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We introduce the notion of combinatorial type of varieties $X$ whichgeneralizes the concept of the dual complex of SNC divisors. It is a unique, upto homotopy, finite simplicial complex $\Sigma(X)$ which is functorial withrespect to morphisms of varieties. Its cohomology $H^i(\Sigma(X),Q)$ forcomplex projective varieties coincide with weight zero part of the Delignefiltration $W_0(H^i(X,Q))$. The notion can be understood as a topologicalmeasure of the singularities of algebaric schemes of finite type. We also provethat any variety in characteristic zero admits the Hironaka desingularizationwith all fibers having SNC. Moreover the dual complexes of the fibers areisomorphic on strata. Also for any morphism $f:X\to Y$ there exists a similardesingularization $\tilde{X}\to X$ for which the induce morphism $\tilde{X}\toY$ 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. Inparticular $\dim(W_0H^i(f^{-1}(y))$ is constructible.

Equisingular resolution with SNC fibers and combinatorial type of varieties