Dimensions of global sections of a divisor and its pullback

Question

I doubt the following claim, but it seems that the proof of Theorem 10.2 (page 301, and one can download the book from libgen.org) in the book "algebraic geometry: an introduction to birational geometry of algebraic varieties" uses it:

Let $V,W$ be smooth projective varieties, and $h: V \to W$ be a dominant morphism. If $Y \subseteq W$ be an effective divisor, then the dimension of global sections $h^0(V, h^*(Y)) = h^0(W,Y)$.

Answer 1

Let me explain why

• the statement in the last paragraph of the question is false as written;
• it is true under the extra assumption that $h$ has connected fibres.

A counterexample for the statement in the question is the following: let $h:C \rightarrow \mathbf P^1$ be a $2:1$ map from an elliptic curve. Let $L=O(2)$. Then $h^0(L)=3$, but $h^*(L)$ has degree 4, so Riemann--Roch says $h^0(h^*(L))=4$.

On the other hand, now suppose $h$ has connected fibres. Then we have $h_* O_V = O_W$. So the projection formula says that for any line bundle $L$ on $W$, we have $h_*(h^*(L))=L \otimes h_* O_V = L$. Since global sections are unchanged by pushforward, we get $H^0(h^*(L))=H^0(h_*(h^*(L))=H^0(L)$.