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
MathSciNet review:
3959070
Full-text PDF
Abstract |
References |
Additional Information
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.
References
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- Oren Ben-Bassat and Kobi Kremnizer, A perspective on the foundations of derived analytic geometry, in progress, personal communication, 2016.
- Carmelo Di Natale and Julian V. S. Holstein, The global derived period map, preprint, arXiv:1607.05984, 2016.
- Hans Grauert and Reinhold Remmert, Coherent analytic sheaves, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 265, Springer-Verlag, Berlin, 1984. MR 755331
- Alexander Grothendieck, Revêtements étales et groupe fondamental. Fasc. II: Exposés 6, 8 à 11, Institut des Hautes Études Scientifiques, Paris, 1963. Troisième édition, corrigée; Séminaire de Géométrie Algébrique, 1960/61. MR 0217088
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- Jacob Lurie, Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009. MR 2522659
- Jacob Lurie, DAG IX: Closed immersions, preprint, 2011.
- Jacob Lurie, DAG V: Structured spaces, preprint, 2011.
- Jacob Lurie, DAG VII: Spectral schemes, preprint, 2011.
- Jacob Lurie, DAG VIII: Quasi-coherent sheaves and Tannaka duality theorems, preprint, 2011.
- Jacob Lurie, DAG X: Formal moduli problems, preprint, 2011.
- Jacob Lurie, DAG XIV: Representability theorems, preprint, 2012.
- Jacob Lurie, Higher algebra, preprint, 2017.
- Jacob Lurie, Spectral algebraic geometry, preprint, 2018.
- Mauro Porta, Comparison results for derived Deligne-Mumford stacks, Pacific J. Math. 287 (2017), no. 1, 177–197. MR 3613438, DOI https://doi.org/10.2140/pjm.2017.287.177
- Mauro Porta, The derived Riemann-Hilbert correspondence, preprint, 2017, arXiv:1703.03907.
- Mauro Porta and Tony Yue Yu, Derived non-archimedean analytic spaces, Selecta Math. (N.S.) 24 (2018), no. 2, 609–665. MR 3782411, DOI https://doi.org/10.1007/s00029-017-0310-1
- Mauro Porta and Tony Yue Yu, Higher analytic stacks and GAGA theorems, Adv. Math. 302 (2016), 351–409. MR 3545934, DOI https://doi.org/10.1016/j.aim.2016.07.017
- Mauro Porta and Tony Yue Yu, Representability theorem in derived analytic geometry, preprint, arXiv:1704.01683, 2017, Version 3, J. European Math. Soc. (to appear).
- J. P. Pridham, Unifying derived deformation theories, Adv. Math. 224 (2010), no. 3, 772–826. MR 2628795, DOI https://doi.org/10.1016/j.aim.2009.12.009
- Jean-Pierre Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier (Grenoble) 6 (1955/56), 1–42 (French). MR 82175
- Carlos Simpson, Geometricity of the Hodge filtration on the $\infty $-stack of perfect complexes over $X_{\rm DR}$, Mosc. Math. J. 9 (2009), no. 3, 665–721, back matter (English, with English and Russian summaries). MR 2562796, DOI https://doi.org/10.17323/1609-4514-2009-9-3-665-721
- Bertrand Toën and Gabriele Vezzosi, Homotopical algebraic geometry. II. Geometric stacks and applications, Mem. Amer. Math. Soc. 193 (2008), no. 902, x+224. MR 2394633, DOI https://doi.org/10.1090/memo/0902
- Tony Yue Yu, Gromov compactness in non-archimedean analytic geometry, J. Reine Angew. Math. 741 (2018), 179–210. MR 3836147, DOI https://doi.org/10.1515/crelle-2015-0077
References
- Federico Bambozzi and Oren Ben-Bassat, Dagger geometry as Banach algebraic geometry, J. Number Theory 162 (2016), 391–462. MR 3448274
- Federico Bambozzi, Oren Ben-Bassat, and Kobi Kremnizer, Stein domains in Banach algebraic geometry, J. Funct. Anal. 274 (2018), no. 7, 1865–1927. MR 3762090, DOI https://doi.org/10.1016/j.jfa.2018.01.003
- Oren Ben-Bassat and Kobi Kremnizer, Non-Archimedean analytic geometry as relative algebraic geometry, Ann. Fac. Sci. Toulouse Math. (6) 26 (2017), no. 1, 49–126 (English, with English and French summaries). MR 3626003
- Oren Ben-Bassat and Kobi Kremnizer, A perspective on the foundations of derived analytic geometry, in progress, personal communication, 2016.
- Carmelo Di Natale and Julian V. S. Holstein, The global derived period map, preprint, arXiv:1607.05984, 2016.
- Hans Grauert and Reinhold Remmert, Coherent analytic sheaves, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 265, Springer-Verlag, Berlin, 1984. MR 755331
- Alexander Grothendieck, Revêtements étales et groupe fondamental. Fasc. II: Exposés 6, 8 à 11, Séminaire de Géométrie Algébrique, vol. 1960/61, Institut des Hautes Études Scientifiques, Paris, 1963. MR 0217088
- Maxim Kontsevich and Yan Soibelman, Homological mirror symmetry and torus fibrations, Symplectic geometry and mirror symmetry (Seoul, 2000) World Sci. Publ., River Edge, NJ, 2001, pp. 203–263. MR 1882331
- Jacob Lurie, Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009. MR 2522659
- Jacob Lurie, DAG IX: Closed immersions, preprint, 2011.
- Jacob Lurie, DAG V: Structured spaces, preprint, 2011.
- Jacob Lurie, DAG VII: Spectral schemes, preprint, 2011.
- Jacob Lurie, DAG VIII: Quasi-coherent sheaves and Tannaka duality theorems, preprint, 2011.
- Jacob Lurie, DAG X: Formal moduli problems, preprint, 2011.
- Jacob Lurie, DAG XIV: Representability theorems, preprint, 2012.
- Jacob Lurie, Higher algebra, preprint, 2017.
- Jacob Lurie, Spectral algebraic geometry, preprint, 2018.
- Mauro Porta, Comparison results for derived Deligne-Mumford stacks, Pacific J. Math. 287 (2017), no. 1, 177–197. MR 3613438
- Mauro Porta, The derived Riemann-Hilbert correspondence, preprint, 2017, arXiv:1703.03907.
- Mauro Porta and Tony Yue Yu, Derived non-archimedean analytic spaces, Selecta Math. (N.S.) 24 (2018), no. 2, 609–665. MR 3782411, DOI https://doi.org/10.1007/s00029-017-0310-1
- Mauro Porta and Tony Yue Yu, Higher analytic stacks and GAGA theorems, Adv. Math. 302 (2016), 351–409. MR 3545934
- Mauro Porta and Tony Yue Yu, Representability theorem in derived analytic geometry, preprint, arXiv:1704.01683, 2017, Version 3, J. European Math. Soc. (to appear).
- Jonathan P. Pridham, Unifying derived deformation theories, Adv. Math. 224 (2010), no. 3, 772–826. MR 2628795
- Jean-Pierre Serre, Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier, Grenoble 6 (1955–1956), 1–42 (French). MR 0082175
- Carlos Simpson, Geometricity of the Hodge filtration on the $\infty$-stack of perfect complexes over $X_\textrm {DR}$, Mosc. Math. J. 9 (2009), no. 3, 665–721, back matter (English, with English and Russian summaries). MR 2562796
- Bertrand Toën and Gabriele Vezzosi, Homotopical algebraic geometry. II. Geometric stacks and applications, Mem. Amer. Math. Soc. 193 (2008), no. 902, x+224. MR 2394633
- Tony Yue Yu, Gromov compactness in non-archimedean analytic geometry, J. Reine Angew. Math. 741 (2018), 179–210. MR 3836147, DOI https://doi.org/10.1515/crelle-2015-0077
Additional Information
Mauro Porta
Affiliation:
Institut de Recherche Mathématique Avancée, 7 rue René Descartes, 67000 Strasbourg, France
MR Author ID:
1177756
Email:
porta@math.unistra.fr
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.