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Almost toric fibrations on K3 surfaces

Pranav Chakravarthy, Yoel Groman

TL;DR

The paper analyzes maximal degenerations of K3 surfaces, establishing that the generic smooth fiber of a type III Kulikov model carries an almost toric fibration over the intersection complex, endowed with a nodal integral affine structure that matches the boundary affine data from Gross–Siebert. A two-step strategy combines a soft preparatory modification with a hard extension across the 1-skeleton, yielding a continuous ATF that extends smoothly through singular loci. In the toric Fano setting, the authors construct symplectic Kulikov models via small resolutions of toric degenerations, and show the resulting integral affine base agrees (up to nodal slides) with Gross–Siebert’s structure on the moment polytope boundary, while providing an explicit ATF. These results connect SYZ-type fibrations for K3 degenerations with Gross–Siebert mirrors, offering tools for understanding Fukaya categories and mirror symmetry in this setting. The work also outlines prospects for higher-dimensional generalizations and further links to non-Archimedean and degenerative mirror constructions.

Abstract

For Kähler K3 surfaces we consider Kulikov models of type III tamed by a symplectic form. Our main result shows that the generic smooth fiber admits an almost toric fibration over the intersection complex, which inherits a natural nodal integral affine structure from almost toric fibrations of the boundary divisors. We prove that a smooth anti-canonical hypersurface in a smooth toric Fano threefold, equipped with a toric Kähler form, admits a symplectic Kulikov model. Moreover, we demonstrate that the induced integral affine structure on the intersection complex is integral affine isomorphic (up to nodal slides) nodal integral affine structure considered by Gross and Siebert on the boundary of the moment polytope.

Almost toric fibrations on K3 surfaces

TL;DR

The paper analyzes maximal degenerations of K3 surfaces, establishing that the generic smooth fiber of a type III Kulikov model carries an almost toric fibration over the intersection complex, endowed with a nodal integral affine structure that matches the boundary affine data from Gross–Siebert. A two-step strategy combines a soft preparatory modification with a hard extension across the 1-skeleton, yielding a continuous ATF that extends smoothly through singular loci. In the toric Fano setting, the authors construct symplectic Kulikov models via small resolutions of toric degenerations, and show the resulting integral affine base agrees (up to nodal slides) with Gross–Siebert’s structure on the moment polytope boundary, while providing an explicit ATF. These results connect SYZ-type fibrations for K3 degenerations with Gross–Siebert mirrors, offering tools for understanding Fukaya categories and mirror symmetry in this setting. The work also outlines prospects for higher-dimensional generalizations and further links to non-Archimedean and degenerative mirror constructions.

Abstract

For Kähler K3 surfaces we consider Kulikov models of type III tamed by a symplectic form. Our main result shows that the generic smooth fiber admits an almost toric fibration over the intersection complex, which inherits a natural nodal integral affine structure from almost toric fibrations of the boundary divisors. We prove that a smooth anti-canonical hypersurface in a smooth toric Fano threefold, equipped with a toric Kähler form, admits a symplectic Kulikov model. Moreover, we demonstrate that the induced integral affine structure on the intersection complex is integral affine isomorphic (up to nodal slides) nodal integral affine structure considered by Gross and Siebert on the boundary of the moment polytope.

Paper Structure

This paper contains 26 sections, 39 theorems, 26 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Statement of the main result
  3. Hypesurfaces in smooth toric Fano
  4. The general context
  5. Outline of the proof of Theorem \ref{['Thm:LTFongoodKulikov']}
  6. Prospects for higher dimensions
  7. The preparatory step
  8. Piecewise nodal integral affine structure on the intersection complex
  9. A fibration of the central fiber over the intersection complex
  10. The nodal integral affine structure on the intersection complex
  11. Continuity of parallel transport
  12. Smoothness of parallel transport
  13. Conclusion of the preparatory step
  14. The hard step
  15. Proof of Theorem \ref{['Thm:LTFongoodKulikov']}
  16. ...and 11 more sections

Key Result

Lemma 1.5

Given a symplectic Kulikov model $\mathcal{X}$ of type III there is a nodal integral affine structure on the intersection complex $\Delta_{\mathcal{X}}$ which is smooth in a neighborhood of the $1$-skeleton and whose restriction to the 2-cell in $\Delta_{\mathcal{X}}$ associated with the component $

Figures (6)

  • Figure 1: Monodromy around $\gamma$
  • Figure 2: Local model of the resolution at a double point $p$ defined by $z_iz_j=\lambda s_1$.
  • Figure 3: Almost toric blow-up
  • Figure 4: The inclusion $Q^{0}\setminus\partial_1Q^{(0)}\to Q^{(1)}\setminus \partial_1Q^{(1)}$
  • Figure 5: Hybrid degeneration after performing small blow up on the face $F_1$
  • ...and 1 more figures

Theorems & Definitions (89)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 1.5
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Remark 1.10
  • ...and 79 more