Equisingular resolution with SNC fibers and combinatorial type of varieties

TL;DR

This paper introduces a topological invariant called the combinatorial type of varieties, generalizing dual complexes of SNC divisors, and demonstrates its functoriality, relation to cohomology, and applications in desingularization and constructibility of fiber types.

Contribution

It defines the combinatorial type of varieties, proves its functoriality and relation to cohomology, and establishes new desingularization results with SNC fibers in characteristic zero.

Findings

The combinatorial type is a homotopy-invariant simplicial complex associated with varieties.

Any variety in characteristic zero admits a desingularization with SNC fibers.

The combinatorial type of fibers is a constructible function in projective morphisms.

Abstract

We introduce the notion of combinatorial type of varieties $X$ which generalizes the concept of the dual complex of SNC divisors. It is a unique, up to homotopy, finite simplicial complex $\Sigma(X)$ which is functorial with respect to morphisms of varieties. Its cohomology $H^i(\Sigma(X),Q)$ for complex projective varieties coincide with weight zero part of the Deligne filtration $W_0(H^i(X,Q))$. The notion can be understood as a topological measure of the singularities of algebaric schemes of finite type. We also prove that any variety in characteristic zero admits the Hironaka desingularization with all fibers having SNC. Moreover the dual complexes of the fibers are isomorphic on strata. Also for any morphism $f:X\to Y$ there exists a similar desingularization $\tilde{X}\to X$ for which the induce morphism $\tilde{X}\to Y$ has SNC fibers. One of the consequence is that for any projective morphism $f:X\to Y$ the combinatorial type of the fiber is a constructible function. In particular $\dim(W_0H^i(f^{-1}(y))$ is constructible.