2017/11/30 by Dmitry Vaintrob, Vaintrob, Dmitry
Computer Science · Mathematics · #11G42 #14B05 #14E15 #14E22 #14F05 #14F30 #14F40 #14J33 #14M25 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Mathematical and Theoretical Analysis #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1712.00045
openalex publication_date 2017/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We write down a new "logarithmic" quasicoherent category Qcohlog(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 Qcohlog(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 Qcohlog(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.