TITLE: Bundle cohomology under a flop
QUESTION [6 upvotes]: This question is about the behaviour of line bundle cohomologies under flops. Specifically, my question is: is the (dimension of the) zeroth cohomology of a line bundle preserved by a flop?
This is motivated by the following example. For a toric variety, a flop corresponds to a change of the triangulation of the associated fan. But the zeroth cohomology of a line bundle is determined by the toric rays / weight system alone, i.e. it is unaffected by the triangulation. Hence it is unaffected by a flop.
Is the same true for more general varieties? Or is this specific to toric varieties?
A couple of notes:

I have only been able to find statements about quantum cohomology being preserved, but this is too unfamiliar, so I do not understand the consequences.
The zeroth line bundle cohomology is the same as the complete linear system of the associated divisor, and I am happy to discuss in either terminology.

REPLY [4 votes]: In the smooth case this follows from Hartogs Theorem, which gives isomorphisms
$$
H^0(X,L) \cong H^0(X-Z,L) = H^0(X'-Z',L) \cong H^0(X',L),
$$
where $X$ and $X'$ are the varieties related by the flop, $Z \subset X$ and $Z' \subset X'$ are the flopping loci, and $L$ is the line bundle in question.