Punctured logarithmic maps

Abstract

We introduce a variant of stable logarithmic maps, which we call punctured logarithmic maps. They allow an extension of logarithmic Gromov-Witten theory in which marked points have a negative order of tangency with boundary divisors. As a main application we develop a gluing formalism which reconstructs stable logarithmic maps and their virtual cycles without expansions of the target, with tropical geometry providing the underlying combinatorics. Punctured Gromov-Witten invariants also play a pivotal role in the intrinsic construction of mirror partners by the last two authors in arXiv:1909.07649, conjecturally relating to symplectic cohomology, and in the logarithmic gauged linear sigma model in upcoming work of the second author with Felix Janda and Yongbin Ruan.

125 pages; v2: minor changes throughout the text, added more figures and an appendix on charts for morphisms of log stacks; v3: added appendix on functorial tropicalization, changed treatment of targets with monodromy, minor corrections and more details provided at many places

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Cited by in corpus (9)

• The canonical wall structure and intrinsic mirror symmetry
• Gromov-Witten theory with maximal contacts
• Intrinsic Mirror Symmetry
• Virtual cycles of stable (quasi)-maps with fields
• The logarithmic gauged linear sigma model
• Non-archimedean quantum K-invariants
• Splitting of Gromov-Witten Invariants with Toric Gluing Strata
• Mirror symmetry for the Tate curve via tropical and log corals
• Towards the Doran-Harder-Thompson conjecture via the Gross-Siebert program