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

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.

v2: Updated references and addresses. Cleaved off a part. 54 pages. v1: Expanded version of the latter half of arXiv:1105.3177. 92 pages. Comments very welcome!

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