Computing zeta functions on log smooth models

Abstract

We establish a formula for the volume Poincaré series of a log smooth scheme. This yields in particular a new expression and a smaller set of candidate poles for the motivic zeta function of a hypersurface singularity and of a degeneration of Calabi-Yau varieties. RésuméCalcul de fonctions zêta à partir de modèles log lisses. Nous établissons une formule pour la série volume de Poincaré d'un schéma log lisse. Ceci nous fournit en particulier une nouvelle expression et un ensemble réduit de candidats pôles pour la fonction zêta motivique d'une singularité d'hypersurface et d'une dégénération de variétés de Calabi-Yau.

Computing motivic zeta functions on log smooth models

“…The main results in this paper form a part of the first author's PhD thesis [Bu15a]. They were announced in [Bu15b].…”

Motivic and Analytic Nearby Fibers at Infinity and Bifurcation Sets

“…In particular, the volume of any locally closed subset of (U K ) an is defined, and the equality Vol HK (F ∞ f,0 ) = Vol HK (U K ) an − Vol HK U R holds naturally in this context. Moreover, the motivic volumes in [17] can be computed in an analogous way as the volumes in [15] and [4] in terms of suitable resolutions of singularities (see [24, Theorem 2.6.1]), and this can be used to show that Vol HK U R = f ! S f , see [24, Corollary 2.6.2] or [13].…”

Motivic Zeta Functions of the Quartic and its Mirror Dual

“…We denote by K 0 (Var C ) the Grothendieck ring of complex varieties, by L = [A 1 C ] ∈ K 0 (Var C ) the class of the affine line, and by M C = K 0 (Var C )[L −1 ] the localized Grothendieck ring. We recall that K 0 (Var C ) is the quotient of the free abelian group on isomorphism classes [X] of complex varieties X modulo the scissor relations…”