is noteworthy that so far we have not used the existence of a Kahler-Einstein metric on X > but have worked with a completely arbitrary Kahler metric. In fact, once the formal solution in the proposition has been chosen as proposed in > then it is automatically convergent provided the chosen Kahler metric is indeed Kahler-Einstein. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=275|Complex Geometry_ An Introduction - Daniel Huybrechts, page 275]] > Proposition 6.2.2 (Ehresmann) Let n : X -> S be a proper family of differentiable manifolds. If S is connected, then all fibres are diffeomorphic. > Corollary 6.2.3 Every proper family of differentiable manifolds is locally diffeomorphic to a product. > It should also be clear from the proof that the arguments do not work in the complex setting. But once we know that the fibres are all isomorphic as complex manifolds, the family is in fact locally trivial. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=276|Complex Geometry_ An Introduction - Daniel Huybrechts, page 276]] > TT : X —> 5* can be viewed as a family of complex structures on the differentiable manifold M underlying the complex fibre X. > Theorem 6.2.5 (Kodaira) Let X be a compact Kahler manifold. If X —> S is a deformation of X = XQ, then any fibre Xtis again Kahler. > Since TT is smooth, we obtain a surjection of the tangent spaces TT : TXX —> TQS for any point x £ X. The kernel of this > tion is the tangent space TXX of the fibre. > The boundary map ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=277|Complex Geometry_ An Introduction - Daniel Huybrechts, page 277]] > The fibre of a morphism ip : (X,Ox) —> (Y,Oy) over a point y £ Y is by definition the complex space ((p^1(y), Ox/<^^1(tny)), where myis the maximal ideal of the local ring Oyy. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=278|Complex Geometry_ An Introduction - Daniel Huybrechts, page 278]] > T0S :=Homc(m0/mg,C). > As soon as the spaces involved are no longer necessarily smooth, one changes the notation from Tx to &x or sometimes Dei {Ox)- > Proposition 6.2.10 Let X be a compact complex manifold. Then there is a natural bijection between classes in H1(X,Tx) and isomorphism classes of deformations X —> Spec(C[e]) of X. > A central question in deformation theory is whether any infinitesimal deformation can be integrated, i.e. given a class v G H1(X,Tx) can one find a deformation over a smooth base S such that v is contained in the image of the Kodaira-Spencer map ToS —» i71(X, Tx). > Definition 6.2.11 A deformation TT : X —> S of the compact complex > fold X = Xo is called complete if any other deformation IT' : X' —> S" of X is obtained by pull-back under some / : S' —> S. If in addition / is always unique, then n : X —> S is called universal. If only its differential Ts'(O') —> Ts(0) is unique, then the deformation is called versal. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=279|Complex Geometry_ An Introduction - Daniel Huybrechts, page 279]] > In fact, in many situations a universal deformation does not exist. > Theorem 6.2.13 (Kuranishi) Any compact complex manifold admits a ver sal deformation. Note that for a versal deformation X —> S the Kodaira-Spencer map TQS —> Hl{X,Tx) is bijective. > If H°(X,Tx) = 0, then any versal deformation is universal and hence unique. > One says that X has unobstructed deformations if X admits a smooth versal deformation, i.e. the base S is an honest complex manifold. This is the case if H2{X,Tx) = 0. In fact, the results explained in the last section (cf. > show that any Calabi-Yau manifold X, i.e. a compact Kahler manifold with trivial canonical bundle Kx, has unobstructed deformations. > The idea why Calabi-Yau manifolds should have unobstructed deforma tions is the following: One observes that for any deformation Xtof a Calabi Yau manifold X of dimension n one has Hl(Xt,TXt) = tf1^, J?^1). The dimension of the latter space does not depend on t, as it occurs as a direct sum mand of the Hodge decomposition of Hn(Xt,C) which in turn only depends on the underlying topological manifold. By iii) this shows that the family is versal and that all tangent spaces of S are of the same dimension. Now, if ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=280|Complex Geometry_ An Introduction - Daniel Huybrechts, page 280]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=281|Complex Geometry_ An Introduction - Daniel Huybrechts, page 281]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=283|Complex Geometry_ An Introduction - Daniel Huybrechts, page 283]] > Since Tx and all exterior powers /\pTx are holomorphic bundles, 8 endows Ax with the structure of a differential graded algebra. > called the Schouten-Nijenhuis bracket. > In particular, it generalizes the Lie bracket [ , ] on A°'*(7~x) > The difference is that for an arbitrary complex manifold X the algebra Ax is not endowed with the ad ditional structure A. This is only the case for a Calabi-Yau manifold, > Corollary 6.A.5 Let X be a complex manifold and let e A Then d^ := d + S^ is an odd derivation on Ax- Furthermore, B^ = 0, i.e. dfj, is a differential, if and only if the master equation ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=284|Complex Geometry_ An Introduction - Daniel Huybrechts, page 284]] > if 4>(t) > A0'1(Tx) describes the deformation I(t) of the complex structure denning X then I(t) is integrable, which is equivalent to (f>(t) satisfying the Maurer Cartan equation (6.2), if and only if \4>{t) G A°'l(Tx) C A°xsatisfies the master equation. In other words, we can study deformations of the complex structure / in terms of deformations of the differential d on A°'*(f\* Tx). > In this sense, solutions of the master equations can be viewed as gene ralized deformations of the given complex structure. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=285|Complex Geometry_ An Introduction - Daniel Huybrechts, page 285]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=287|Complex Geometry_ An Introduction - Daniel Huybrechts, page 287]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=288|Complex Geometry_ An Introduction - Daniel Huybrechts, page 288]] > In this vain, one defines the cotangent bundle f\M as the dual (TM)* = Hom(TM, Mxl ) and the bundles of /b-forms ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=290|Complex Geometry_ An Introduction - Daniel Huybrechts, page 290]] ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=291|Complex Geometry_ An Introduction - Daniel Huybrechts, page 291]] > A := d*d + dd*. > (a, /3) := I g(a, 0) • vol(Afiff)= / a A > i.e. d* is the adjoint operator of d with respect to ( , ) and A is self-adjoint. > Note that for the standard metric on M.mthe Laplacian A as defined above applied to a function / : IRm—> M is A(f) = — ^2d2f/dxf, > A(a) = 0, if and only if da = d*a = 0. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=292|Complex Geometry_ An Introduction - Daniel Huybrechts, page 292]] > Ak(M) = d{Ak~x(M)) 8 Hk(M,g) © d*{Ak+1(M)). > Equivalently, every cohomology class has a unique harmonic representative. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=293|Complex Geometry_ An Introduction - Daniel Huybrechts, page 293]] > ruy '• 3~(V) —> J-{U) > i) ru,u = id;F(i/)- ii) For open subsets U C V C W one has ruy °fv,w = rjj,w > J-{0) = 0. In > s\u instead of ruy(s). ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=294|Complex Geometry_ An Introduction - Daniel Huybrechts, page 294]] > iv) If functions fi £ C°M{Ui) are given for all i such that rUjnu^UrUitiUjVjifj) f°r anY 3i then there exists a continuous function / G C^(U)with rjji,u(f)=fi f°rall i > constant sheaf(!) JR., which, on an open set U C M, yields the set of all continuous functions / : U —> M, where K is endowed with the discrete topology. > sheaf £ of sections of a (topological) vector bundle ir : E —> M. > £(U) is the set of all (continuous) maps s : U —> -E with TT o s = idjy > £ is a sheaf of C^-modules, i.e. each £(U) is a C^(£/)-module and the restriction maps are compatible with the module structures on the different open subsets. > If

Definition B.0.25 The sheaf T^ T^ associated to a pre-sheaf T is > J- (S)ii Q of two 7£-modules J- and Q is defined as the sheafification of U i—> J-{U) ®iz{U) G{U)- >

ip might be surjective without ipu being surjective for all/any open subset. However, both properties can be detected by their stalks. > A complex > is exact if and only if the induced complex of stalks ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=296|Complex Geometry_ An Introduction - Daniel Huybrechts, page 296]] > A sheaf J- is called flasque if for any open subset U C M the restriction map ru,M '• 3~(M) —> J-(U) is surjective. Why flasque sheaves are the right ones is explained by the following > is a short exact sequence and !F° is flasque, then the induced sequence > is exact for any open subset U C M. Next, one has to ensure that any sheaf can be resolved by flasque sheaves. > Proposition B.0.32 Any sheaf J- on M admits a resolution ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=298|Complex Geometry_ An Introduction - Daniel Huybrechts, page 298]] > acyclic sheaves, > sheaves with trivial higher cohomology groups, > Definition B.0.38 A sheaf T is called soft if the restriction F(M,Jr) > F(K, J7) is surjective for any closed subset K C M. The space of sections F(K, F) of T over the closed set K is defined as the direct limit of the spaces of sections over all open neighbourhoods of K. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=299|Complex Geometry_ An Introduction - Daniel Huybrechts, page 299]] > More precisely, if M = 1J4£/, is refined by an open covering M = |J • Vj, then there exists a natural map and one defines the Cech cohomology of a sheaf without specifying an open covering as > Examples B.0.41 i) Let us compute the Cech cohomology of S1. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=300|Complex Geometry_ An Introduction - Daniel Huybrechts, page 300]] > Proposition B.0.43 The natural map H1(M,Jr) —> ^(M,^) is always bi jective. > Let 7T : L —> M be a line bundle that can be trivialized over open subsets U{ of an open cover M = (JC/J by maps ipi : L\tj. = Ut x M. Then {tpij := V'i ° TpJ1€ Chilli C\Uj)} can be considered as a cocycle and thus gives rise to an element in H1({Ui},Cli), ), where C^ is the sheaf of differentiable functions without zeros. > Moreover, any class in Hl{{Ui},C*M) can be interpreted as a cocycle of a line bundle trivialized over the open subsets Ui. ### [[Complex Geometry_ An Introduction - Daniel Huybrechts.pdf#page=301|Complex Geometry_ An Introduction - Daniel Huybrechts, page 301]] > E.g. the direct image sheaf f*J- of a sheaf J- on M under a continuous map f : M —> N isthe sheaf U ^ f(f- > We rarely use direct images (let alone higher direct images), but we do use the pull-back of a sheaf. > inverse image f~xJ- is the sheaf on M > [f~1J-){U) is the set of all maps s : U > Uzst/^/O)suc nthat for any x E U there exist open subsets x S UxC U, f(Ux) C V c N and an element t € T(V) with s(y) = t(f(y)) for all y e U > {f~1J-)x= Ff{x)- > Then the pull-back f*!F of any 7?.jv-niodule F on N is the 7?-M-module given by f~1Jr®f~iriNTZ > one can define the fibre of any 7?.M-niodule J- at a point x £ M as J-{x) := J- ®-R.M xTZM,X/™-, where m C HM,X is the maximal ideal.