Obstructions to Semiorthogonal Decompositions for Singular Threefolds I: K-Theory

Base Change for Semiorthogonal Decompositions

Let X be an algebraic variety over a base scheme S and φ : T → S a faithful base change. Given an admissible subcategory A in D(X), the bounded derived category of coherent sheaves on X, we construct an admissible subcategory AT in D b(X×ST ), called the base change of A, in such a way that the following base change theorem holds: if a semiorthogonal decomposition of D(X) is given then the base...

Hochschild Homology and Semiorthogonal Decompositions

We investigate Hochschild cohomology and homology of admissible subcategories of derived categories of coherent sheaves on smooth projective varieties. We show that the Hochschild cohomology of an admissible subcategory is isomorphic to the derived endomorphisms of the kernel giving the corresponding projection functor, and the Hochschild homology is isomorphic to derived morphisms from this ke...

Compactifications of F-Theory on Calabi–Yau Threefolds – I

We study compactifications of F-theory on certain Calabi–Yau threefolds. We find that N = 2 dualities of type II/heterotic strings in 4 dimensions get promoted to N = 1 dualities between heterotic string and F-theory in 6 dimensions. The six dimensional heterotic/heterotic duality becomes a classical geometric symmetry of the Calabi–Yau in the F-theory setup. Moreover the F-theory compactificat...

1 4 A pr 1 99 9 K 3 - fibered Calabi - Yau threefolds II , singular fibers

Continuation of Singular Value Decompositions

In this work we consider computing a smooth path for a (block) singular value decomposition of a full rank matrix valued function. We give new theoretical results and then introduce and implement several algorithms to compute a smooth path. We illustrate performance of the algorithms with a few numerical examples.