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Institute for the Physics and Mathematics of the UniverseWPI

Exceptional collections and degenerations of varieties

Date: September 1 - 5, 2008
Place: IPMU, Seminar room at Prefab.B
  • Alexey Bondal (IPMU)
  • Kyoji Saito (IPMU)
  • Kentaro Hori (Toronto / IPMU)
  • Yukinobu Toda (IPMU)
  • Klaus Altmann (Free University)
  • Sergey Galkin (Steklov Mathematical Institute)
  • Kentaro Hori (Toronto/IPMU)
  • Hiroshige Kajiura (Chiba University)
  • Bumsig Kim (Korea Institute for Advanced Study)
  • Alexander Kuznetsov (Steklov Mathematical Institute)
  • Markus Perling (Ruhr-Universita"t Bochum)
  • Victor Przhiyalkovskiy (Steklov Mathematical Institute)
  • Atsushi Takahashi (Osaka University)
  • Yukinobu Toda (IPMU)
  • Monday, September 1
    Registration fee 1000yen, collect: 13:30(Mon)
     13:30-14:00 registration
     14:00-15:00 M. PERLING 1
     15:00-15:30 tea break
     15:30-16:30 Yu. TODA 1
     18:00-  informal dinner

  • Tuesday, September 2
     9:30-10:00 breakfast
     10:00-11:00 M. PERLING 2
     11:00-11:30 coffee break
     11:30-12:30 B. KIM
     12:30-14:00 lunch
     14:00-15:00 H. KAJIURA
     15:00-15:30 tea time

  • Wednesday, September 3
     9:30-10:00 breakfast
     10:00-11:00 Y.TODA 2
     11:00-11:30 coffee break
     11:30-12:30  S. GALKIN
     15:00-15:30 tea time
     (afternoon is free)

  • Thursday, September 4
     9:30-10:00 breakfast
     10:00-11:00 K. ALTMANN
     11:00-11:30 coffee break
     11:30-12:30  K. HORI 1
     14:00-15:00  V. PRZHYJALKOWSKI
     15:00-15:30 tea time
     18:00-  party at cafeteria

  • Friday, September 5
     9:30-10:00 breakfast
     10:00-11:00 A. TAKAHASHI
     11:00-11:30 coffee break
     11:30-12:30  A. KUZNETSOV
     12:30-14:00  lunch
     14:00-15:00  K. HORI 2
     15:00-15:30 tea time

Abstract: Klaus Altmann, Free Universitaet, Berlin, Germany

Title: Deformation of flag quivers (with Duco van Straten)

Connected quivers without oriented cycles give rise to Fano varieties being the moduli space of stable quiver representations. Via Batyrev's construction, they host Calabi-Yau spaces with the same type of singularities.
We show that for a certain type of quivers, the Fano (and hence the CY varieties) are smoothable in codimension three. Since the obstruction spaces T2 are, in general, non-trivial, this requires a careful analysis of the multidegrees carrying non-zero T2 parts.
The name flag quiver arises from the fact that the toric degeneration of flag varieties is a special case of this situation.

Sergei Galkin, free artist, Moscow, Russia

Title: Degenerations of Fano threefolds (towards exceptional collections)

We will discuss small degenerations of smooth Fano threefolds (like V5) to toric nodal Fano threefolds and some applications of these for computing Gromov-Witten invariants and constructing exceptional collections on the primary smooth threefolds.

Kentaro Hori 1, IPMU, Kashiwa, Japan

Title: Mirror Symmetry in 2+1 and 1+1 dimensions

I discuss how mirror symmetry in 1+1 dimensions and 2+1 dimensions are related. I report on a progress, as an application, to find the mirrors of Grassmannians and other flag varieties that fix a problem in the mirror conjecture by Eguchi, Xiong and myself. The latter conjecture was also considered by Batyrev, Ciocan-Fontanine, Kim and van Straten using toric degenerations. This is based on a work with Aganagic, Karch and Tong.

Kentaro Hori 2, IPMU, Kashiwa, Japan

Title: Witten index of 2d supersymmetric QCD

We compute the Witten index of 2d (2,2) supersymmetric QCD with twisted masses. This is based on a work with David Tong. Possible relation to exceptional collections in Grassmannians may be discussed, possibly.

Hiroshige Kajiura, Chiba University, Japan

Title: Strongly exceptional collections associated to regular systems of weights with ε = -1.

I plan to talk about strongly exceptional collections of the triangulated categories of graded matrix factorizations associated to regular system of weights with ε = -1, where the strongly exceptional collections give examples of quivers of wild type. I will discuss special mutations between these(strongly) exceptional collections and the Auslander-Reiten triangles among them.

Bumsig Kim, KIAS, Seoul, Korea

Title: Logarithmic stable maps

A new compactification of maps from curves, using only finite maps, will be presented, which is a joint work with A. Kresch and Y.-G. Oh. After this, I will apply it to provide a modular desingularization of the main component of moduli of stable maps to any projective space. A notion of logarithmic stable maps will be needed for the purpose as well as the other general reason.

Alexander Kuznetsov, Steklov Institute, Moscow, Russia

Title: Exceptional collections on SGr(2,2n) and OGr(2,2n+1)

I will use the notion of a Lefschetz exceptional collection to construct exceptional collections on SGr(2,2n) and OGr(2,2n+1) - the Grassmanians of 2-dimensional subspaces isotropic for a symplectic form on a 2n-dimensional vector space and for a quadratic form on a (2n+1)-dimensional vector space respectively.

Markus Perling 1, 2, Ruhr-Universität, Bochum, Germany

Title: Exceptional sequences of invertible sheaves on rational surfaces

We present an easy procedure to construct exceptional and strongly exceptional sequences of invertible sheaves on smooth complete rational surfaces. We show that on every such surface there exists an exceptional sequence of invertible sheaves, and we give a criterion for the existence of full strongly exceptional sequences of this type. We present a complete classification of full strongly exceptional sequences of invertible sheaves for toric surfaces.

Viktor Przhyalkowski, Steklov Institute, Moscow, Russia

Title: Weak Landau--Ginzburg models of Fano varieties

Mirror symmetry conjectures states that for any smooth Fano variety there exists a dual Landau--Ginzburg model, that is, a pencil of algebraic varieties whose symplectic properties transforms to algebraic ones of the initial Fano, and vice versa, whose algebraic properties transforms to symplectic ones. One of the basic problems of mirror symmetry is to find such models. We discuss a purely computational method of finding natural candidates for dual models (called weak Landau-Ginzburg models). We discuss their properties and their relations with toric degenerations.

Atsushi Takahashi, Osaka University, Japan

Title: Homological Mirror Symmetry for Cusp Singularities

I will formulate and prove the Homological mirror symmetry conjecture for cusp singularities, the equivalence between the derived category of coherent sheaves on a weight projective line and the derived directed Fukaya category of a cusp singularity.

Yokinobu Toda 1, IPMU, Kashiwa, Japan

Title: Degenerations of del Pezzo surfaces (after Hacking, Prokhorov, Manetti)

The Manetti surface is a complex surface with quotient singularities which admits a smoothing to the projective plane. I will give the classification of such surfaces. Suprisingly, Manetti surfaces are classified by the solution of Markov equation, which also appears in the theory of exceptional collections on the projective plane. This talk is an introduction to the paper of Hacking and Prokhorov, ``Degenerations of del Pezzo surfaces I, math.AG/0509529'' and Manetti, ''Normal degenerations of the complex plane, J. reine angew, Math. 419, 89-118''.

Yokinobu Toda 2, IPMU, Kashiwa, Japan

Title:Stable pairs and wall-crossings in the derived category.

The notion of stable pairs on Calabi-Yau 3-folds and counting invariants of them are introduced by Pandharipande-Thomas. This is a variant of Donaldson-Thomas theory, a kind of curve counting via coherent sheaves. In this talk I introduce new invariants which count certain stable perverse coherent sheaves, and show that they generalize Pandharipande-Thomas theory. Then I will show that the rationality conjecture of stable pair invarians is deduced from the wall-crossing phenomena of our new invariant.