TY - JOUR
T1 - Sheaves of maximal intersection and multiplicities of stable log maps
AU - Choi, Jinwon
AU - van Garrel, Michel
AU - Katz, Sheldon
AU - Takahashi, Nobuyoshi
N1 - Funding Information:
We wish to thank Mark Gross and Dhruv Ranganathan for a discussion on multiplicities of stable log maps that motivated parts of this work. We are grateful to Alexei Oblomkov for discussions on multiplicities of stable maps to surfaces. We thank Helge Ruddat and Travis Mandel for a discussion clarifying the relationship between stable map multiplicity and tropical multiplicity. We especially thank Helge Ruddat for suggesting to us the example of Sects. and . JC is supported by the Korea NRF Grant NRF-2018R1C1B6005600. JC would like to thank Korea Institute for Advanced Study for the support where some of the work for this paper was completed. MvG is supported by the EC REA MSCA-IF-746554. SK is supported in part by NSF Grant DMS-1502170 and NSF Grant DMS-1802242. NT is supported by JSPS KAKENHI Grant Number JP17K05204. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No 746554.
Funding Information:
We wish to thank Mark Gross and Dhruv Ranganathan for a discussion on multiplicities of stable log maps that motivated parts of this work. We are grateful to Alexei Oblomkov for discussions on multiplicities of stable maps to surfaces. We thank Helge Ruddat and Travis Mandel for a discussion clarifying the relationship between stable map multiplicity and tropical multiplicity. We especially thank Helge Ruddat for suggesting to us the example of Sects. 2.2.4 and 2.3.1. JC is supported by the Korea NRF Grant NRF-2018R1C1B6005600. JC would like to thank Korea Institute for Advanced Study for the support where some of the work for this paper was completed. MvG is supported by the EC REA MSCA-IF-746554. SK is supported in part by NSF Grant DMS-1502170 and NSF Grant DMS-1802242. NT is supported by JSPS KAKENHI Grant Number JP17K05204. This project has received funding from the European Union?s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie Grant Agreement No 746554.
Publisher Copyright:
© 2021, The Author(s).
PY - 2021/9
Y1 - 2021/9
N2 - A great number of theoretical results are known about log Gromov–Witten invariants (Abramovich and Chen in Asian J Math 18:465–488, 2014; Chen in Ann Math (2) 180:455–521, 2014; Gross and Siebert J Am Math Soc 26: 451–510, 2013), but few calculations are worked out. In this paper we restrict to surfaces and to genus 0 stable log maps of maximal tangency. We ask how various natural components of the moduli space contribute to the log Gromov–Witten invariants. The first such calculation (Gross et al. in Duke Math J 153:297–362, 2010, Proposition 6.1) by Gross–Pandharipande–Siebert deals with multiple covers over rigid curves in the log Calabi–Yau setting. As a natural continuation, in this paper we compute the contributions of non-rigid irreducible curves in the log Calabi–Yau setting and that of the union of two rigid curves in general position. For the former, we construct and study a moduli space of “logarithmic” 1-dimensional sheaves and compare the resulting multiplicity with tropical multiplicity. For the latter, we explicitly describe the components of the moduli space and work out the logarithmic deformation theory in full, which we then compare with the deformation theory of the analogous relative stable maps.
AB - A great number of theoretical results are known about log Gromov–Witten invariants (Abramovich and Chen in Asian J Math 18:465–488, 2014; Chen in Ann Math (2) 180:455–521, 2014; Gross and Siebert J Am Math Soc 26: 451–510, 2013), but few calculations are worked out. In this paper we restrict to surfaces and to genus 0 stable log maps of maximal tangency. We ask how various natural components of the moduli space contribute to the log Gromov–Witten invariants. The first such calculation (Gross et al. in Duke Math J 153:297–362, 2010, Proposition 6.1) by Gross–Pandharipande–Siebert deals with multiple covers over rigid curves in the log Calabi–Yau setting. As a natural continuation, in this paper we compute the contributions of non-rigid irreducible curves in the log Calabi–Yau setting and that of the union of two rigid curves in general position. For the former, we construct and study a moduli space of “logarithmic” 1-dimensional sheaves and compare the resulting multiplicity with tropical multiplicity. For the latter, we explicitly describe the components of the moduli space and work out the logarithmic deformation theory in full, which we then compare with the deformation theory of the analogous relative stable maps.
KW - Log Calabi–Yau surfaces
KW - Log Gromov–Witten theory
KW - Moduli spaces of sheaves
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U2 - 10.1007/s00029-021-00671-0
DO - 10.1007/s00029-021-00671-0
M3 - Article
AN - SCOPUS:85108978214
SN - 1022-1824
VL - 27
JO - Selecta Mathematica, New Series
JF - Selecta Mathematica, New Series
IS - 4
M1 - 61
ER -