A category of kernels for equivariant factorizations, II: further implications

TL;DR

This paper explores advanced topics in algebraic geometry and category theory, connecting kernels, derived categories, and mirror symmetry, and introduces new invariants for triangulated categories.

Contribution

It extends previous work on kernels for equivariant factorizations, proposes a geometric framework for Homological Mirror Symmetry, and introduces Noether-Lefschetz spectra as new invariants.

Findings

New cases of Orlov's conjecture on Rouquier dimension

Introduction of Noether-Lefschetz spectra as invariants

Connections between algebraic classes and triangulated categories

Abstract

We leverage the results of the prequel in combination with a theorem of D. Orlov to yield some results in Hodge theory of derived categories of factorizations and derived categories of coherent sheaves on varieties. In particular, we provide a conjectural geometric framework to further understand M. Kontsevich's Homological Mirror Symmetry conjecture. We obtain new cases of a conjecture of Orlov concerning the Rouquier dimension of the bounded derived category of coherent sheaves on a smooth variety. Further, we introduce actions of $A$-graded commutative rings on triangulated categories and their associated Noether-Lefschetz spectra as a new invariant of triangulated categories. They are intended to encode information about algebraic classes in the cohomology of an algebraic variety. We provide some examples to motivate the connection.