Equisingular resolution with SNC fibers and combinatorial type of varieties

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.