Stability Conditions on An-Singularities

Abstract

We study the spaces of locally finite stability conditions on the derived categories of coherent sheaves on the minimal resolutions of A n -singularities supported at the exceptional sets. Our main theorem is that they are connected and simply-connected. The proof is based on the study of spherical objects in [30] and the homological mirror symmetry for A n -singularities.

Bridgeland stability conditions on threefolds II: An application to Fujita’s conjecture

“…the total space of the canonical bundle of a surface. For instance, see [Bri06,BM11,IUU10,Tod08,Tod09b]. The case of non-projective complex tori has been studied in [Mei07].…”

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.…”

Stability conditions and Calabi–Yau fibrations

“…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.…”