--- created: 2023-03-26T11:58 updated: 2024-06-02T22:38 aliases: - short exact sequences for cohomology of sheaves - short exact sequences links: tags: - dissertation --- # Curves - Arithmetic genus: $p_a(C) = {1\over 2}(C^2 + C\cdot K_X)$ for $C \subseteq X$ an irreducible curve. - Riemann-Roch: $h^0(D) + h^0(K_X-D) = h^1(D) + {1\over 2}D\cdot (D-K_X) + \chi(\OO_X)$. - For K3s: $h^0(D) + h^0(-D) = h^1(D) + {1\over 2}D^2 + 2$. # Blowups - For $\pi: \Bl_1 X\to X$ a blowup of a point on a surface: - $(\pi^* C).(\pi^* D) = C.D$ - $\pi^*C \in \gens{E}^{\perp \Pic (\Bl_1 X) }$. - $E^2 = -1$ - $\Pic(\Bl_1 X) \cong \Pic(X) \oplus E$. - $(\pi^* C). D = C.(\pi_* D)$ - $\pi^* C = \tilde C + rE$ - where $C \approx V(f)$ locally $\implies r\da \mult_p(C) \da \max\ts{n \st f\in \mfm_p^n \subseteq \OO_{X, p}}$. - $\tilde C. E = r$ - $p_a(\tilde C) = p_a(C)- {1\over 2}r(r-1)$ - Genus formula for singular curves: $g(C) \da g(C^\nu) = p_a(C) + {1\over 2}\sum_{p\in C^\sing} r_p(r_p-1)$ - Canonicals for blowups: - $K_{\Bl_1 X} = \pi^* K_X + E$ - $K_{\Bl_1 X}^2 = K_X^2-1$. - $\omega_{\Bl_n X} \cong \pi^* \omega_X \tensor_{\OO_{\Bl_n X}} \OO_{\Bl_n X}(\sum E_i)$? - Structure sheaves and partial Hodge diamonds for blowups: - $\RR\pi_* \OO_{\Bl_1 X} \cong \OO_X$ and $\tau_{\geq 1}\RR\pi_* \OO_{\Bl_1 X} = 0$. - $H^*(\OO_{\Bl_1 X}) \cong H^*(\OO_X)$. - Birational invariants: - **Not** $K_X$ - $p_a(X)$ - $p_g(X)$ - $q(X)$ - If $f: X\birational Y$ with $X, Y$ smooth and $E\in \Div(X)$ an exceptional divisor, $f_* \OO_X(E) \cong f_* \OO_X \cong \OO_Y$. - Canonicals - $\kappa\inv(\Bl_1 X) \leq \kappa\inv (X)$ where $\kappa\inv$ is the Itaka dimension of $-K_X$. # Divisors - For $D\in \Div(X)$: - $\mcl(D) \da \mcl\tensor_{\OO_X}\OO_X(D)$. - $\OO_X(f^* D) \cong f^*\OO_Y(D)$ - $f: X\birational Y \implies f_* \OO_X = \OO_Y$ and $\tau_{\geq 1} \RR f_* \OO_X = 0$ when $X, Y$ are smooth. - $H^{p, p}(\Bl_1 X) = H^{p, p}(X) \oplus \CC$ for all $1\leq p\leq n-1$. - If $f: X\birational Z$ is an isomorphism outside of $\codim \geq 2$ closed subsets, then $\tau_{\leq 2} H(X; \ZZ)\cong \tau_{\leq 2} H(Z; \ZZ)$; similarly for $\pi_*$. # Sheaf Cohomology - **Projection formula:** ? - For $f:X\to Y$ and $\mcl_i \in \Pic(Y)$: $f^*(\mcl_1\tensor_{\OO_Y} \mcl_2) = f^*\mcl_1 \tensor_{\OO_X} f^* \mcl_2$. - Defining pullbacks: $f^* \mcl \da f\inv(\mcl)\tensor_{f\inv(\OO_Y)} \OO_X$. - Short exact sequences: - Structure sheaves and twists: - $\mci_D \injects \OO_X \surjects i_*\OO_D$. - $\ul{\ZZ} \injects \OO_X \surjects \OO_X\units$ - Used to relate singular cohomology $H^*(X; \ZZ)$ to sheaf cohomology. - $\OO_X(-D) \injects \OO_X \surjects i_*\OO_D$ for $i: D\to X$ a closed embedding of a divisor. - $\OO_C(D-p) \injects \OO_C(D) \surjects \kappa(p)$ - Differentials - **Relative cotangent**: $f_* \Omega_Y \to \Omega_X \surjects \Omega_{X/Y}$ - Right exact in general - Exact when $f:X\to Y$ is a finite morphism of curves - $\Omega_{\PP^{n+1}} \injects \Omega_{\PP^{n+1}}(\log X) \surjects \Omega_X[-1]$ for $X\subseteq \PP^{n}$ # Branched Covers - **Definition (branched covers)**: finite surjective proper holomorphic maps between normal varieties $f: X\to Y$ of generic degree $d$. - **Theorem**: if $s\in H^0(Y; L^n)\smz$ with $Z(s) \in \abs{nL}$ then there exists a finite flat $f: X\to Y$ and $\exists \tilde s\in H^0(X; f^* L)$ such that $(\tilde s)^n = f^* s$ and $$f_* \OO_X \cong \tau_{\leq n-1} \Sym L\inv = \OO_Y \oplus L\inv \oplus \cdots \oplus L^{-(n-1)}$$ - If $\mcl \in \Pic(Y)$ with $f^* \mcl \cong \OO_X$ then $\mcl^{\tensor d} \cong \OO_Y$. - $\OO_X(R) = f^* \mcl$ - $f^* B = nR$ - $\omega_X = f^*(\omega_Y \tensor_{\OO_Y} \mcl^{\tensor (d-1)})$. - $f_* \OO_X =\displaystyle\bigoplus_{k=0}^{d-1}\mcl^{-k}$. - If $H^i(f^* \mce) = 0$ for some $i$, then $H^i(\mce) = 0$. - $f^*(B) = \displaystyle\sum_{x\in f\inv(B)} e_x x$. - $R \da \displaystyle\sum_{x\in X} \len(\Omega_{X/Y})_x x = \sum_{x\in X}(e_x - 1) x$. - **Hurwitz formula**: $\omega_X \cong \pi^*(\omega_Y)\tensor_{\OO_X} \OO_X(R)$. - $K_X \sim \pi^* K_Y + R$. - $\Omega_{X/Y} \cong \OO_R$. - **Riemann-Hurwitz**: $2g_X - 2 = \deg(f)\cdot(2g_Y - 2) + \sum_{x\in X} (e_x - 1)$ for $f:X\to Y$ a finite morphism of curves. - If $f$ is dominant, $g_X \geq g_Y$. # Hodge theory - General hodge diamond: ![](2024-03-03.png) - Hodge Symmetries for compact Kahlers $X$: dihedral $D_4$. - Hodge duality: $H^{p, q}(X) \cong \overline{H^{q, p}}(X) \implies h^{p,q}(X) = h^{q, p}(X)$. - $H^{q, 0}(X) = \overline{H^{0, q}}(X)$ are birational invariants. - Serre duality: If $X$ is compact Kahler, $H^{p, q}(X) \cong H^{n-p, n-q}(X)\dual$; invariant under $\pi$ rotation. - Together imply horizontal symmetry $h^{p, q}(X) = h^{n-p, q}$. Summary: - ![](2024-03-03-2.png) - Vanishing theorems: - For Kahlers, if $c_1(F) < 0$, then $H^q\left(\Omega_X^p(F)\right)=0$ when $q+p\leq n$. - For compact Kahlers and $L$ is a positive line bundle, $\tau_{\geq 1}H\left(\omega_X \otimes L\right)=0$ - Equivalently, $\tau_{\geq 1} H(\OO_X(K_X + D)) = 0$ and $\tau_{\leq n} H(\OO_X(-D)) = 0$. - For complex manifolds $X$ with $E$ a holomorphic vector bundle which is ample, $H^{q}(\Omega_X^p \tensor E) = 0$ for $p+q \geq n+r \da \dim_\CC X + \rank_\CC E$. - **Nakano vanishing**: $H^{p, q}(L) \da H^q(\Omega^p \tensor L) = 0$ for $p+q\geq n+1$ and $H^{p, q}(L\inv) = 0$ for $p+q \leq n-1$. # Misc - **Euler characteristics of surfaces**: - For $\mcl\in \Pic(X)$, one has $\chi(\mcl) = {1\over 2}(L^2 \tensor \omega_X\dual) + \chi(\OO_X)$ - For a K3: $\chi(\mcl) = {1\over 2}L^2 + 2$ - If $\mcl\in \Pic(X)^\amp$ then $h^{\geq 1}(\mcl) = 0$, so $h^0(\mcl) = {1\over 2}(L^2 \tensor \omega_X\dual) + \chi(\OO_X)$, - **Noether's formula**: $\chi(\OO_X) = {1\over 12}(c_1^2 + c_2)$. - **Lefschetz hyperplane**: for $X \subseteq \PP^n$ projective and $Y = H\intersect X$ a hyperplane section with $U \da X\sm Y$ smooth, $H_{i\leq n-1}(X,Y;\ZZ) = 0$, and so $H^{i\leq n-1}(X; \ZZ) \iso H^{i\leq n-1}(Y; \ZZ)$. - If $X$ is smooth projective and $D\in \Div(X)^{\eff, \amp}$ is smooth, then $H^{p, q}(X) \iso H^{p, q}(D)$ for $p+q\leq n-2$ and injective for $p+q = n-1$. - **Hard Lefschetz**: $h_{n-k}(X) = h^{n+k}(X)$ for $X \subseteq \PP^n$ smooth projective.