Autoequivalences of derived categories on the minimal resolutions of An-singularities on surfaces

Abstract

Grothendieck proved that any locally free sheaf on a projective line over a field (uniquely) decomposes into a direct sum of line bundles. Ishii and Uehara construct an analogue of Grothendieck's theorem for pure sheaves on the fundamental cycle of the Kleinian singularity An. We first study the analogue for the other Kleinian singularities. We also study the classification of rigid pure sheaves on the reduced scheme of the fundamental cycles. The classification is related to the classification of spherical objects in a certain Calabi-Yau 2-dimensional category.

The symplectic topology of some rational homology balls

“…We will use the understanding provided by Ritter ([34]) and Ishii, Ueda and Uehara ( [21], [22]) to prove the following theorem:…”

“…We now use Ishii, Ueda and Uehara's results from [21], [22] discussed above to replace V with an isomorphic object V c in the exact Fukaya category of S p−1 where V c is a matching sphere for a possibly quite complicated path c. Now, R r is the antipodal map, R r (x, y, z) = (−x, −y, −z). Hence, R r V is represented by the matching sphere over the path −c.…”

“…On the other hand, the non-existence of closed exact Lagrangians is proved using a detailed understanding of closed exact Lagrangians in the A n Milnor fibres based on twisted symplectic cohomology applied by Ritter in [34] which suffices for p odd. For p even, we utilize a deeper understanding coming from homological mirror symmetry and calculations on the B-model provided by Ishii, Ueda, Uehara [21], [22]. It is remarkable that algebro-geometric calculations on the mirror side can be utilized profitably towards an application to symplectic topology.…”

Fourier-Mukai transforms

“…It remains to classify the auto-equivalences of the derived category D b (X) for a surface X. Orlov solved this problem for an abelian surface [58] (and more generally for abelian varieties). Ishii and Uehara [36] solve the problem for the minimal resolutions of A n -singularities on a surface (so this is a local result).…”

Stability conditions and Calabi–Yau fibrations

“…Since Y has A n singularity, C is a chain of rational curves Note that Auteq(X /Y ) is regarded as a subgroup of autoequivalences on D(X /Y ). The similarly defined group Auteq(X/Y ) was studied in [9] and the purpose of this section is to study Auteq(X /Y ) via the space Stab(X /Y ). Recently the detailed study of the space Stab(X/Y ) has been done by [10], using the technique of [9] and local mirror symmetry.…”