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Cyclic homology of categories of matrix factorizations

Alexander I. Efimov

TL;DR

This work establishes a precise link between noncommutative invariants of matrix factorization categories and classical vanishing-cycle cohomology. It develops a framework of curved DG categories and mixed complexes with $u$-connections to transfer information between periodic cyclic homology $HP_{ullet}(MF_{coh}(X,W))$ and vanishing cohomology, while identifying Hochschild homology with twisted de Rham hypercohomology $H^{\bullet}(X, (\Omega_X^{\bullet}, dW))$. The main achievement is proving, first in the affine case and then globally, that $HP_{ullet}(MF_{coh}(X,W))$ and $HH_{ullet}(MF_{coh}(X,W))$ correspond to twisted de Rham data and vanishing cohomology under a Riemann–Hilbert-type correspondence, including a monodromy twist by sign. These results yield a robust bridge between noncommutative algebro-geometric invariants and the theory of vanishing cycles, with implications for the categorical Chern character and conjectural Hodge-theoretic surjectivity onto vanishing-cycle Hodge classes. The work thus generalizes classical HKR-type identifications to the setting of matrix factorizations and curved DG categories, providing tools for studying degenerations, Thom–Sebastiani decompositions, and Hodge-theoretic conjectures in a derived categorical context.

Abstract

In this paper, we will show that for a smooth quasi-projective variety over $\C,$ and a regular function $W:X\to \C,$ the periodic cyclic homology of the DG category of matrix factorizations $MF(X,W)$ is identified (unde Riemann-Hilbert correspondence) with vanishing cohomology $H^{\bullet}(X^{an},φ_W\C_X),$ with monodromy twisted by sign. Also, Hochschild homology is identified respectively with the hypercohomology of $(Ω_X^{\bullet},dW\wedge).$ One can show that the image of the Chern character is contained in the subspace of Hodge classes. One can formulate the Hodge conjecture stating that it is surjective ($\otimes\Q$) onto Hodge classes. For W=0 and $X$ smooth projective this is precisely the classical Hodge conjecture.

Cyclic homology of categories of matrix factorizations

TL;DR

This work establishes a precise link between noncommutative invariants of matrix factorization categories and classical vanishing-cycle cohomology. It develops a framework of curved DG categories and mixed complexes with -connections to transfer information between periodic cyclic homology and vanishing cohomology, while identifying Hochschild homology with twisted de Rham hypercohomology . The main achievement is proving, first in the affine case and then globally, that and correspond to twisted de Rham data and vanishing cohomology under a Riemann–Hilbert-type correspondence, including a monodromy twist by sign. These results yield a robust bridge between noncommutative algebro-geometric invariants and the theory of vanishing cycles, with implications for the categorical Chern character and conjectural Hodge-theoretic surjectivity onto vanishing-cycle Hodge classes. The work thus generalizes classical HKR-type identifications to the setting of matrix factorizations and curved DG categories, providing tools for studying degenerations, Thom–Sebastiani decompositions, and Hodge-theoretic conjectures in a derived categorical context.

Abstract

In this paper, we will show that for a smooth quasi-projective variety over and a regular function the periodic cyclic homology of the DG category of matrix factorizations is identified (unde Riemann-Hilbert correspondence) with vanishing cohomology with monodromy twisted by sign. Also, Hochschild homology is identified respectively with the hypercohomology of One can show that the image of the Chern character is contained in the subspace of Hodge classes. One can formulate the Hodge conjecture stating that it is surjective () onto Hodge classes. For W=0 and smooth projective this is precisely the classical Hodge conjecture.

Paper Structure

This paper contains 13 sections, 27 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. (Curved) DG categories
  3. DG categories
  4. Matrix factorizations
  5. Curved DG categories
  6. Mixed complexes with a connection
  7. Mixed complexes.
  8. $u$-connections on mixed complexes.
  9. Examples of $u$-connections.
  10. Sheaves of mixed complexes with $u$-connections.
  11. Affine case
  12. General case
  13. Concluding remarks

Key Result

Theorem 1.1

Let $X$ be a smooth quasi-projective algebraic variety over $\mathbb{C},$ and $W:X\to\mathbb{C}$ a regular function. Then we have an identification of $\mathbb{Z}/2$-graded vector bundles with connection on the formal punctured disk: where $T$ is the monodromy automorphism.

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • ...and 52 more