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Tyurin Degenerations, Derived Lagrangians and Categorification of DT Invariants

Jacob Kryczka, Artan Sheshmani

TL;DR

This work develops a derived-geometric degeneration framework for Calabi–Yau threefolds undergoing Tyurin degenerations $X o X_1igcup_S X_2$, where $X_i$ meet along an anti-canonical divisor $S$. By proving that the total moduli space carries a relative Lagrangian foliation, the authors obtain a flat Gauss–Manin connection on the fiberwise periodic cyclic homology of matrix-factorization categories and show the Fano pieces induce derived Lagrangians inside the restriction to $S$, enabling a derived intersection interpretation of categorified DT invariants. The paper establishes a Calabi–Yau structure on the vertical (fiber-supported) category, constructs a global shifted $(-1)$-shifted potential, and demonstrates deformation invariance of the categorified DT invariants across the degeneration; the special fiber DT information is captured by a derived Lagrangian intersection in $M(S)$, computed via deformation-quantization and spectral sequences. The framework thus links shifted symplectic geometry, derived intersection theory, and matrix-factorization categories to yield a deformation-invariant, computable categorification of DT invariants for Tyurin-degenerate Calabi–Yau threefolds, with explicit local models for Ext-algebras and potential functions. The results provide both a conceptual and computational route to understanding how DT-type invariants behave under Tyurin degenerations and pave the way for explicit calculations via derived Lagrangian intersections and HP of MF categories.

Abstract

We consider the moduli space of rigidified perfect complexes with support on a general complete intersection Calabi-Yau threefold $X$ and its Tyurin degeneration $X\rightsquigarrow X_1\cup_SX_2$ to a complete intersection of Fano threefolds $X_1,X_2$ meeting along their anti-canonical divisor $S$. The corresponding derived dg moduli scheme over the generic fiber degenerates to the (Fano) moduli spaces $\mathcal{M}_{1}, \mathcal{M}_{2},$ of perfect complexes supported on each Fano which glue after derived restriction to the relative divisor $S$. We prove that the total moduli space of the degeneration family carries a relative Lagrangian foliation structure, which implies the existence of a flat Gauss-Manin connection on periodic cyclic homology of the category of the matrix factorizations associated with fiber-wise moduli spaces, realized locally as the derived critical loci of suitable potential functions. The Fano moduli spaces each define derived Lagrangians in the (ambient) moduli space of restricted complexes to the relative divisor $S$. The flatness of the Gauss-Manin connection implies the derived geometric deformation invariance of the categorified DT-invariants associated to fiberwise matrix factorization categories, hence, the categorified DT-invariants of the generic fiber are expressed in terms of a derived intersection cohomology of the corresponding Fano moduli spaces on the special fiber.

Tyurin Degenerations, Derived Lagrangians and Categorification of DT Invariants

TL;DR

This work develops a derived-geometric degeneration framework for Calabi–Yau threefolds undergoing Tyurin degenerations , where meet along an anti-canonical divisor . By proving that the total moduli space carries a relative Lagrangian foliation, the authors obtain a flat Gauss–Manin connection on the fiberwise periodic cyclic homology of matrix-factorization categories and show the Fano pieces induce derived Lagrangians inside the restriction to , enabling a derived intersection interpretation of categorified DT invariants. The paper establishes a Calabi–Yau structure on the vertical (fiber-supported) category, constructs a global shifted -shifted potential, and demonstrates deformation invariance of the categorified DT invariants across the degeneration; the special fiber DT information is captured by a derived Lagrangian intersection in , computed via deformation-quantization and spectral sequences. The framework thus links shifted symplectic geometry, derived intersection theory, and matrix-factorization categories to yield a deformation-invariant, computable categorification of DT invariants for Tyurin-degenerate Calabi–Yau threefolds, with explicit local models for Ext-algebras and potential functions. The results provide both a conceptual and computational route to understanding how DT-type invariants behave under Tyurin degenerations and pave the way for explicit calculations via derived Lagrangian intersections and HP of MF categories.

Abstract

We consider the moduli space of rigidified perfect complexes with support on a general complete intersection Calabi-Yau threefold and its Tyurin degeneration to a complete intersection of Fano threefolds meeting along their anti-canonical divisor . The corresponding derived dg moduli scheme over the generic fiber degenerates to the (Fano) moduli spaces of perfect complexes supported on each Fano which glue after derived restriction to the relative divisor . We prove that the total moduli space of the degeneration family carries a relative Lagrangian foliation structure, which implies the existence of a flat Gauss-Manin connection on periodic cyclic homology of the category of the matrix factorizations associated with fiber-wise moduli spaces, realized locally as the derived critical loci of suitable potential functions. The Fano moduli spaces each define derived Lagrangians in the (ambient) moduli space of restricted complexes to the relative divisor . The flatness of the Gauss-Manin connection implies the derived geometric deformation invariance of the categorified DT-invariants associated to fiberwise matrix factorization categories, hence, the categorified DT-invariants of the generic fiber are expressed in terms of a derived intersection cohomology of the corresponding Fano moduli spaces on the special fiber.
Paper Structure (37 sections, 33 theorems, 242 equations)

This paper contains 37 sections, 33 theorems, 242 equations.

Table of Contents

  1. Introduction
  2. Li-Wu machinery of degenerations and obstructions for its generalizations
  3. Properness of the moduli spaces
  4. Structure of the paper and main results
  5. Shifted symplectic structures over degenerating families
  6. Calabi-Yau hypersurfaces and good degenerations
  7. $\mathcal{O}$-compactness over good degenerations
  8. Dg schemes of rigidified simple perfect complexes
  9. Basic example
  10. The derived quasi Fano Lagrangian moduli spaces
  11. Shifted symplectic geometry
  12. Calabi-Yau category structure on $\mathsf{Perf}^{vert}(\mathbb{P})$
  13. Calabi-Yau categories and shifted Lagrangian morphisms
  14. Relative Calabi-Yau structure on a generic fiber
  15. Calabi-Yau structure on the special fiber
  16. ...and 22 more sections

Key Result

Lemma 1

Consider $\mathcal{M}(\mathbb{P})$ as above. Thus, when $\mathcal{F} \simeq \mathcal{O}_Z$ we have $Ext^0_\mathbb{P}(\mathcal{F}, \mathcal{F}) = Ext^3_P (\mathcal{F}, \mathcal{F}) = \mathbb{C}.$ Furthermore, setting $Q_h:= V^\vee/\mathbb{C} h$, we have The first terms of the direct sums are also isomorphic to $Ext^{1, resp.\ 2}_X(\mathcal{F}, \mathcal{F})$, while the second terms give the fiber

Theorems & Definitions (93)

  • Definition 1
  • Definition 2
  • Remark 1: Obstructions to generalizations
  • Remark 2
  • Lemma 1
  • proof
  • Remark 3
  • Definition 3
  • Definition 4
  • Lemma 2
  • ...and 83 more