III.7: Serre Duality
III.7.1. Special case of Kodaira vanishing. #to_work
Let \(X\) be an integral projective scheme of dimension \(\geqslant 1\) over a field \(k\), and let \(\mathcal{L}\) be an ample invertible sheaf on \(X\). Then \begin{align*} H^0\left(X, \mathcal{L}^{-1}\right)=0 .\end{align*}
III.7.2. #to_work
Let \(f: X \rightarrow Y\) be a finite morphism of projective schemes of the same dimension over a field \(k\), and let \(\omega_Y^{\circ}\) be a dualizing sheaf for \(Y\).
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Show that \(f^{\prime} \omega_Y^{\circ}\) is a dualizing sheaf for \(X\), where \(f^{\prime}\) is defined as in (Ex. 6.10).
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If \(X\) and \(Y\) are both nonsingular, and \(k\) algebraically closed, conclude that there is a natural trace map \(t: f_* \omega_X \rightarrow \omega_Y\).
III.7.3. #to_work
Let \(X=\mathbf{P}_k^n\). Show that \(H^q\left(X, \Omega_X^p\right)=0\) for \(p \neq q\), \(k\) for \(p=q, 0 \leqslant p, q \leqslant n\).
III.7.4. * The Cohomology Class of a Subvariety. #to_work
Let \(X\) be a nonsingular projective variety of dimension \(n\) over an algebraically closed field \(k\). Let \(Y\) be a nonsingular subvariety of codimension \(p\) (hence dimension \(n-p\) ). From the natural map \(\Omega_X \otimes\) \(\mathcal{O}_Y \rightarrow \Omega_Y\) of \((\mathrm{II}, 8.12)\) we deduce a map \(\Omega_X^{n-p} \rightarrow \Omega_Y^{n-p}\). This induces a map on cohomology \begin{align*} H^{n-p}\left(X, \Omega_X^{n-p}\right) \rightarrow H^{n-p}\left(Y, \Omega_Y^{n-p}\right) .\end{align*} Now \(\Omega_Y^{n-p}=\omega_Y\) is a dualizing sheaf for \(Y\), so we have the trace map \begin{align*} t_Y: H^{n-p}\left(Y, \Omega_Y^{n-p}\right) \rightarrow k .\end{align*} Composing, we obtain a linear map \(H^{n-p}\left(X, \Omega_X^{n-p}\right) \rightarrow k\). By (7.13) this corresponds to an element \(\eta(Y) \in\) \(H^p\left(X, \Omega_X^p\right)\), which we call the cohomology class of \(Y\).
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If \(P \in X\) is a closed point, show that \(t_X(\eta(P))=1\), where \(\eta(P) \in H^n\left(X, \Omega^n\right)\) and \(t_X\) is the trace map.
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If \(X=\mathbf{P}^n\), identify \(H^p\left(X, \Omega^p\right)\) with \(k\) by (Ex. 7.3), and show that \(\eta(Y)=(\operatorname{deg} Y) \cdot 1\), where deg \(Y\) is its degree as a projective variety (I, \(\S\) 7). 1
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For any scheme \(X\) of finite type over \(k\), we define a homomorphism of sheaves of abelian groups \(d \log : \mathcal{O}_X^* \rightarrow \Omega_X\) by \(d \log (f)=f^{-1} \, df\). Here \(\mathcal{O}^*\) is a group under multiplication, and \(\Omega_X\) is a group under addition. This induces a map on cohomology \begin{align*} \operatorname{Pic}X=H^1\left(X, \mathcal{O}_X^*\right) \rightarrow H^1\left(X, \Omega_X\right) \end{align*} which we denote by \(c\). See (Ex. 4.5).
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Returning to the hypotheses above, suppose \(p=1\). Show that \(\eta(Y)=c(\mathcal{L}(Y))\), where \(\mathcal{L}(Y)\) is the invertible sheaf corresponding to the divisor \(Y\).