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Moduli of products of stable varieties

Part of:Families, fibrationsSurfaces and higher-dimensional varieties

Published online by Cambridge University Press:  06 September 2013

Bhargav Bhatt,
Wei Ho,
Zsolt Patakfalvi and
Christian Schnell
Bhargav Bhatt*
Affiliation:
School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA
Wei Ho
Affiliation:
Department of Mathematics, Columbia University, NY 10027, USA email who@math.columbia.eduDepartment of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA email who@math.columbia.edupzs@princeton.edu
Zsolt Patakfalvi
Affiliation:
Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08544-1000, USA email who@math.columbia.edupzs@princeton.edu
Christian Schnell
Affiliation:
Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA email christian.schnell@stonybrook.edu
*
bhargav.bhatt@gmail.com
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Abstract

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We study the moduli space of a product of stable varieties over the field of complex numbers, as defined via the minimal model program. Our main results are: (a) taking products gives a well-defined morphism from the product of moduli spaces of stable varieties to the moduli space of a product of stable varieties; (b) this map is always finite étale; and (c) this map very often is an isomorphism. Our results generalize and complete the work of Van Opstall in dimension $1$. The local results rely on a study of the cotangent complex using some derived algebro-geometric methods, while the global ones use some differential-geometric input.

Keywords

stable varietiesmoduli spacesdeformation theorycotangent complexderived algebraic geometrystable vector bundles

MSC classification

Secondary:14D05: Structure of families (Picard-Lefschetz, monodromy, etc.)14D06: Fibrations, degenerations14D15: Formal methods; deformations14D20: Algebraic moduli problems, moduli of vector bundles14D22: Fine and coarse moduli spaces14D23: Stacks and moduli problems14J15: Moduli, classification14J17: Singularities

Information

Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 12 , December 2013, pp. 2036 - 2070
Copyright
© The Author(s) 2013 

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