Semisimple Quantum Cohomology and Blow-ups

Using results of Gathmann, we prove the following theorem: If a smooth projective variety X has generically semisimple (p,p)-quantum cohomology, then the same is true for the blow-up of X at any number of points. This a successful test for a modified version of Dubrovin's conjecture from the ICM 1998.Comment: 13 page

Projectivity and Birational Geometry of Bridgeland moduli spaces

We construct a family of nef divisor classes on every moduli space of stable complexes in the sense of Bridgeland. This divisor class varies naturally with the Bridgeland stability condition. For a generic stability condition on a K3 surface, we prove that this class is ample, thereby generalizing a result of Minamide, Yanagida, and Yoshioka. Our result also gives a systematic explanation of the relation between wall-crossing for Bridgeland-stability and the minimal model program for the moduli space. We give three applications of our method for classical moduli spaces of sheaves on a K3 surface: 1. We obtain a region in the ample cone in the moduli space of Gieseker-stable sheaves only depending on the lattice of the K3. 2. We determine the nef cone of the Hilbert scheme of n points on a K3 surface of Picard rank one when n is large compared to the genus. 3. We verify the "Hassett-Tschinkel/Huybrechts/Sawon" conjecture on the existence of a birational Lagrangian fibration for the Hilbert scheme in a new family of cases.Comment: v2: 50 pages, exposition substantially improved based on referee's comments. Accepted for publication in Journal of the AM

The space of stability conditions on the local projective plane

We study the space of stability conditions on the total space of the canonical bundle over the projective plane. We explicitly describe a chamber of geometric stability conditions, and show that its translates via autoequivalences cover a whole connected component. We prove that this connected component is simply-connected. We determine the group of autoequivalences preserving this connected component, which turns out to be closely related to Gamma1(3). Finally, we show that there is a submanifold isomorphic to the universal covering of a moduli space of elliptic curves with Gamma1(3)-level structure. The morphism is Gamma1(3)-equivariant, and is given by solutions of Picard-Fuchs equations. This result is motivated by the notion of Pi-stability and by mirror symmetry.Comment: 56 pages, 3 figures. v2: various improvements based on referees' feedback (e.g. in Prop. 3.3); appendix C removed; new Prop. B.4 clarifies relation between support property and "full" stability condition

Stability Conditions, Wall-crossing and weighted Gromov-Witten Invariants

We extend B. Hassett's theory of weighted stable pointed curves ([Has03]) to weighted stable maps. The space of stability conditions is described explicitly, and the wall-crossing phenomenon studied. This can be considered as a non-linear analog of the theory of stability conditions in abelian and triangulated categories. We introduce virtual fundamental classes and thus obtain weighted Gromov-Witten invariants. We show that by including gravitational descendants, one obtains an \LL-algebra as introduced in [LM04] as a generalization of a cohomological field theory.Comment: 28 pages; v2: references added and updated, addressed referee comments; to appear in Moscow Math Journa

Bridgeland Stability conditions on threefolds I: Bogomolov-Gieseker type inequalities

We construct new t-structures on the derived category of coherent sheaves on smooth projective threefolds. We conjecture that they give Bridgeland stability conditions near the large volume limit. We show that this conjecture is equivalent to a Bogomolov-Gieseker type inequality for the third Chern character of certain stable complexes. We also conjecture a stronger inequality, and prove it in the case of projective space, and for various examples. Finally, we prove a version of the classical Bogomolov-Gieseker inequality, not involving the third Chern character, for stable complexes.Comment: 45 pages, 4 figures. Comments are welcome. v2: Referee comments incorporated. To appear in JA

A short proof of the deformation property of Bridgeland stability conditions

The key result in the theory of Bridgeland stability conditions is the property that they form a complex manifold. This comes from the fact that given any small deformation of the central charge, there is a unique way to correspondingly deform the stability condition. We give a short direct proof of a strong version of this deformation property.Comment: 14 pages, 6 figures. v2: Exposition overhauled. Addressed inaccuracy (thanks to Martin Gulbrandsen for pointing it out) in treating the topology. v3: addressed referee comments improving the exposition. To appear in Math. Annale

Semisimple Quantum Cohomology, deformations of stability conditions and the derived category

The introduction discusses various motivations for the following chapters of the thesis, and their relation to questions around mirror symmetry. The main theorem of chapter 2 says that if the quantum cohomology of a smooth projective variety V yields a generically semisimple product, then the same holds true for its blow-up at any number of points (theorem 3.1.1). This is a positive test for a conjecture by Dubrovin, which claims that quantum cohomology of V is generically semisimple if and only if its bounded derived category Db(V) has a complete exceptional collection. Chapter 3 generalizes Bridgeland's notion ofstability condition on a triangulated category. The generalization, a polynomial stability condtion (definition 2.1.4), allows the central charge to have values in complex polynomials instead of complex numbers. We show that polynomial stability conditions have the same deformation properties as Bridgeland's stability conditions (theorem 3.2.5). In section 4, it is shown that every projective variety has a canonical family of polynomial stability conditions. In chapter 4, we define and study the notion of a weighted stable map (definition 2.1.2), depending on a set of weights. We show the existence of moduli spaces of weighted stable maps as proper Deligne-Mumford stacks of finite type (theorem 2.1.4), and study in detail their birational behaviour under changes of weights (section 4). We introduce a category of weighted marked graphs in section 6, and show that it keeps track of the boundary components of the moduli spaces, and natural morphisms between them. We introduce weighted Gromov-Witten invariants by defining virtual fundamental classes, and prove that these satisfy all properties to be expected (sections 5 and 7). In particular, we show that Gromov-Witten invariants without gravitational descendants do not depend on the choice of weights.Halbeinfache Quanten-Kohomologie, Deformation von Stabilitätsbedingungen und die Derivierte Kategorie Die Einleitung erläutert verschiedene Ausgangspunkte für die nachfolgenden Kapitel, und ihre Verbindungen zu Fragen rund um Spiegelsymmetrie. Hauptaussage von Kapitel 2 ist Satz 3.1.1: wenn das Produkt der Quantenkohomologie einer glatten projektiven Varietät V generisch halbeinfach ist, dann gilt dasgleiche für die Aufblasung von V an beliebig vielen Punkten. Dies ist ein erfolgreicher Test für eine Vermutung von Dubrovin, die besagt, dass die Quantenkohomologie von V genau dann generisch halbeinfach ist, wenn die beschränkte derivierte Kategorie Db(V) ein vollständiges exzeptionelles System besitzt. Kapitel 3 verallgemeinert Bridgelands Begriff einer Stabilitätsbedingung in einer triangulierten Kategorie. Diese Verallgemeinerung, eine polynomiale Stabilitätsbedingung (Definition 2.1.4), lässt eine zentrale Ladung mit Werten in komplexen Polynomen statt komplexen Zahlen zu. Es wird gezeigt, dass polynomiale Stabilitätsbedingungen dieselben Deformationseigenschaften wie Bridgelands Stabilitätsbedingungen haben (Satz 3.2.5). Abschnitt 4 zeigt, dass es für jede projektive Varietät V eine kanonische Familie von polynomialen Stabilitätsbedingungen in Db(V) gibt. Kapitel 4 führt den Begriff einer gewichteten stabilen Abbildung ein (Definition 2.1.2), in Abhängigkeit einer Menge von Gewichten. Satz 2.1.4 zeigt die Existenz der Modulräume gewichter stabiler Abbildung als eigentliche Deligne-Mumford-Stacks endlichen Typs, und Abschnitt 4 beschäftigt sich im Detail mit dem birationalen Verhalten der Modulräume bei Änderungen der Gewichte. In Abschnitt 6 führen wir eine Kategorie gewichteter markierter Graphen ein, und zeigen, dass sie natürlicherweise Randkomponenten der Modulräume und die natürliche Morphismen zwischen ihnen indiziert. Gewichtete Gromov-Witten-Invarianten werden durch die Definition von virtuellen Fundamentalklassen eingefürt, und wir zeigen, dass diese alle zu erwartenden Eigenschaften erfüllen (Abschnitte 5 und 7). Insbesondere zeigen wir, dass Gromov-Witten-Invarianten ohne Kopplung an Gravitation nicht von der Wahl der Gewichte abhängen