The Hodge structure on the singularity category of a complex hypersurface
Copyright policy )The Hodge structure on the singularity category of a complex hypersurface
Given a complex affine hypersurface with isolated singularity determined by a homogeneous polynomial, we identify the noncommutative Hodge structure on the periodic cyclic homology of its singularity category with the classical Hodge structure on the primitive cohomology of the associated projective hypersurface. As a consequence, we show that the Hodge conjecture for the projective hypersurface is equivalent to a dg-categorical analogue of the Hodge conjecture for the singularity category.
31 pages
- Auburn University at Montgomery United States
- Auburn University United States
- University of Nebraska-Lincoln United States
- Mathematical Sciences Research Institute United States
- University of Nebraska System United States
Commutative Algebra, 13D03, 13D09, 14C30, 14F08, 14J70, 19D55, noncommutative Hodge theory, Derived categories and commutative rings, K-Theory and Homology (math.KT), singularity category, matrix factorization, Commutative Algebra (math.AC), hypersurface, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects), FOS: Mathematics, Hypersurfaces and algebraic geometry, \(K\)-theory and homology; cyclic homology and cohomology, Hodge conjecture, K-Theory and Homology, Algebraic Geometry, Algebraic Geometry (math.AG)
Commutative Algebra, 13D03, 13D09, 14C30, 14F08, 14J70, 19D55, noncommutative Hodge theory, Derived categories and commutative rings, K-Theory and Homology (math.KT), singularity category, matrix factorization, Commutative Algebra (math.AC), hypersurface, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects), FOS: Mathematics, Hypersurfaces and algebraic geometry, \(K\)-theory and homology; cyclic homology and cohomology, Hodge conjecture, K-Theory and Homology, Algebraic Geometry, Algebraic Geometry (math.AG)
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