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.
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.
The case of algebraic curves was already implicit in the Riemann-Roch theorem. For a curve C the coherent groups Hi vanish for i > 1; but H1 does enter implicitly. In fact, the basic relation of the theorem involves l(D) and l(K−D), where D is a divisor and K is a divisor of the canonical class. After Serre we recognise l(K−D) as the dimension of H1(D), where now D means the line bundle determined by the divisor D. That is, Serre duality in this case relates groups H1(D) and H0(KD*), and we are reading off dimensions (notation: K is the canonical line bundle, D* is the dual line bundle, and juxtaposition is the tensor product of line bundles).
In this formulation the Riemann-Roch theorem can be viewed as a computation of the Euler characteristic of a sheaf
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