2011/05/16 by Matthew R. Ballard, Ballard, Matthew, David Favero +3 · 4 citations
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1105.3177
openalex publication_date 2011/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We provide a factorization model for the continuous internal Hom, in the\nhomotopy category of k-linear dg-categories, between dg-categories of\nequivariant factorizations. This motivates a notion, similar to that of\nKuznetsov, which we call the extended Hochschild cohomology algebra of the\ncategory of equivariant factorizations. In some cases of geometric interest,\nextended Hochschild cohomology contains Hochschild cohomology as a subalgebra\nand Hochschild homology as a homogeneous component. We use our factorization\nmodel for the internal Hom to calculate the extended Hochschild cohomology for\nequivariant factorizations on affine space.\n Combining the computation of extended Hochschild cohomology with the\nHochschild-Kostant-Rosenberg isomorphism and a theorem of Orlov recovers and\nextends Griffiths' classical description of the primitive cohomology of a\nsmooth, complex projective hypersurface in terms of homogeneous pieces of the\nJacobian algebra. In the process, the primitive cohomology is identified with\nthe fixed subspace of the cohomological endomorphism associated to an\ninteresting endofunctor of the bounded derived category of coherent sheaves on\nthe hypersurface. We also demonstrate how to understand the whole Jacobian\nalgebra as morphisms between kernels of endofunctors of the derived category.\n Finally, we present a bootstrap method for producing algebraic cycles in\ncategories of equivariant factorizations. As proof of concept, we show how this\nreproves the Hodge conjecture for all self-products of a particular K3 surface\nclosely related to the Fermat cubic fourfold.\n