A category of kernels for equivariant factorizations and its implications for Hodge theory

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.

Updates references and addresss. comments still very welcome!

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Cited by in corpus (9)

• Adjunctions and defects in Landau-Ginzburg models
• Orbifold equivalent potentials
• Homological Projective Duality via Variation of Geometric Invariant Theory Quotients
• Matrix Factorizations for Complete Intersections and Minimal Free Resolutions
• Extending Landau-Ginzburg models to the point
• Calabi-Yau completions and orbifold equivalences
• A category of kernels for equivariant factorizations, II: further implications
• Lefschetz type formulas for dg-categories
• Double solids, categories and non-rationality