## 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 
\[
H^0\left(X, \mathcal{L}^{-1}\right)=0
.\]

### 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 $Y$.

a. Show that $f^{\prime} \omega_Y^{\circ}$ is a dualizing sheaf for $X$, where $f^{\prime}$ is defined as in (Ex. 6.10).

b. 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 
\[
H^{n-p}\left(X, \Omega_X^{n-p}\right) \rightarrow H^{n-p}\left(Y, \Omega_Y^{n-p}\right)
.\]
Now $\Omega_Y^{n-p}=\omega_Y$ is a dualizing sheaf
for $Y$, so we have the trace map 
\[
t_Y: H^{n-p}\left(Y, \Omega_Y^{n-p}\right) \rightarrow k
.\]
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 $Y$**.

a. 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.

b. 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).[^hint.3.7.4.b]

c. 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 
\[
\Pic X=H^1\left(X, \mathcal{O}_X^*\right) \rightarrow H^1\left(X, \Omega_X\right)
\]
which we denote by $c$. See (Ex. 4.5).

d. 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 $Y$.

[^hint.3.7.4.b]: 
Hint: Cut with a hyperplane $H \subseteq X$, and use Bertini's theorem (II, 8.18) to reduce to the case $Y$ is a finite set of points.