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Forschungsseminar Komplexe Analysis - Summer Semester 2015

Bridgeland stability conditions on derived categories

Wednesdays 4:30-6:00pm - Arnimallee 3, SR 005/A3. ( 15th April 2015 - 15th July 2015 )


During this semester we will study Bridgeland stability conditions on derived categories. In [Bri07], Bridgeland introduces stability conditions on triangulated categories and proves that the set of stability conditions has a natural topology and is a (complex) manifold, possibly infinite dimensional. The goal of the seminar is to construct the stability manifold for the bounded derived category of coherent sheaves on a smooth projective complex variety. In particular, we will describe the stability manifold for curves and partially for K3 surfaces [Bri08].

References:
[Bri07] Tom Bridgeland,Stability conditions on triangulated categories, Ann. Math. (2007), 166:317-345.
[Bri08] Tom Bridgeland,Stability conditions on K3 surfaces, Duke Math. J. (2008), 141(2):241-291.

A more detailed description of the talks can be found here.


If you want to participate, please don’t hesitate to contact us: Eva Martínez and Alejandra Rincón.


Schedule

♠ 15.04.2015 // Derived categories of coherent sheaves and t-structures - Eva Martínez
The aim of this talk is to recall basic results about the derived category associated to an abelian category and t-structures that we will need all along the semester. After introducing briefly the outline of the seminar, we will recall a few important facts about the bounded derived category of coherent sheaves on smooth projective varieties. In the second part of the talk we will introduce t-structures as the derived version of torsion structures for abelian categories. Finally, we introduce tilting of an abelian category, which gives a way of constructing new t-structures on triangulated categories, and see the relation between tilts of the heart of a t-structure on a triangulated category D and new t-structures on D.

♠ 22.04.2015 // Derived categories and instanton bundles on Fano threefolds - Giangiacomo Sanna
One of the few classes of vector bundles on threefolds whose moduli space is well understood is that of instantons on P3. In this case, the simple structure of the derived category of coherent sheaves provides a GIT-quotient description for the moduli space. This point of view generalises to other Fano threefolds. The first part of the talk will be an introduction to the derived categories of coherent sheaves on smooth projective varieties and to their decompositions. In the second part, we will describe the moduli space of instantons on a linear section of Gr(2,5): we will relate the geometry of the moduli space of instantons on this specific Fano threefold to that of rational curves in the threefold itself and to the structure of its derived category.

♠ 29.04.2015 // Stability functions and slicings - Anna Wißdorf
We will recall the definition of slope-stability for coherent sheaves on smooth projective curves. Then, we will define stability functions on abelian categories and slicings on triangulated categories and relate these to the Harder-Narasimhan filtration.

♠ 06.05.2015 // Stability conditions - Joana Cirici
We will introduce the notion of stability condition on a triangulated category and identify it with the heart of a bounded t-structure satisfying certain conditions. We will then see that the set of such stability conditions has a natural topology, defining a new invariant of triangulated categories.

♠ 13.05.2015 // Deformations of stability conditions - Nikolai Beck
The aim of this talk is to prove that the space of stability conditions of a triangulated category is a complex manifold naturally induced by a metric. We also study the subspace of numerical stability conditions and we prove that is a finite dimensional submanifold.

♠ 20.05.2015 // Stability conditions on curves - Victoria Hoskins
The aim of this talk is to explicitly describe the (numerical) stability manifold of a smooth complex projective curve; this description is due to Bridgeland (g=1), Macri (g >1) and Okada (g=0), and builds on work of Gorodentscev, Kuleshov and Rudakov. For genus greater than zero, we show that the stability manifold is isomorphic to the product of the complex plane with the (open) upper half plane, by using the group actions introduced in the previous talk. The case of the projective line is more involved but more interesting (for example, we see wall-crossing phenomena) and will require a second talk: the stability manifold is determined by studying a holomorphic action of the additive group on the stability manifold combined with the action given by tensoring by line bundles.

♠ 27.05.2015 // Stability conditions and unfoldings of simple singularities - Tom Sutherland (University of Pavia)
We will describe how the unfolding space of a simple singularity parameterises stability conditions on the derived categories of finite-dimensional representations of the corresponding ADE quiver as well as its Calabi-Yau completions in positive dimensions. The relationship between these spaces of stability conditions is encoded in the geometry on the unfolding space first described by Saito and later axiomatised by Dubrovin as a Frobenius manifold structure.
We will focus on giving geometric interpretations of the description of the spaces of stability conditions of small Calabi-Yau dimension. In dimension 1 we recover the usual description of the space of stability conditions of coherent sheaves on an elliptic curve. In dimension 2 we obtain Bridgeland's description of the spaces of stability conditions on non-compact K3 surfaces obtained as resolutions of Kleinian singularities. Finally in dimension 3 there is an interesting relationship to the hyperkaehler geometry of certain moduli spaces of solutions to Hitchin's equations.

♠ 03.06.2015 // First examples of stability conditions on K3 surfaces - Alejandra Rincón
The aim of this talk is to construct explicit examples of stability conditions on the derived category of coherent sheaves on a complex algebraic K3 surface. We will show that the Coh(X) cannot be the heart of a stability condition on X, then we tilt a torsion theory on Coh(X), obtaining a suitable heart and a stability function on it. Finally, we explain how to describe a connected component of Stab(X) and we mention Bridgeland's conjecture.

♠ 10.06.2015 // Stability conditions on K3 surfaces (2): the open subset U(X) of good stability conditions - Ángel Muñoz
The aims of this talk are to prove the covering map property and to describe a locally finite chamber structure in the connected components of Stab(X). If time permits we will also show that we can find distinguished points in the orbits of the action of the group Cov(GL+(2,R)) on the open subset of good stability conditions.

♠ 17.06.2015 // Stability conditions on K3 surfaces (3): the open subset U(X) and its boundary - Giangiacomo Sanna
The aim of this talk is to introduce an open subset U of the space of stability conditions which will provide us with a fundamental domain for the action of derived auto-equivalences. We will first describe the orbits of U under a natural action of a linear group and then provide a description for its boundary.

♠ 24.06.2015 // Stability conditions on K3 surfaces (4): The main theorem and final remarks - Eva Martínez
The aim of this talk is to prove the main theorem, [Bri08, Theorem 1.1]. The crucial point is that the images of the closure of the open subset U(X) under the action of the group of derived auto-equivalences cover the entire connected component.

♠ 01.07.2015 // Stability conditions on the projective line - Victoria Hoskins
We present the description of the stability manifold of the projective line due to Okada. We carefully analyse the action of the additive group of complex numbers and the action of Picard group on the stability manifold to find and describe a fundamental domain for this action. From this we obtain an explicit description of the stability manifold itself.

♠ 08.07.2015 // Lagrangian fibrations on moduli spaces of stable sheaves on a K3 surface - Alejandra Rincón
The aim of this talk is to prove the that the existence of a birational Lagrangian fibration on a moduli space of stable sheaves M on a K3 surface (after choosing appropriate Mukai vectors and a polarisation) is equivalent to the existence of an integral divisor of square zero with respect to the Beauville-Bogomolov quadratic form. First of all we show that we can recover a moduli space of stable sheaves on a K3 in terms of a moduli space of Bridgeland stable objects. Then, we use wall-crossing with respect Bridgeland stability conditions to study the birational geometry of M. From this we deduce the result.