Hodge index theorem
In mathematics, the Hodge index theorem for an algebraic surface V determines the signature of the intersection pairing on the algebraic curves C on V. It says, roughly speaking, that the space spanned by such curves has a one-dimensional subspace on which it is positive definite, and decomposes as a direct sum of some such one-dimensional subspace, and a complementary subspace on which it is negative definite.
In a more formal statement, specify that V is a non-singular projective surface, and let H be the divisor class on V of a hyperplane section of V in a given projective embedding. Then the intersection
where d is the degree of V. Let D be the vector space of rational divisor classes on V, up to algebraic equivalence. The dimension of D is finite and is usually denoted by ρ. The Hodge index theorem says that the subspace spanned by H in D has a complementary subspace on which the intersection pairing is negative definite. Therefore the signature is.
The abelian group of divisor classes up to algebraic equivalence is now called the Néron-Severi group; it is known to be a finitely-generated abelian group, and the result is about its tensor product with the rational number field. Therefore ρ is equally the rank of the Néron-Severi group.
This result was proved in the 1930s by W. V. D. Hodge, for varieties over the complex numbers, after it had been a conjecture for some time of the Italian school of algebraic geometry. Hodge's methods were the topological ones brought in by Lefschetz. The result holds over general fields.
