Categorical Logarithmic Hodge Theory, I

Abstract

We write down a new "logarithmic" quasicoherent category ${\mathrm{Qcoh}}_{log}(U,X,D)$ attached to a smooth open algebraic variety $U$ with toroidal compactification $X$ and boundary divisor $D$. This is a (large) symmetric monoidal Abelian category, which we argue can be thought of as the categorical substrate for logarithmic Hodge theory of $U$. We show that its Hochschild homology theory coincides with the theory of log-forms on $X$ with logarithmic structure induced by $D$, and in particular, that the noncommutative Hodge-to de Rham sequence on ${\mathrm{Qcoh}}_{log}(U,X,D)$ recovers known log Hodge structure on the de Rham cohomology of the open variety $U$. As an application, we compute the Hochschild homology of the category of coherent sheaves on the infinite root stack of Talpo and Vistoli in the toroidal setting. We prove a derived invariance result for this theory: namely, that strictly toroidal changes of compactification do not change the derived category of ${\mathrm{Qcoh}}_{log}(U,X,D)$. The definition is motivated by the coherent object appearing in the author's microlocal mirror symmetry result [20]. In this paper, the first in a series, we work over an algebraically closed field of characteristic zero. The next installment will develop the characteristic p and mixed-characteristic theories.

arXiv comment20 pages

References in corpus (7)

• Localization theorems in topological Hochschild homology and topological cyclic homology
• Parabolic sheaves on logarithmic schemes
• Integral p-adic Hodge theory
• Wrapped microlocal sheaves on pairs of pants
• Motivic structures in non-commutative geometry
• Berkovich Spaces and Tubular Descent
• The Gauss-Manin connection on the periodic cyclic homology

Cited by in corpus (2)

• Moduli of framed formal curves
• Formality of little disks and algebraic geometry