Abstract
A complex variety, degenerating over a punctured disk, carries a limit mixed Hodge structure on its cohomology, encoding the action of monodromy. The associated limit mixed Hodge–Deligne polynomial can be expressed in terms of the motivic nearby fiber. Using techniques from tropical geometry, we present a new formula for the motivic nearby fiber and concentrate on the case of degenerating families of complex hypersurfaces, generalizing work of Danilov and Khovanskiĭ and Batyrev and Borisov. If these families satisfy a natural smoothness condition, called schönness, their limit mixed Hodge–Deligne numbers can be expressed in terms of new, combinatorial invariants of a polyhedral subdivision of the associated Newton polytope. These invariants are multivariable, Ehrhart-theoretic extensions of Stanley’s invariants of subdivisions and are situated in his theory in a companion combinatorial paper.
Highlights
Let O be the ring of germs of analytic functions in C in a neighborhood of the origin, and let K be its field of fractions
Our goal is to compute an important invariant of X called the motivic nearby fiber ψX = ψf, which was introduced by Denef and Loeser [23] and contains information about the extension of f to a family over the whole complex disk D
The motivic nearby fiber specializes to the limit Hodge–Deligne polynomial of X and to both the χy-characteristic and Euler characteristic of Xgen: uvw2Est (Xgen)
Summary
Let O be the ring of germs of analytic functions in C in a neighborhood of the origin, and let K be its field of fractions. We observe that a motivic formula holds for the refined limit Hodge–Deligne polynomials for intersection cohomology with compact support (Theorem 6.1) and deduce that the following corollary is equivalent to Theorem 1.5 (see Lemma 6.2). (−1)dim γ 0 if γ is bounded otherwise This follows directly from Lemma 2.1 if we do the following: Let Cγ be the cone over γ × 1 in Rn × R; choose H to be an affine hyperplane such that P = Cγ ∩ H is a polytope not containing the origin; set Q to be the intersection of P with Rn × {0}; and let S be polyhedral subdivision of P induced by the fan refinement of Cγ induced by.
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