Adjunction formula (Griffiths & Harris proof)

Question

I'm having trouble understanding the proof of the adjunction formula on Griffiths & Harris book (p. 146).

The formula states that if $V \subset M$ is a smooth analytic hypersurface then we have an isomorphism $N^*_V \simeq [-V]|_V$, where $N_V$ is the normal bundle of $V$ and $[-V]$ the line bundle associated to the divisor $-V$.

The strategy is to show that $N_V^* \otimes [V] \simeq \mathcal{O}_Y$ (the trivial line bundle over $Y$) by constructing a nonvanishing global section.

If $V$ is defined by $f_i$ on $U_i$ then the cocicles of $[V]$ are $f_{ij}=f_i/f_j$ and $df_i$ is a section of $N^*_V$. On the other hand, using the product rule for the derivative one gets that $df_i = f_{ij} df_j$ and hence glue to a section of $[V]$. The book states then that the $df_i$ give a global section of $N_V^* \otimes [V]$. Why is that?

Is this statement true? I see that if one has sections $s$ of $L$ and $s'$ of $L'$ then $s \otimes s'$ is a section of $L \otimes L'$. In our case we know that $df_i$ is a section of both $N_V^*$ and $[V]$ and so we get that $df_i \cdot df_i$ (and not $df_i$) is a section of $N_V^* \otimes [V]$. What am I missing here?

Answer 1

The normal bundle
Choose on each $U_i$ a coordinate system $(z_1^{i},...,z_n^{i})$ such that $U_i\cap V$ is given by the equation $z_n^{i}=0$, so that $f_i=z_n^{i}$ .
The normal bundle $N_V$ on $V \;$ is then given by the cocycle

$$n_{ij}=\frac {\partial z_n^{i}}{\partial z_n^{j}} \in \mathcal O^*(V \cap U_{ij})$$ Note carefully that this bundle is defined only on $V$ , and not on $M$.

The bundle associated to $V$
In the same notation the bundle $\mathcal O(V)=[V]$ is defined by a cocycle $g_{ij}\in \mathcal O^*(U_{ij})$ satisfying
$z^i_n=z^j_n.g_{ij}$ on $U_{ij}$.
Taking partial derivatives with respect to $z^j_n$ yields $\frac {\partial z_n^{i}}{\partial z_n^{j}} = g_{ij}+z^j_n.\frac {\partial g_{ij}}{\partial z^j_n} $.
Restricting this to $V \cap U_{ij}$, we get a cocycle for $[V]|V$ ( remember that $z^j_n=0$ on $V\cap U_j$) $$\frac {\partial z_n^{i}}{\partial z_n^{j}} = g_{ij} \in \mathcal O^*(V \cap U_{ij})$$

The two displayed equations prove that $n_{ij}=g_{ij}$ and thus that $$N_V \simeq[V]|V$$

(As you see, I prefer to avoid differential forms and dualization : no $N_V^*$, no $[-V]$. )

Answer 2

The $df_i$ define local frames for $N_V^*$. These induce holomorphic local trivializations of $N_V^*$ which, because $df_i = f_{ij}df_j$ on V, have transition functions $f_{ij}^{-1}$, suitably restricted. This indicates that $N_V^*\simeq [-V]|_V$.

Answer 3

Let me explain this problem more explicitly.

We first should observe the following fact:Let $\varphi=(\varphi^1,...,\varphi^n)$ be a holomorphic map $U\rightarrow\mathbb C^n$ of some connected open neighbourhood $U\subset\mathbb C^n$of the origin,such that $\varphi ^n(z_1,...,z_{n-1},0)=0$ for all $(z_1,...,z_{n-1},0)\in U$.Then using identify theorem,we have $\varphi ^n(z)=z_n\cdot h(z_1,...,z_n)$,where $h(z_1,...,z_n)$ is a power series in $z_1,...,z_n$.Hence,

$$\frac {\partial\varphi^n} {\partial z_k} (z_1,...,z_{n-1},0)=\begin{cases} 0&\mbox{for } k\mbox { $=1,...,n-1$}\\h(z_1,...,z_{n-1},0) &\mbox{for }k=n \end{cases} $$

Now,let us apply this to our situation. We may fix local coordinates $\varphi _i:U_i\cong\varphi_{i}(U_i)\subset\mathbb C^n$,such that $\varphi_{i}(U_i\cap V)= \left\{\ (z_1,...,z_n)\in\varphi_i(U_i)\, |\,z_n=0 \right\}$.Since $V$ is defined by $f_i$ on $U_i$,we get $f_i=\varphi_i^{n}$.

Also,it is easy to verify that the transition maps $\varphi_{ij}:\varphi_j(U_i\cap U_j)\cong\varphi_i(U_i\cap\ U_j)$ have the above property!

So,on $V\cap\ U_i\cap\ U_j$,$$df_i(x)=d(\varphi_{ij}\circ\varphi_j)^n(x)=\frac {\partial\varphi_{ij}^{n}} {\partial z_n}dz_n\circ \varphi_j(x)=h(z_1,...,z_{n-1},0)dz_n\circ \varphi_j(x)$$ is a section of $\mathcal N_V^*$, we have $f_i/f_j=f _{ij}$,so $df_i=f_{ij}df_j+f_{j}df_{ij}=f_{ij}df_j$.Using cocycle condition,we can solve this problem.

Answer 4

Here is yet another way to look at this.

Since $df_i = f_{ij} df_j$ on $V\cap U_i\cap U_j$, we have that $f_i^{-1} df_i = f_j^{-1} df_j$ on $V\cap U_i\cap U_j$. This equality allows us to build a global meromorphic section of $N_V^*$, say $s: V\to \mathcal{M}\otimes_{\mathcal{O}} N_V^*$ with $s|_{V\cap U_i}=f_i^{-1}df_i$.

Now, the defining functions $\{f_i,U_i\}$ of $V$ are holomorphic and trivially respect the cocycle condition $f_i=f_{ij}f_j$ on $U_i\cap U_j$ so they induce a global section of $[V]$ in $M$. Call it $f : M\to [V]$. Note that $f|_V$ is a global section of $[V]|_V$. Also, recall that the way the section is induced is by composing each $f_i$ with the inverse of the trivialization on $U_i$, say $\phi_i^{-1}$, so that $f|_{U_i} = \phi_i^{-1} f_i$

We combine the sections $s$ and $f|_V$ into a meromorphic section of $N_V^* \otimes [V]|_V$ by tensoring : $s\otimes_{\mathcal{O}}f|_V$. The claim is that the section is holomorphic and non-vanishing. This should be clear as on each patch $U_i$, we have $s\otimes_{\mathcal{O}}f|_V = f_i^{-1}df_i \otimes_{\mathcal{O}} \phi^{-1}_if_i = df_i\otimes_\mathcal{O} \phi_{i}^{-1}$ (Since $f_i$ is holomorphic and $\phi_i$ is $\mathbb{C}$-linear on each fiber). Finally, $\phi_i^{-1}$ is non-vanishing (since it's invertible) and holomorphic.

So we have found a global section of $N_V^* \otimes [V]|_V$ that is both holomorphic and non-vanishing, implying that it must be the trivial bundle.