Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

GAGA theorems in derived complex geometry


Author: Mauro Porta
Journal: J. Algebraic Geom. 28 (2019), 519-565
DOI: https://doi.org/10.1090/jag/716
Published electronically: April 18, 2019
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Abstract: In this paper, we expand the foundations of derived complex analytic geometry introduced by Jacob Lurie in 2011. We start by studying the analytification functor and its properties. In particular, we prove that for a derived complex scheme locally almost of finite presentation $ X$, the canonical map $ X^{\mathrm {an}} \to X$ is flat in the derived sense. Next, we provide a comparison result relating derived complex analytic spaces to geometric stacks. Using these results and building on the previous work of the author and Tony Yue Yu, we prove a derived version of the GAGA theorems. As an application, we prove that the infinitesimal deformation theory of a derived complex analytic moduli problem is governed by a differential graded Lie algebra.


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Mauro Porta
Affiliation: Institut de Recherche Mathématique Avancée, 7 rue René Descartes, 67000 Strasbourg, France
Email: porta@math.unistra.fr

DOI: https://doi.org/10.1090/jag/716
Received by editor(s): June 3, 2017
Received by editor(s) in revised form: July 23, 2017, August 12, 2017, and October 9, 2017
Published electronically: April 18, 2019
Additional Notes: This research was partially supported by the Simons Foundation grant No. 347070.
Article copyright: © Copyright 2019 University Press, Inc.