RESEARCH SEMINAR — BRIDGELAND STABILITY CONDITIONS

DANIEL GREB

1. Introduction The aim of the seminar is to study stability conditions on triangulated categories, more precisely the bounded of a given projective variety X. The objects in this cat- egory are complexes of coherent sheaves on X, and we consider them up to quasi-isomorphism, i.e., morphisms that induce isomorphisms in cohomology. As every sheaf can be seen as a complex with just one non-zero entry (say in degree zero), this is closely related to studying sheaves on X, their resolutions, cohomology, moduli, etc. The derived category turns out to be a very reasonable invariant of X, for example it determines X whenever its canonical bundle is either ample or anti-ample. Moreover, the following is expected, and proven in dimension three (we will see Bridgelands proof): if X and Y are two K-equivalent varieties (that means connected by a birational correspondence that preserves the canonical bundles) then the derived categories of X and Y are equivalent. Therefore, the study of the derived category is closely intertwined with the study of the birational geometry of a given X, the main theme of the so-called minimal model program, to which we will also see an introduction in the seminar. For example, one may ask whether two “minimal models” of a given variety X are always K-equivalent. Again, the answer will be provided during the seminar. The techniques employed by Bridgeland in order to prove the above mentioned result are interesting in their own right, since they introduced what is nowadays called a Bridgeland- stability condition. This notion generalises the more well-known notion of stability for coher- ent sheaves on X to the derived category, i.e., to complexes. We will study it in the second part of the seminar, and see how the results presented in the first part can be interpreted in this language.

2. Detailed program A general reference that should be consulted by everybody is Bridgeland’s ICM article [Bri06]. 9.4. Daniel Greb: Introduction. 16.4. NN: Triangulated categories. The bounded derived category studied later in the seminar is an example of a . In this talk the abstract, but rather straight- forward theory of triangulated categories should be introduced. The prime example of a tri- angulated category is the (bounded) derived category of an abelian category, which should be introduced and shown to be a triangulated category. References: [Huy06, Chapter 1 and 2 (up to p. 39)]

Date: April 8, 2015.

1 2 D. GREB

23.4. NN: Derived categories of coherent sheaves. As prime example of a triangulated category, we will study the bounded derived category of complexes of sheaves on a given projective variety. The aim of the talk is to introduce this category (which in some sense we all know) and to remind us of the basic properties of this category in the language of triangulated categories introduced last week. In particular, derived functors should be recalled. References: [Huy06, Second part of Chapter 2 and Chapter 3].

30.4. NN: Fourier-Mukai transforms. A Fourier-Mukai transform is the derived analogue of a correspondence (which have been studied for all kinds of cohomology theories, e.g. Chow groups, singular cohomology). After introducing what a Fourier-Mukai transorm is, it should be proven that any equivalence between the derived categories of two smooth projective varieties is isomorphic to a Fourier-Mukai transform. This implies a geometric proof of the fact that two smooth projective varieties with eqivalent derived categories have the same dimension. References: [Huy06, Chapter 5]

21.5. NN: Derived category and canonical bundle - II. Here, a short summary of the main objects of study in the minimal model program (Kodaira dimension, canonical ring, ...) should be given, before proving that these notions are invariant under derived equivalence; i.e., two varieties whose derived categories are equivalent have isomorphic canonical rings. Afterwards, a result of Kawamata should be presented that says that two smooth projective varieties of maximal Kodaira dimension which are derived-equivalent are in fact K-equivalent, i.e., there exists a birational correspondence preserving the canonical bundles. Afterwards, the main conjecture stating the converse for varieties of pure type (i.e. with ample, anti-ample or trivial canonical bundle) should be stated, and the main results concerning K-equivalent minimal models should be discussed. In particular, it should be discussed that two birational smooth Calabi-Yau threefolds are related by a sequence of special flops. References: [Huy06, Chapter 6].

28.5. NN: Bridgeland’s proof of the 3-fold case - I. This and the next talk are devoted to the proof of the main conjecture in dimension three due to Bridgeland. References: [Bri02] and [Bri06]

11.6. NN: Bridgeland’s proof of the 3-fold case - II. see previous talk

In the next three talks the technical definitions and properties of Bridgeland stability conditions will be introduced. [Huy14] motivated and explains everything quite slowly starting with a discussion of sheaves on curves and surfaces.

18.6. NN: Torsion theories and t-structures. Reference: [Huy14, Chapter 4.1]

25.6. NN: Stability conditions: definition and examples. Reference: [Huy14, Chapter 4.2]

9.7. NN: Stability conditions on surfaces. Reference: [Huy14, Chapter 4.3]

16.7. NN: Stability conditions and threefold flops. In this last talk, the result of Bridgeland discussed earlier will be explained using stability conditions, following [Bri06] and [Tod08]. RESEARCH SEMINAR — BRIDGELAND STABILITY CONDITIONS 3

References [Bri02] Tom Bridgeland, Flops and derived categories, Inv. Math. 147 (2002), 613–632. [Bri06] Tom Bridgeland, Derived categories of coherent sheaves, International Congress of Mathematicians, vol. II, 563–582, Eur. Math Soc., Z¨urich, 2006. [Huy06] Daniel Huybrechts, Fourier-Mukai Transforms in , , 2006. [Huy14] Daniel Huybrechts, Introduction to Stability Conditions, in: Moduli Spaces, London Math. Soc. Lec- ture Note Series, vol. 411, 2014. [Tod08] Yakinobu Toda, Stability conditions and crepant small resolutions, Trans. Amer. Math. Soc. 360 (2008), 6149–6178.

Essener Seminar fur¨ Algebraische Geometrie und Arithmetik, Fakultat¨ fur¨ Mathematik, Uni- versitat¨ Duisburg-Essen, 45117 Essen, Germany E-mail address: [email protected]