Abstract
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.
Original language | English |
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Article number | 61 |
Number of pages | 51 |
Journal | Selecta Mathematica, New Series |
Volume | 27 |
Issue number | 4 |
Early online date | 28 Jun 2021 |
DOIs | |
Publication status | Published - Sept 2021 |
Keywords
- Log Calabi–Yau surfaces
- Log Gromov–Witten theory
- Moduli spaces of sheaves
ASJC Scopus subject areas
- Mathematics(all)
- Physics and Astronomy(all)