Grassmannian twists on the derived category via spherical functors

Abstract

We construct new examples of derived autoequivalences for a family of higher‐dimensional Calabi–Yau varieties. Specifically, we take the total spaces of certain natural vector bundles over Grassmannians double-struckGdouble-struckr(r,d) of r‐planes in a d‐dimensional vector space, and define endofunctors of the bounded derived categories of coherent sheaves associated to these varieties. In the case r = 2, we show that these are autoequivalences using the theory of spherical functors. Our autoequivalences naturally generalize the Seidel–Thomas spherical twist for analogous bundles over projective spaces.

Window shifts, flop equivalences and Grassmannian twists

“…For r ≤ 2 these functors were proven to be equivalences in [Don11b]: it is an immediate corollary of the following theorem that in fact T F is an equivalence for all r < d.…”

“…Theorem 3.13 was proved in [Don11b] for the r ≤ 2 case. We now temporarily reinstate the d's and r's into our notation, and state these theorems in a slightly different way.…”

“…The relevant correspondences for the r = 2 case were described by the first author in [Don11b]: we describe the general case in Section 3.1. Then we have functors…”

“…The geometric construction. In [Don11b], the first author constructed an endofunctor of D b (X + ) using more geometric techniques, and proved that it was an autoequivalence when r ≤ 2. In this section we will show that this endofunctor agrees with the window-shift ω 0,1 , and hence that it is in fact an autoequivalence for all r.…”

The derived category of a GIT quotient

“…We will call such a direct system a Koszul system for S ⊂ X Proof. First assume X is smooth in a neighborhood of S. Then O X /I i S is perfect, so (12) implies that the derived duals K q i = (O X /I i S ) ∨ satisfy properties (1) and (2) with K q i → O X the dual of the map O X → O X /I i S . We compute the mapping cone…”