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Logarithmic Quasimaps

Qaasim Shafi

TL;DR

This work develops a comprehensive theory of logarithmic quasimaps for pairs $(X,D)$ with simple normal crossings divisors, constructing a proper Deligne–Mumford moduli stack $\mathcal{Q}_{g,\alpha}^{\mathrm{log}}(X|D,\beta)$ and a virtual fundamental class of the expected dimension. The construction proceeds by encoding tangency via maps to $[\mathbb{A}^r/\mathbb{G}_m^r]$ and forming a fiber product with the moduli of absolute quasimaps, enabling a canonical obstruction theory and computable invariants. In genus zero, for toric $X$ with $D$ a smooth ample divisor, the theory recovers Battistella–Nabijou’s relative quasimap theory, and the authors prove a compatibility via a virtual-pullback framework using embeddings into projective toric models. The results position logarithmic quasimaps as a natural, modular approach to relative and logarithmic curve counting, with potential implications for wall-crossing, mirror symmetry, and logarithmic Gromov–Witten theory beyond smooth divisors, especially in the SNC setting.

Abstract

We construct a proper moduli space which is a Deligne-Mumford stack parametrising quasimaps relative to a simple normal crossings divisor in any genus using logarithmic geometry. We show this moduli space admits a virtual fundamental class of the expected dimension leading to numerical invariants which agree with the theory of Battistella-Nabijou where the latter is defined.

Logarithmic Quasimaps

TL;DR

This work develops a comprehensive theory of logarithmic quasimaps for pairs with simple normal crossings divisors, constructing a proper Deligne–Mumford moduli stack and a virtual fundamental class of the expected dimension. The construction proceeds by encoding tangency via maps to and forming a fiber product with the moduli of absolute quasimaps, enabling a canonical obstruction theory and computable invariants. In genus zero, for toric with a smooth ample divisor, the theory recovers Battistella–Nabijou’s relative quasimap theory, and the authors prove a compatibility via a virtual-pullback framework using embeddings into projective toric models. The results position logarithmic quasimaps as a natural, modular approach to relative and logarithmic curve counting, with potential implications for wall-crossing, mirror symmetry, and logarithmic Gromov–Witten theory beyond smooth divisors, especially in the SNC setting.

Abstract

We construct a proper moduli space which is a Deligne-Mumford stack parametrising quasimaps relative to a simple normal crossings divisor in any genus using logarithmic geometry. We show this moduli space admits a virtual fundamental class of the expected dimension leading to numerical invariants which agree with the theory of Battistella-Nabijou where the latter is defined.
Paper Structure (14 sections, 23 theorems, 46 equations)

This paper contains 14 sections, 23 theorems, 46 equations.

Table of Contents

  1. Results
  2. Why Quasimaps?
  3. Outline
  4. History
  5. Absolute quasimaps with divisors
  6. Moduli space of logarithmic quasimaps
  7. Moduli space of logarithmic maps to $[\mathbb{A}^r/\mathbb{G}_m^r]$
  8. Moduli space of logarithmic quasimaps
  9. Examples
  10. Logarithmic quasimap invariants
  11. Obstruction theory
  12. Perfectness of the obstruction theory
  13. Toric Divisors
  14. Comparison with Battistella--Nabijou theory

Key Result

Theorem 1

Fix non-negative integers $g,n$, an effective degree $\beta$ and $\alpha = (\alpha_{i,j})_{i,j}$ a matrix of non-negative integers with $\sum_{j=1}^n \alpha_{i,j} = D_{i} \cdot \beta$. The moduli space $\mathcal{Q}_{g, \alpha}^{\mathrm{log}}(X|D,\beta)$ parametrising logarithmic quasimaps to $(X,D)$

Theorems & Definitions (64)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Example 1.5
  • Definition 2.1
  • Theorem 2.2: abramovich2018birational
  • ...and 54 more