Derived Autoequivalences and a Weighted Beilinson Resolution

Exceptional Sequences and Derived Autoequivalences

We prove a general theorem that gives a non trivial relation in the group of derived autoequivalences of a variety (or stack) X, under the assumption that there exists a suitable functor from the derived category of another variety Y admitting a full exceptional sequence. Applications include the case in which X is Calabi-Yau and either X is a hypersurface in Y (this extends a previous result b...

The Beilinson Complex and Canonical Rings of Irregular Surfaces

An important theorem by Beilinson describes the bounded derived category of coherent sheaves on P, yielding in particular a resolution of every coherent sheaf on P in terms of the vector bundles Ω Pn (j) for 0 ≤ j ≤ n. This theorem is here extended to weighted projective spaces. To this purpose we consider, instead of the usual category of coherent sheaves on P(w) (the weighted projective space...

Groups of Autoequivalences of Derived Categories of Smooth Projective Varieties

Window Shifts, Flop Equivalences and Grassmannian Twists

We introduce a new class of autoequivalences that act on the derived categories of certain vector bundles over Grassmannians. These autoequivalences arise from Grassmannian flops: they generalize Seidel-Thomas spherical twists, which can be seen as arising from standard flops. We first give a simple algebraic construction, which is well-suited to explicit computations. We then give a geometric ...

The Beilinson Equivalence for Differential Operators and Lie Algebroids

Let D be the ring of differential operators on a smooth irreducible affine variety X over C; or, more generally, the enveloping algebra of any locally free Lie algebroid on X. The category of finitely-generated graded modules of the Rees algebra e D has a natural quotient category qgr( e D) which imitates the category of modules on Proj of a graded commutative ring. We show that the derived cat...