648 research outputs found

    Motivic zeta function via dlt modification

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    Given a smooth variety XX and a regular function ff on it, by considering the dlt modification, we define the dlt motivic zeta function Zmotdlt(s)Z^{\rm dlt}_{\rm mot}(s) which does not depend on the choice of the dlt modification.Comment: 11 page

    The essential skeleton of a degeneration of algebraic varieties

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    In this paper, we explore the connections between the Minimal Model Program and the theory of Berkovich spaces. Let kk be a field of characteristic zero and let XX be a smooth and proper k((t))k((t))-variety with semi-ample canonical divisor. We prove that the essential skeleton of XX coincides with the skeleton of any minimal dltdlt-model and that it is a strong deformation retract of the Berkovich analytification of XX. As an application, we show that the essential skeleton of a Calabi-Yau variety over k((t))k((t)) is a pseudo-manifold.Comment: To appear in American Journal of Mathematic

    Poles of maximal order of motivic zeta functions

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    We prove a 1999 conjecture of Veys, which says that the opposite of the log canonical threshold is the only possible pole of maximal order of Denef and Loeser's motivic zeta function associated with a germ of a regular function on a smooth variety over a field of characteristic zero. We apply similar methods to study the weight function on the Berkovich skeleton associated with a degeneration of Calabi-Yau varieties. Our results suggest that the weight function induces a flow on the non-archimedean analytification of the degeneration towards the Kontsevich-Soibelman skeleton.Comment: to appear in Duke Mathematical Journa