A category of kernels for graded matrix factorizations and Hodge theory

We provide a matrix factorization model for the derived internal Hom
(continuous), in the homotopy category of k-linear dg-categories,
between categories of graded matrix factorizations. This description is
used to calculate the derived natural transformations between twists
functors on categories of graded matrix factorizations. Furthermore, we
combine our model with a theorem of Orlov to establish a geometric
picture related to Kontsevich's Homological Mirror Symmetry Conjecture.
As applications, 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 and a proof of the Hodge conjecture for
n-fold products of a K3 surface closely related to the Fermat cubic
fourfold. We also introduce Noether-Lefschetz spectra as a new Morita
invariant of dg-categories. They are intended to encode information
about algebraic classes in the cohomology on an algebraic variety. This
is joint with M. Ballard and L. Katzarkov.