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On a logarithmic version of the derived McKay correspondence

Part of: Families, fibrations (Co)homology theory Birational geometry

Published online by Cambridge University Press:  08 November 2018

Sarah Scherotzke ,
Nicolò Sibilla  and
Mattia Talpo
Sarah Scherotzke
Affiliation:
Mathematical Institute of the University of Münster, Einsteinstrasse 62, 48149 Münster, Germany email [email protected]
Nicolò Sibilla
Affiliation:
Max Planck Institute for Mathematics, Bonn, Germany School of Mathematics, Statistics and Actuarial Sciences, University of Kent, Canterbury, Kent CT2 7NF, UK email [email protected]
Mattia Talpo
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK email [email protected]
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Abstract

We globalize the derived version of the McKay correspondence of Bridgeland, King and Reid, proven by Kawamata in the case of abelian quotient singularities, to certain logarithmic algebraic stacks with locally free log structure. The two sides of the correspondence are given respectively by the infinite root stack and by a certain version of the valuativization (the projective limit of every possible logarithmic blow-up). Our results imply, in particular, that in good cases the category of coherent parabolic sheaves with rational weights is invariant under logarithmic blow-up, up to Morita equivalence.

Keywords

McKay correspondencesemistable degenerationsparabolic sheaves

MSC classification

Primary: 14F05: Sheaves, derived categories of sheaves and related constructions 14D06: Fibrations, degenerations 14E16: McKay correspondence
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 12 , December 2018 , pp. 2534 - 2585
Copyright
© The Authors 2018 

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References

Abramovich, D. and Chen, Q., Stable logarithmic maps to Deligne–Faltings pairs II , Asian J. Math 18 (2014), 465–488.Google Scholar
Abramovich, D., Chen, Q., Gillam, D., Huang, Y., Olsson, M., Satriano, M. and Sun, S., Logarithmic geometry and moduli (English summary) , in Handbook of moduli, Vol. I, Advanced Lectures in Mathematics (ALM), vol. 24, eds Farkas, G. and Morrison, H. (International Press, Somerville, MA, 2013), 1–61.Google Scholar
Abramovich, D., Chen, Q., Marcus, S., Ulirsch, M. and Wise, J., Skeletons and fans of logarithmic structures , in Nonarchimedean and tropical Geometry, Simons Symposia, eds Baker, M. and Payne, S. (Springer, Cham, 2016), 287–336.Google Scholar
Abramovich, D., Chen, Q., Marcus, S. and Wise, J., Boundedness of the space of stable logarithmic maps , J. Eur. Math. Soc. (JEMS) 19 (2017), 2783–2809.Google Scholar
Abramovich, D. and Fantechi, B., Orbifold techniques in degeneration formulas , Ann. Sci. Norm. Super. Pisa Cl. Sci. (5) 16 (2016), 519–579.Google Scholar
Abramovich, D., Matsuki, K. and Rashid, S., A note on the factorization theorem of toric birational maps after Morelli and its toroidal extension , Tohoku Math. J. (2) 51 (1999), 489–537.Google Scholar
Abramovich, D. and Vistoli, A., Compactifying the space of stable maps , J. Amer. Math. Soc. 15 (2002), 27–75 (electronic).Google Scholar
Abramovich, D. and Wise, J., Birational invariance in logarithmic Gromov–Witten theory , Compos. Math. 154 (2018), 595–620.Google Scholar
Altmann, K., The versal deformation of an isolated toric Gorenstein singularity , Invent. Math. 128 (1997), 443–479.Google Scholar
Baranovsky, V. and Pecharich, J., On equivalences of derived and singular categories , Cent. Eur. J. Math. 8 (2010), 1–14.Google Scholar
Ben-Zvi, D., Francis, J. and Nadler, D., Integral transforms and Drinfeld centers in derived algebraic geometry , J. Amer. Math. Soc. 23 (2010), 909–966.Google Scholar
Ben-Zvi, D., Nadler, D. and Preygel, A., Integral transforms for coherent sheaves , J. Eur. Math. Soc. (JEMS) 19 (2017), 3763–3812.Google Scholar
Biswas, I., Parabolic bundles as orbifold bundles , Duke Math. J. 88 (1997), 305–326.Google Scholar
Borne, N., Fibrés paraboliques et champ des racines , Int. Math. Res. Not. IMRN 2007 (2007), Art. ID rnm049, 38.Google Scholar
Borne, N. and Vistoli, A., Parabolic sheaves on logarithmic schemes , Adv. Math. 231 (2012), 1327–1363, oct.Google Scholar
Bridgeland, T., King, A. and Reid, M., The McKay correspondence as an equivalence of derived categories , J. Amer. Math. Soc. 14 (2001), 535–554 (electronic).Google Scholar
Burban, I. and Kreussler, B., Fourier-Mukai transforms and semi-stable sheaves on nodal weierstrass cubics , J. Reine Angew. Math. 2005 (2005), 45–82.Google Scholar
Carchedi, D., Scherotzke, S., Sibilla, N. and Talpo, M., Kato–Nakayama spaces, infinite root stacks and the profinite homotopy type of log schemes , Geom. Topol. 21 (2017), 3093–3158.Google Scholar
Chen, Q., Stable logarithmic maps to Deligne–Faltings pairs I , Ann. of Math. (2) 180 (2014), 455–521.Google Scholar
Chiodo, A., Néron models of Pic0 via Pic0, Preprint (2015), arXiv:1509.06483.Google Scholar
Cohn, L., Differential graded categories are k-linear stable ∞-categories, Preprint (2013), arXiv:1308.2587.Google Scholar
Cox, D. A., Toric Varieties and Toric Resolutions (Birkhäuser Basel, Basel, 2000), 259–284.Google Scholar
Drinfeld, V., Dg quotients of dg categories , J. Algebra 272 (2004), 643–691.Google Scholar
Drinfeld, V. and Gaitsgory, D., On some finiteness questions for algebraic stacks , Geom. Funct. Anal. 23 (2013), 149–294.Google Scholar
Grothendieck, A. and Dieudonné, J. (eds), Éléments de géométrie algébrique: IV. Étude locale des schémas et des morphismes de schémas, Troisième partie, Publ. Math. Inst. Hautes Études Sci., 28 (1966).Google Scholar
Gaitsgory, D., Notes on geometric langlands: quasi-coherent sheaves on stacks (2011),http://abel.math.harvard.edu/∼gaitsgde/GL/QCohtext.pdf.Google Scholar
Gaitsgory, D., Ind-coherent sheaves (2012), www.math.harvard.edu/∼gaitsgde/GL/IndCohtext.pdf.Google Scholar
Geraschenko, A. and Satriano, M., Toric stacks I: the theory of stacky fans , Trans. Amer. Math. Soc. 367 (2015), 1033–1071.Google Scholar
Gieseker, D., A degeneration of the moduli space of stable bundles , J. Differential Geom. 19 (1984), 173–206.Google Scholar
Gross, M. and Siebert, B., Logarithmic gromov-witten invariants , J. Amer. Math. Soc. 26 (2013), 451–510.Google Scholar
Holmes, D., A Néron model of the universal jacobian, Preprint (2014), arXiv:1412.2243.Google Scholar
Illusie, L., Logarithmic spaces (according to K. Kato) , in Barsotti symposium in algebraic geometry, Perspectives in Mathematics, vol. 15 (Academic Press, San Diego, CA, 1994), 183–203.Google Scholar
Illusie, L., An overview of the work of K. Fujiwara, K. Kato, and C. Nakayama on logarithmic étale cohomology , in Cohomologies p-adiques et applications arithmétiques, II, Astérisque, 279 , (2002), 271–322.Google Scholar
Kapranov, M. and Vasserot, E., Kleinian singularities, derived categories and Hall algebras , Math. Ann. 316 (2000), 565–576.Google Scholar
Kato, K., Logarithmic degeneration and Dieudonné theory, Preprint.Google Scholar
Kato, K., Logarithmic structures of Fontaine–Illusie , in Algebraic analysis, geometry, and number theory, Baltimore, MD, 1988 (Johns Hopkins University Press, Baltimore, MD, 1989), 191–224.Google Scholar
Kato, K. and Nakayama, C., Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over C , Kodai Math. J. 22 (1999), 161–186.Google Scholar
Kawamata, Y., Log crepant birational maps and derived categories , J. Math. Sci. Univ. Tokyo 12 (2005), 211–231.Google Scholar
Keller, B., On differential graded categories , in Proceedings of the International Congress of Mathematicians, Madrid, August 22–30, 2006 (European Mathematical Society, Zürich, 2007), 151–190.Google Scholar
Kempf, G., Knudsen, F. F., Mumford, D. and Saint-Donat, B., Toroidal embeddings, Lecture Notes in Mathematics, vol. 1 (Springer, Berlin, Heidelberg, 1973).Google Scholar
Li, J., A degeneration formula of gw-invariants , J. Differential Geom. 60 (2002), 199–293.Google Scholar
Li, J. and Baosen, W., Good degeneration of quot-schemes and coherent systems , Comm. Anal. Geom. 23 (2015), 841–921.Google Scholar
Lurie, J., Higher Algebra (2016), http://www.math.harvard.edu/∼lurie/papers/HA.pdf.Google Scholar
Martens, J. and Thaddeus, M., Compactifications of reductive groups as moduli stacks of bundles , Compos. Math. 152 (2016), 62–98.Google Scholar
Niles, A., Moduli of elliptic curves via twisted stable maps , Algebra Number Theory 7 (2013), 2141–2202.Google Scholar
Nizioł, W., Toric singularities: log-blow-ups and global resolutions , J. Algebraic Geom. 15 (2006), 1–29.Google Scholar
Nizioł, W., K-theory of log-schemes. I , Doc. Math. 13 (2008), 505–551.Google Scholar
Nizioł, W., Semistable conjecture via K-theory , Duke Math. J. 141 (2008), 151–178.Google Scholar
Olsson, M., Log algebraic stacks and moduli of log schemes, PhD thesis, University of California, Berkeley (ProQuest LLC, Ann Arbor, MI, 2001).Google Scholar
Olsson, M., Logarithmic geometry and algebraic stacks , Ann. Sci. Éc. Norm. Supér. (4) 36 (2003), 747–791.Google Scholar
Olsson, M., Universal log structures on semi-stable varieties , Tohoku Math. J. (2) 55 (2003), 397–438.Google Scholar
Olsson, M., The logarithmic cotangent complex , Math. Ann. 333 (2005), 859–931.Google Scholar
Pahnke, V., Uniformisierung log-abelscher Varietäten, Schriftenreihe des Mathematischen Instituts der Universität Münster 3 Serie (Universität Ulm, 2005).Google Scholar
Pascaleff, J. and Sibilla, N., Topological Fukaya category and mirror symmetry for punctured surfaces, Preprint (2016), arXiv:1604.06448.Google Scholar
Rozenblyum, N., Filtered colimits of ∞-categories, Preprint (2012),http://www.math.harvard.edu/∼gaitsgde/GL/colimits.pdf.Google Scholar
Berthelot, P., Grothendieck, A. and Illusie, L. (eds), Théorie des intersections et théoreme de Riemann–Roch, Séminaire de Géométrie Algébrique (SGA 6), Bois Marie, 1966–67, Lecture Notes in Mathematics, vol. 225 (Springer, Berlin, Heidelberg, 1971).Google Scholar
Scherotzke, S., Sibilla, N. and Talpo, M., Additive invariants of logarithmic schemes, Preprint (2018), arXiv:1803.06398.Google Scholar
Sibilla, N., Mirror symmetry in dimension 1 and Fourier-Mukai transforms , in Homological mirror symmetry and tropical geometry, Lecture Notes of the Unione Matematica Italiana, vol. 15 (Springer, Cham, 2014), 407–428.Google Scholar
Sibilla, N., A note on mapping class group actions on derived categories , Proc. Amer. Math. Soc. 142 (2014), 1837–1848.Google Scholar
Solis, P, A complete degeneration of the moduli of G-bundles on a curve, Preprint (2013), arXiv:1311.6847.Google Scholar
The Stacks Project Authors. Stacks project (2018), http://stacks.math.columbia.edu.Google Scholar
Tabuada, G, Une structure de catégorie de modèles de Quillen sur la catégorie des dg-catégories , C. R. Math. Acad. Sci. Paris 340 (2005), 15–19.Google Scholar
Talpo, M., Moduli of parabolic sheaves on a polarized logarithmic scheme , Trans. Amer. Math. Soc. 369 (2017), 3483–3545.Google Scholar
Talpo, M., Parabolic sheaves with real weights as sheaves on the Kato–Nakayama space , Adv. Math. 336 (2018), 97–148.Google Scholar
Talpo, M. and Vistoli, A., A general formalism for logarithmic structures , Boll. Unione Mat. Ital (2017), doi:10.1007/s40574-017-0149-6.Google Scholar
Talpo, M. and Vistoli, A., The Kato–Nakayama space as a transcendental root stack , Int. Math. Res. Not. IMRN (2017), rnx079; doi:10.1093/imrn/rnx079.Google Scholar
Talpo, M. and Vistoli, A., Infinite root stacks and quasi-coherent sheaves on logarithmic schemes , Proc. Lond. Math. Soc. (3) 117 (2018), 1187–1243.Google Scholar
Thomason, R. W. and Trobaugh, T., Higher algebraic K-theory of schemes and of derived categories , in The Grothendieck Festschrift, Vol. III, Progress in Mathematics, vol. 88 (Birkhäuser, Boston, MA, 1990), 247–435.Google Scholar
Toën, B., The homotopy theory of dg-categories and derived morita theory , Invent. Math. 167 (2007), 615–667.Google Scholar
Toën, B., Lectures on dg-categories , in Topics in algebraic and topological K-theory, Lecture Notes in Mathematics, vol. 2008 (Springer, Berlin, Heidelberg, 2011), 243–302.Google Scholar
Włodarczyk, J., Decomposition of birational toric maps in blow-ups and blow-downs , Trans. Amer. Math. Soc. 349 (1997), 373–411.Google Scholar