A category of kernels for equivariant factorizations and its implications for Hodge theory
Matthew Ballard, David Favero, Ludmil Katzarkov
TL;DR
The paper develops a factorization model for the internal Hom in the dg-category of equivariant factorizations, yielding an explicit equivalence $\mathbf{R}\!\mathbf{Hom}_c(\mathsf{Inj}(X,G,w),\mathsf{Inj}(Y,H,v)) \simeq \mathsf{Inj}(\operatorname{U}(\mathcal{L}) \times \operatorname{U}(\mathcal{L}'),G \times H \times \mathbb{G}_m,(-f_w)\boxplus f_v)$ that grounds computations of extended Hochschild cohomology and connects to Griffiths’ description of primitive cohomology via Orlov’s singularity framework and the HKR isomorphism. It further develops a bootstrap method for constructing algebraic cycles in equivariant factorizations and proves a Hodge-theoretic consequence for self-products of a K3 surface related to a Fermat cubic fourfold. The work integrates generation results for equivariant derived categories, Morita theory for factorization categories, and integral-transform style functors, providing a robust toolkit for linking noncommutative Hodge theory with classical algebraic geometry. These developments yield computational power for extended Hochschild cohomology, categorical realizations of Jacobian algebras, and categorical routes to Hodge-theoretic conjectures in an equivariant setting.
Abstract
We provide a factorization model for the continuous internal Hom, in the homotopy category of $k$-linear dg-categories, between dg-categories of equivariant factorizations. This motivates a notion, similar to that of Kuznetsov, which we call the extended Hochschild cohomology algebra of the category of equivariant factorizations. In some cases of geometric interest, extended Hochschild cohomology contains Hochschild cohomology as a subalgebra and Hochschild homology as a homogeneous component. We use our factorization model for the internal Hom to calculate the extended Hochschild cohomology for equivariant factorizations on affine space. Combining the computation of extended Hochschild cohomology with the Hochschild-Kostant-Rosenberg isomorphism and a theorem of Orlov recovers and extends Griffiths' classical description of the primitive cohomology of a smooth, complex projective hypersurface in terms of homogeneous pieces of the Jacobian algebra. In the process, the primitive cohomology is identified with the fixed subspace of the cohomological endomorphism associated to an interesting endofunctor of the bounded derived category of coherent sheaves on the hypersurface. We also demonstrate how to understand the whole Jacobian algebra as morphisms between kernels of endofunctors of the derived category. Finally, we present a bootstrap method for producing algebraic cycles in categories of equivariant factorizations. As proof of concept, we show how this reproves the Hodge conjecture for all self-products of a particular K3 surface closely related to the Fermat cubic fourfold.