A numerical obstruction to smooth fillings of Calabi-Yau degenerations

Abstract: In a previous work with D. Eriksson and C. Mourougane, we introduced a real valued invariant of Calabi-Yau manifolds, based on the analytic torsion of holomorphic differentials, generalizing work of Fang, Lu and Yoshikawa in dimension 3. We call it the BCOV invariant. For one-parameter degenerations, we established the logarithmic divergence of the BCOV invariant. It follows that the rate of logarithmic divergence is an obstruction to smooth fillings of such families. Recently, in joint work with D. Eriksson, we have considered the problem of effectively computing this obstruction. We obtain simple expressions for degenerations with isolated singularities, depending on Hodge theoretic invariants of the Milnor fiber. Numerical experiments motivated us to conjecture that the obstruction is strictly positive for isolated terminal singularities. This can be proven in several instances, covering in particular the ADE singularities. As a result, Calabi-Yau degenerations with such singularities do not admit any smooth fillings. This generalizes a result by Claire Voisin from 1990, in the case of Lefschetz pencils of hypersurfaces (A_1 singularities), which was one of the few known results.

algebraic geometryanalysis of PDEsalgebraic topologycomplex variablesdifferential geometrygeneral topologygeometric topologyK-theory and homologymetric geometrysymplectic geometry

Audience: researchers in the topic

CRM - Séminaire du CIRGET / Géométrie et Topologie

Series comments: Hybrid seminar of geometry and topology. Laboratory : CIRGET - www.cirget.uqam.ca The homepage of the seminar is www.cirget.uqam.ca/fr/seminaires.html

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