Algebraic Varieties Which Are a Disjoint Union of Subvarieties

The purpose of this note is to prove the following result: Theorem 1

Let X and Y be smooth projective complex varieties, and suppose that X = X₁ ∪ ··· ∪ X_n and Y = Y₁ ∪ ··· ∪ Y_n are a disjoint union of quasiprojective subvarieties. Suppose that X_i is algebraically isomorphic to Y_i, for all i. Then the Betti numbers of X and Y are equal, and in fact their Hodge numbers are equal.

The result for the Betti numbers is part of the mathematical folklore; it has been used, for instance, in [1] and implicitly in [4]. Apparently it was originally noticed by Serre, who proved it by reducing the variety modulo p, counting points and using the solution to the Weil conjectures. The proof given in this paper uses mixed Hodge theory rather than the Weil conjectures; the general yoga of Hodge theory and the Weil conjectures suggest that such a proof should exist.

The theorem is in fact a simple consequence of a sum formula for the Hodge numbers of the mixed Hodge structure of an arbitrary variety in terms of those of the pieces of a decomposition. (Throughout this paper, the term "variety" will mean "quasiprojective complex variety.") First, some definitions: According to [2], the rational cohomology groups of a complex algebraic variety have a weight filtration W and a Hodge filtration F, and so do its rational cohomology groups with compact support. The associated graded objects of these filtrations are denoted Gr^W and Gr_F, respectively. We refine the concept of Euler characteristic by using these filtrations: Let χ ( X ) = ∑ ( − 1 ) k dim H k ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq635.tif"/> χ m ( X ) = ∑ ( − 1 ) k dim G r m W H k ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq636.tif"/> χ p q ( X ) = ∑ ( − 1 ) k dim G r F p G r p + q W H k ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq637.tif"/>

100Similarly, define χ^c, χ m c https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq638.tif"/> and χ c pq https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq639.tif"/> using H c k ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq640.tif"/> (cohomology with compact supports) in place of H^k(X). These graded Euler characteristics satisfy χ ( X ) = ∑ m X m ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq641.tif"/> χ m ( X ) = ∑ p + q = m X p q ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq642.tif"/>

and so forth. If X is smooth of dimension n, the Poincaré duality implies that χ c p q ( X ) = χ n − p , n − q ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq643.tif"/>

If X is smooth and projective, then χ m ( X ) = ( − 1 ) m dim H m ( X ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072386/f0e84800-6123-461d-a9d3-5a5098a93bca/content/eq644.tif"/>