Serre Duality

In algebraic geometry, a branch of mathematics, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n (and in greater generality for vector bundles and further, for coherent sheaves). It shows that a cohomology group Hi is the dual space of another one, Hn−i. If the variety is defined over the complex numbers, this yields different information from Poincaré duality, which relates Hi to H2n−i, considering V as a real manifold of dimension 2n.

In the case for holomorphic vector bundle E over a smooth compact complex manifold V, the statement is in the form:

in which V is not necessarily projective.