--- created: 2024-04-19T16:15 updated: 2024-04-26T13:31 --- # Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox > [!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 ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=20|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 20]] > representing the particle by a point leads to a singularity, while the string repre sentation is a smooth 2-manifold with boundary: > many infini ties which require renormalization. > supersymmetry can eliminate many of these difficulties. Supersymmetry transforms bosons (particles with integer spins and symmetric wavefunctions) into fermions (particles with half-integer spins and antisymmetric wavefunctions) and > the world sheet ¥ has a conformal struc ture, and our supersymmetric string theory needs to be equivalent under conformal equivalence. Hence this theory is a superconformal field theory (SCFT for short). The Lie algebra of the symmetry group of such a theory is a superconformal alge bra. This algebra contains the conformal algebra (the Lie algebra of the group of conformal transformations of the world sheet) as as a subalgebra, and it also contains the supersymmetry transformations. The superstring theories come in four basic types: type I, type IIA, type 1IB, and, of greatest interest to us, heterotic. Heterotic string theory is an N = 2 SCFT because there are two supersymmetries. In such a theory, the equations of motion for the fermions decouple into left- and right-moving solutions, which means that there are actually four supersymmetries, two left-moving and two right-moving. > heterotic string theories are more properly called (2, 2) theories, as there are two independent supersymmetries in each of the left- and right-moving sectors of the theory. ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=22|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 22]] > where mirror symmetry comes from. > (Q,Q) are only well-defined up to sign, yet the sigma model coming from (V,w) makes a very specific choice. If we changed Q to —Q and left Q as is, we would interchange the (p,q) and (—p, ¢) eigenspaces, > would interchange H9(V, APTy) and H9(V,%,). This is not possible since these are vector spaces of different dimensions in general. Yet from the physical point of view, such a sign change is reasonable. This asymmetry suggests that maybe the sign change corresponds to the sigma model arising from a different pair (V°,w®). If such a pair (V°,w®) exists, we say that (V,w) and (V°,w°) are a mirror pair. > (V,w) and (V°,w®) form a mirror pair if their sigma models induce isomorphic superconformal field theories whose N = 2 super conformal representations are the same up to the above sign change. > To see what mirror symmetry tells us about V and V°, > HY(V,NPTy) > HY(V°,Qf.) HY(V, Q) ~ HY(V° APTys). Since V' is Calabi-Yau, it has a nonvanishing holomorphic 3-form €2, and cup product with ( gives a (noncanonical) isomorphism H¢(V, APTy) = HI(V, Q?,_ P). ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=24|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 24]] > using two different constructions for the SCFT. > The best example is given by the correlation functions of the SCFT, > stable under the flow of time (i.e., is not unitary), > Early evidence for mirror symmetry of Calabi-Yau threefolds was given by lists of Calabi-Yau hypersurfaces in weighted projective spaces > The Hodge numbers of these hypersurfaces exhibited a striking {but far from perfect) symmetry. > all of these weighted projective hypersurfaces are a subclass of those that arise from Batyrev’s reflexive polytope construction > It is conjectured > that the larger class of toric complete inter sections [Borisov1] is mirror symmetric. > correlation functions ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=26|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 26]] > to be considered, which in this case are the holomorphic instantons. These are nonconstant holomorphic maps ¥ — V, where T is a compact Riemann surface. > ¥ can have nodal singularities and more than one component. In the A-model correlation function, the only instantons needed are those where L has genus 0. Naively, these are what the ng count in formula (1.7). > we call ng an instenton number. The second Yukawa coupling to consider comes from H!(V, Ty). > Alternatively, one can think of this as > where V is the Gauss-Manin connection.? > This Yukawa coupling is clearly inde pendent of the complexified Kahler class w but depends on the complex structure of V' (since § is a holomorphic 3-form). The Yukawa coupling (1.9) is sometimes called the B-model correlation func tion, since it is identical with the corresponding three-point function in a different twisted theory, the B-model ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=28|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 28]] > is to approach a nonlinear sigma model by first considering other theories which are more elementary > flat metric. ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=30|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 30]] > H?(V,R). > However, in order to determine the structure of the Kahler moduli space Kc(V)/Aut(V), we need to know how the automorphism group Aut(V') acts on the Kahler cone. > by a result of [Reidl], a Gorenstein orbifold has canonical singularities. > there are even physical theories (the gauged linear sigma models of [Witten5) to ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=34|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 34]] > we can then define the A-model correlation function to be the formal sum > equation (2.2). > degree 5 ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=36|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 36]] > =13 is a local coordinate for the complex moduli (we > It follows that ¢ = 2TM induces an isomorphism > Thus g is a local parameter for Kihler moduli space, with ¢ = 0 as a boundary point. > Thus the A-model correlation function (2.1) is naturally a function in the local parameter ¢ for Kidhler moduli. > Since q is a local coordinate at a boundary point, > Back on the quintic threefold V', cup product with H gives an endomorphism UH of ef,=0HP'P(V) which is maximally nilpotent (meaning (UH)3 # 0 but (UH)* = 0). We will see > that under mirror symmetry, we expect UH to correspond to the logaritbm of the monodromy about the point of the complex moduli space of V*® corresponding to ¢ = 0. Hence, mirror symmetry tells us to look for mazimally unipotent monodromy, and of the above boundary points, this occurs only at z = 0. ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=40|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 40]] > Then (H, H, H) = (8,0, 6) gives the equation ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=42|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 42]] > also completely determine the variation of Hodge structure on H3(V° C). ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=50|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 50]] ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=54|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 54]] > servables, ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=56|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 56]] > variety X, ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=58|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 58]] > The symplectic structure converts functions into vector fields as follows: if f is a C* function on the manifold, then there is a unique vector field X ¢ on the ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=60|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 60]] > is an orbifold diffeomorphism. Furthermore, the symplectic form w on C", when restricted to uz'(a), descends to a symplectic form on 7 1(a)/Gr whose cohomology class is identified with a € H*(X,R) via the above diffeomorphism. > note that w is not symplectic when restricted to ugl(a). Going to the quotient is exactly what is needed to make it nondegenerate. This process is called symplectic reduction. > gauged linear sigma models, which take as their starting point a toric variety as described by symplectic reduction. > two Calabi Yau manifolds related by a flop have “adjacentTM Kihler cones, which will enable us to glue together the corresponding Kihler moduli spaces. ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=62|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 62]] > set of elements {e : v € =7} is called the Gale transform of Z*. > An especially nice case is when |=+| = n+3 ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=64|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 64]] > Cohen-Macaulay variety V has a dualizing sheaf 7, where n = dim(V'), and that 7, is a line bundle if and only if V' is Gorenstein. Being Fano means that the dual of the dualizing sheaf is ample. > this indicates that we might want fi"', to be a line bundle, so that such a variety is Gorenstein. > DerFINITION 3.5.1. A complete n-dimensional Gorenstein variety V is Fano if the dual of the line bundle Q} is ample. > the dualizing sheaf on an arbitrary toric variety X has the simple description > LEMMA 3.5.2. A complete toric variety X of dimension n is Fano if and only if > o Do 18 Cartier and ample. > DEFINITION 3.5.3. A n-dimenstonal integral polytope A C Mg ~ R"TM is reflex ive if the following two conditions hold: > Al facets T of A are supported by an affine hyperplane of the form {m € Mg : {m,vr) = -1} for somewvr € N. > Int(A)N M = {0}. > LeMMA 3.5.4. A is reflezive if and only if A° is reflezive. ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=66|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 66]] > If X is a simplicial complete toric variety, then the three classes of automorphisms coming from the torus, roots, and fan symmetries gen erate Aut(X). > dim(Aut(PTM)) = (n+1)2—1 by ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=68|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 68]] > These vectors satisfy the relation vp + v; + 2vg + 2v3 4+ 2v4 = 0. Using the fan > subsets of {vg, v1, v, v3,v4}, we get the weighted projective space P(1,1,2,2,2), which is Fano by Lemma 3.5.6. > where (ag, 61, a2,63,64) €0—2%Z% maps to—2°ag +—Z-—09,a; + 262 +2a3 + 2a4. Thus, the ho mogeneous coordinate ring is Clzo, Z1, 22, 23, 24, where 7o, 71 have degree 1 and T9,Z3, 24 have degree 2. To resolve the singularities of P(1, 1, 2,2, 2), we take the singular cone generated by vg,v; and add the generator > Furthermore, the only two primitive collections are {vg,v1} and {vg, vs,v4,vs}. > Z(X) = {zo = 21 = 0} U {za = 3 = 74 = 75 = 0}. Thus > the moment map is given by > The GKZ decomposition of A7 (Xz) ® R C A3(Xt) ® R = R? is especially simple since there are only two projective simplicial fans with generators contained in £(1): ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=70|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 70]] > GKZ decomposition of X°. > For each face, this can be done in two ways, so that the GKZ decomposition has 26 = 64 cones. ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=74|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 74]] > Xy has terminal sin gularities. ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=76|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 76]] > Fix a reflexive polytope A, and let £ be a maximal projective subdivision of its normal fan, as in Definition 4.1.1. This gives the family of Calabi-Yau hypersurfaces V ¢ Xy for V e |- Kxy|, > each member of this family comes equipped with a complexified Kéhler cone, so that we obtain a family {(V, w)} consisting of all possible pairs of a Calabi-Yau hypersurface and a complexified Kihler class. > repeat the above construction using A°. > The families {(V,w)} and {(V°,w°)} induce iso morphic superconformal field theories whose N = 2 superconformal representations are the same, up to the sign change discussed in Section 1.1. > The Greene-Plesser [GPY] construction gives a physics proof that {(V,w)} and {(V°,w°)} form a mirror pair in the special case of Fermat hypersur faces of the appropriate degree in certain weighted projective spaces. The general case is still open. > The cohomology group H'!(V) > should be isomorphic to H"=21(V°) > the Kihler moduli of V > should be locally isomorphic > to the complex moduli of V° > The A-model correlation function of V' > should correspond > to the suitably normalized B-model correlation function of V° ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=80|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 80]] > For a threefold, this is all that is needed, but ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=96|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 96]] > Finally, a Kahler class w on X determines the primitive cohomology group HE(X,C) or PH*(X) in H*(X,C), and the pairing > The spaces F?(X,) may be seen to fit together to form a locally free subsheaf F®. By construction, F° contains a local system, the locally constant sheaf R*r.C. This uniquely determines a flat connection V on F°, the Gauss-Manin connec tion V, whose flat (or horizontal) sections coincide with the local system R*r,.C. Concretely, V : ¢ — 70 ® QL is defined by V(s® f)=s®df, where s is a sectior’ of R*m,C and f is a function on S. The Gauss-Manin connection satisfies Griffiths transversality V(FP) ¢ FP~1 ® QL. If we set 7 = F°, then ¥ has the locally constant subsheaf +c = R*r.C, and this in turn has the subsheaf +z of integer sections (the image of R*7.Z — R*m.C). We call (H,V,Hz, F") a variation of Hodge structure. > Let (t) be a fixed local section of FTM at a point of p € 5, and let D be the sheaf of linear differential operators on S. > where C{z1,...,2.} is the ring of convergent power series > the Gauss-Manin connection V determines an Og-homomorphism ¢ : D — F° determined by the rule > for vector fields X; on S. This gives F? the structure of a D-module. The ideal I = ker(¢) consists of the differential operators annihilating $(s). We call I the Picard-Fuchs ideal. ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=98|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 98]] > Another important ingredient is the monodromy weight filtration > F; induces a Hodge structure of weight kK on W, /W;_,. The last item says that (W,, F};,) is a mized Hodge structure. ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=102|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 102]] ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=104|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 104]] > use quantum cohomology in Chapter 8 to construct a flat connection on the Kihler moduli space of V° called the A-model connection. ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=134|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 134]] > when the Calabi-Yau threefold V is a toric hypersurface and V*° is its Batyrev mirror, the Hodge-Theoretic Toric Mirror Conjectures discussed ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=140|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 140]] > radius limit points > on the Kihler moduli space of its mirror V°, In practice, starting from a particular choice of M, one usually blows down as much as possible until we get a smooth normal crossings compactification which can’t be blown down any further. This makes the equivalence classes as small > possible. > Finally, notice that we’ve said nothing about the ezistence of maximally unipo tent boundary points. This is still conjectural, though they are present in every example computed to date, > the moduli of toric hypersur faces have naturally occurring compactifications with distinguished boundary points conjectured to be maximally unipotent > mirror symmetry predicts the existence of very special compactifications of M which not only have maximally unipotent boundary points > the structure of the Kihler moduli of the mirror. > “polynomial” part Moty C M consisting of those complex structures which can be realized as hypersurfaces in X. > the polynomial moduli space is the quotient > Moy = P(L(A N M))/Aut(X). > and we also need to worry about the existence of the quotient. The latter concern is serious since Aut(X) need not be reductive. ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=148|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 148]] ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=152|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 152]] > For this purpose, fix a maximal dimensional simplicial cone o whose inte rior lies in the Kahler cone K (V), and let D, = (H*(V,R) +ilnt(0)) /im H2(V,Z) C Kc(V). ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=172|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 172]] ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=176|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 176]] ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=194|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 194]] > Kontsevich [Kontsevich2] made the key observation that curves C C X in (7.1) should be replaced with n pointed curves (C,p1,... ,p,) and holomorphic maps f : C — X. > (7.2) f:C — X such that f,[C] = B and f(p;)€ Z; fori=1,... .n. > To get a compact moduli space, we will allow certain reducible curves C of genus g. > Thus, Gromov-Witten classes are a system of maps > Ig.n,fi B H'(Xv Q)®n — H*(Hg,fivQ)' > Note that (7.4) vanishes unless the Gromov Witten class I ., g{ay,...,on) has a component of top degree in H*(M,., Q). ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=196|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 196]] > If X is projective and G is a hamology class in in Hy(X,Z), then the functor My (X, 3) is an algebraic stack which is proper over C. > the stack ./\_A_o,,.(FT, B) is a smooth stack. > point of M, (X, ). This strictly speaking makes no sense, because functors do not have points. Depending on context, we might mean the corresponding point of the algebraic space associated to '/\_A_g,,.(X ,B), or the corresponding element of My (X, B) (Spec(C)). > we can write 3 = df, where ¢ is the class of a line. Then Mo.(PT,df) is smooth > which means that the underlying algebraic space My . (P",df) is an orbifold. > We will frequently write M, »(P", d) in place of M, (P, db). ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=198|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 198]] > the fundamental class €= [Myn(X,0)| € H.(M,,n(X,0),Q), > Tpmp)(@r,.. an) = /g ei(@) U--- Uel(an). > A class £ with the desired properties is usually called the virtual fundamental class. ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=200|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 200]] > This is a smooth Calabi-Yau threefold with infinitely many lines. > multiplicity 2 at its generic point, and multiplicity 5 at each of the 375 special points corresponding to the 375 lines common to more than one cone > each component of C lying over a component of Mg o(V,£) contributes 20 to the virtual fundamental class (an > while each component over the special points contributes 5. > 20 x 50 + 5 x 375 = 2875, ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=202|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 202]] > Let S be an affine scheme over C and let V be an Os-module. Let Sy denote the trivial extension of § by A, which means that Sy = Spec(I'(Os) & T'(NV)), where I'(Og) ®T(N) is the trivial ring extension of I'(Os) by T'(NV). In particular, we get an infinitesimal extension § <= Sy. The tangent functor TF of F is the following collection of functors. To each § as above and element a € F(S), we have a contravariant functor > taking A to the set of all elements of F(Sx/) which restrict to o under the natural “restriction” map F(Sx) — F(&). > Thus the tangent functor encodes the data of all “extensions” of a to all infinitesimal extensions of S. > A tangent-obstruction complex for F consists of a complex of functors T'F- T*F > there is associated to the data (o, S, ) an obstruction class ob(a, S,NV) € T(T2F(a) ®os N) > A tangent-obstruction complex is perfect if for each (e, §) as above, there is, atleast locally on §, a 2-term complex of locally free sheaves of @g-modules £! — £2such that for any N, T* F(a)(N) is the i*" sheaf cohomology of the induced complexE Qo N. ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=204|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 204]] ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=210|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 210]] ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=212|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 212]] > One of the nice features of the symplectic approach is that many of the com plications of the algebraic case (e.g., moduli spaces being singular or having the wrong dimension) go away when we use a generic almost complex structure. > However, in more recent approaches to symplectic Gromov-Witten invariants (for which semi-positivity can be dropped), analogs of the virtual fundamental class are developed. > For technical reasons, we need to assume that fis simple, meaning that f doesn’t factor as C — €’ — M, where C — C' has degree > 1. (If f is not simple, we say that it is a multiple-cover.) > set M(C, J, B) has the natural structure of a real manifold of dimension > (1 - g)dimgM + 2[ ¢, (TM). To prove this, one represents M(C, J,3) as the fiber of a map between infinite dimensional manifolds. For generic J. this map has a Fredholm linearization, and then the implicit function theorem implies that the fiber M(C, J, 8) is a finite di mensiopal manifold. The dimension of the fiber is given by the Fredholm index, which by the Hirzebruch-Riemann-Roch theorem gives (7.25). > similar analyses are used to construct the moduli spaces appearing in Donaldson theory and Seiberg-Witten theory. ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=218|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 218]] > With this in mind, we now proceed to the axioms. Linearity Axiom. The first axiom asserts that I,5 is linear in each variable. Naively, this is because a sum of cycles is simply their union. Effectivity Axiom. The pext axiom says that for a smooth projective variety X, I,ng = 0 if 3 is not an effective class. > Degree Axiom. This axiom asserts that for a,... , @, € H*(X, Q)®", the coho mology class I , s(a1,... ,an) € H* (Mg.r,Q) has degree 2(g—1)dimX + 2 fpwx + 30, degas > The Degree Axiom implies that Iy, g(an,... ,an) is a top degree class if and only if if > Sor dega; = 2(1 — g)dimX ~ 2f5wx +2(3g—3+n). > we can always assume that (7.33) is satisfied (since the invariant is zero otherwise). > This axiom asserts that the > Ig.n.fi cH” (Xa Q)®n — H* (Mg‘nu Q) is S,-equivariant. ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=224|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 224]] > The numbers N can alternatively be defined as the degree of the Severi va~riety of degree d rational plane curves. However, traditional methods in algebraicgeometry had previously only yielded the first few numbers. The new feature hereis that by interpreting Ny as a a Gromov-Witten invariant, one gets the recursion ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=226|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 226]] ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=228|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 228]] > CONJECTURE 7.4.3. Let V C P* be a generic quintic threefold. Then for each degree d > 1, we have: > There are only finitely many irreducible rational curves C Cc V of degree d. > These curves, as we vary over all degrees, are disjoint from each other. > Note that it is not claimed that the rational curves are smooth. That assertion is false, since 6-nodal rational plane quintic curves C C V exist on a generic V > It is known that there are finitely many rational curves on V of degree less than or equal to 9. > This shows that My 0(V,d) has positivedimension, even though the expected dimension (7.7) is 0 since V' is a Calabi-Yau threefold. > Then, if ny is the total number of such curves of degree d, the above discussion suggests that ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=244|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 244]] > DEFINITION 8.1.1. Let w be a complezified Kihler class on a smooth projective variety X. Then, for a,b € H*(X,C), define axb=3 3 (loas)a,bT) g7 ¢° Ty, > where gf = 2TM Js“. We call a + b the small quantum product of a and b. > T =3¢7T, > so that T9,... ,TTM form the dual basis of Ty, ... , Ty, satisfying [, T°UT, = §;. Then we can write the small quantum product more simply as > axb=Y" > (loap)a,bT)¢" T, > CONJECTURE 8.1.2. For a Calabi-Yau manifold, the sum in Definition 8.1.1 converges provided the imaginary part of w is sufficiently large. > Fortunately, there are some projective manifolds where the above sum is known to behave nicely, giving a rigorous meaning to the small quantum product in terms of Kihler moduli. > Let X be a smooth projective variety such that either > X is Fano, or ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=248|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 248]] ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=258|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 258]] ### [[Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox.pdf#page=304|Algebraic Mirror symmetry and Geometry - Sheldon Katz David A. Cox, page 304]] > put X = X x¢ EG,