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Intrinsic Mirror Symmetry

Mark Gross, Bernd Siebert

TL;DR

This work constructs an intrinsic mirror ring for a log Calabi–Yau pair (X,D) or a maximally unipotent degeneration X→B by counting punctured Gromov–Witten invariants N^A_{pqr} via tropicalization data. The central object R is built from the tropical lattice on the dual intersection complex with multiplication governed by these invariants, yielding Spec R (or Proj R) as a mirror to X∖D in the log Calabi–Yau case (and to the degeneration in the relative case). A key achievement is proving associativity of the resulting product under two broad nef/CY-type conditions, using a logarithmic WDVV-style argument that harnesses tropical geometry, transversality, and a refined gluing theory for punctured maps. The paper also develops the invariant construction and shows that the resulting mirror data is robust under compactification changes, connecting to symplectic cohomology and providing a flexible framework for relative quantum-like structures. Overall, it advances a general algebro-geometric realization of mirror symmetry via punctured invariants, tropical methods, and log-geometry, with broad potential for computation and applications in higher dimensions.

Abstract

We associate a ring R to a log Calabi-Yau pair (X,D) or a degeneration of Calabi-Yau manifolds X->B. The vector space underlying R is determined by the tropicalization of (X,D) or X->B, while the product rule is defined using punctured Gromov-Witten invariants, defined in joint work with Abramovich and Chen. In the log Calabi-Yau case, if D is maximally degenerate, then we propose that Spec R is the mirror to X\D, while in the Calabi-Yau degeneration case, if the degeneration is maximally unipotent, the mirror is expected to be Proj R. The main result in this paper is that R as defined is an associative, commutative ring with unit, with associativity the most difficult part.

Intrinsic Mirror Symmetry

TL;DR

This work constructs an intrinsic mirror ring for a log Calabi–Yau pair (X,D) or a maximally unipotent degeneration X→B by counting punctured Gromov–Witten invariants N^A_{pqr} via tropicalization data. The central object R is built from the tropical lattice on the dual intersection complex with multiplication governed by these invariants, yielding Spec R (or Proj R) as a mirror to X∖D in the log Calabi–Yau case (and to the degeneration in the relative case). A key achievement is proving associativity of the resulting product under two broad nef/CY-type conditions, using a logarithmic WDVV-style argument that harnesses tropical geometry, transversality, and a refined gluing theory for punctured maps. The paper also develops the invariant construction and shows that the resulting mirror data is robust under compactification changes, connecting to symplectic cohomology and providing a flexible framework for relative quantum-like structures. Overall, it advances a general algebro-geometric realization of mirror symmetry via punctured invariants, tropical methods, and log-geometry, with broad potential for computation and applications in higher dimensions.

Abstract

We associate a ring R to a log Calabi-Yau pair (X,D) or a degeneration of Calabi-Yau manifolds X->B. The vector space underlying R is determined by the tropicalization of (X,D) or X->B, while the product rule is defined using punctured Gromov-Witten invariants, defined in joint work with Abramovich and Chen. In the log Calabi-Yau case, if D is maximally degenerate, then we propose that Spec R is the mirror to X\D, while in the Calabi-Yau degeneration case, if the degeneration is maximally unipotent, the mirror is expected to be Proj R. The main result in this paper is that R as defined is an associative, commutative ring with unit, with associativity the most difficult part.

Paper Structure

This paper contains 64 sections, 72 theorems, 328 equations, 16 figures.

Table of Contents

  1. The construction and its variants
  2. Algebras associated to pairs
  3. The log Calabi-Yau case
  4. Mirrors, examples, basic properties, and auxilliary constructions
  5. Preliminaries and review
  6. Generalities on log geometry
  7. Tropicalization
  8. Artin fans
  9. Idealized log structures
  10. Fine saturated fibre products
  11. Transversality and flatness
  12. Stable punctured maps
  13. Punctures and punctured maps
  14. Contact orders for punctured points
  15. Moduli of punctured maps
  16. ...and 49 more sections

Key Result

Theorem 1.9

Let $X$ be a Zariski log scheme with a projective log smooth morphism $g:X\rightarrow S$ with $S\cong\operatorname{Spec}\Bbbk$ or $S$ a non-complete curve carrying a divisorial log structure coming from a single point $s_0\in S$. If $c_1(\Theta_{X/\Bbbk})$ is nef or anti-nef, the structure constants

Figures (16)

  • Figure 1.1: Cones and vectors in the second case of Example \ref{['runningexample3']}. The shaded areas are the two cones, and $B$ identifies the rays $\rho_1$, $\rho'_1$ and the tangent vectors $v'_1$ and $v_1$.
  • Figure 1.2: The geometry of Example \ref{['ex:relative']}.
  • Figure 1.3: The polyhedral affine manifold for the mirror geometry of Invitation.
  • Figure 3.1: A tropical map without and with point constraints. We note in all depictions of tropical punctured maps, we only draw the legs corresponding to punctured points as arrows. By pre-stability, in fact these legs should extend as far as possible in the cone of $\Sigma(X)$ containing the leg, but this will make the figures harder to parse.
  • Figure 4.1: Punctured map at a general point of $\mathscr{M}_1$
  • ...and 11 more figures

Theorems & Definitions (216)

  • Example 1.2
  • Example 1.4
  • Example 1.7
  • Theorem 1.9
  • Remark 1.10
  • Definition 1.11
  • Remark 1.12
  • Remark 1.13
  • Remark 1.14
  • Theorem 1.15
  • ...and 206 more