Cyclic homology of categories of matrix factorizations

TL;DR

This paper establishes a deep connection between the cyclic homology of matrix factorization categories and vanishing cohomology, linking algebraic and topological invariants in complex geometry.

Contribution

It identifies the periodic cyclic homology of matrix factorizations with vanishing cohomology and relates Hochschild homology to hypercohomology, extending the Hodge conjecture framework.

Findings

Periodic cyclic homology matches vanishing cohomology with monodromy.

Hochschild homology corresponds to hypercohomology of differential forms.

The Chern character image lies within Hodge classes, relating to the Hodge conjecture.

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 $H^{\bullet}(X^{an},\phi_W\C_X),$ with monodromy twisted by sign. Also, Hochschild homology is identified respectively with the hypercohomology of $(\Omega_X^{\bullet},dW\wedge).$ 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.