Categorical Methods inAlgebraic Geometry and Mirror Symmetry
Date: March 28 ~ 30, 2016 Place: Rm. 8101, KIAS |
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Speaker: Taro Hayashi (Tokyo)
Title: Universal Covering Calabi-Yau Manifolds of the Hilbert Schemes of n Points of Enriques Surfaces
Abstract: For an Enriques surface E, let E[n] be the Hilbert scheme of n points of Enriques surface. E[n] has a Calabi-Yau manifold X as the universal covering space π : X → E[n] of degree 2. I will show that for n ≥ 3, the number of distinct Enriques quotients of X is one. Furthermore X admits no Enriques quotient of X other than the original.
Speaker: Naoki Koseki (Tokyo)
Title: A functorial construction of moduli of sheaves.
Abstract: I will explain the paper of L.A-Consul and A.King, "A functorial construction of moduli of sheaves". We will show that the moduli space of coherent sheaves can be constructed as a closed subscheme of the moduli space of quiver representations. In the first talk, we recall two moduli problems; moduli of sheaves and moduli of quiver representations. Stability conditions are important notion in these moduli theory. After that, we recall the construction of the moduli space of quiver representations. In the second talk, we construct the morphism from the moduli functor of sheaves to the moduli functor of quiver representations. The key point is to show that the morphism preserves stability.
Speaker: Genki Ouchi (IPMU)
Title: Derived automorphisms of K3 surfaces of Picard number one inspired by complex dynamics
Abstract: The notion of categorical entropy of endofunctors of triangulated categories is introduced by Dimitrov-Haiden-Katzarkov-Kontsevich. Recently, Kikuta and Takahashi proved that if an endofunctor is a derived pullback of surjective endomorphism of a smooth projective variety, then the categorical entropy is equal to the topological entropy. In this talk, I would like to talk about derived automorphisms of K3 surfaces of Picard number one from point of view of complex dynamics. It is well known that K3 surfaces of Picard number one have only trivial automorphisms. However, we can show that there are derived automorphisms of positive entropy for any algebraic K3 surface. We will see the relation between entropy (or logarithm of spectral radius) of derived automorphisms and dynamical property of induced autmorphisms of the space of stability conditions for K3 surfaces of Picard number one. As an application, we will discuss the existence problem of Gepner type stability conditions on triangulated categories of matrix factorizations for defining polynomials of special cubic fourfolds with associated K3 surfaces. Moreover, we will discuss relation between derived automorphisms of K3 surfaces and (birational) automorphisms of moduli spaces of stable objects on them via some examples. This talk is based on work in progress.
Speaker: Xun Yu (POSTECH)
Title: McKay correspondence and new Calabi-Yau threefolds
Abstract: Classification of topological types of Calabi-Yau threefolds is still an open problem. It turns out that we can obtain some new Calabi-Yau threefolds by considering crepant resolutions of the quotient varieties of smooth quintic threefolds by Gorenstein group actions. In the first lecture, we will discuss the classical McKay correspondences (which relate the topology of crepant resolutions to representation theory of finite groups) and classification of automorphism groups of smooth quintic threefolds. In the second lecture, we will explain how to determine all possible Hodge numbers of crepant resolutions of quotients of smooth quintic threefolds and will relate such crepant resolutions to known Calabi-Yau threefolds constructed by other methods.
Title: Universal Covering Calabi-Yau Manifolds of the Hilbert Schemes of n Points of Enriques Surfaces
Abstract: For an Enriques surface E, let E[n] be the Hilbert scheme of n points of Enriques surface. E[n] has a Calabi-Yau manifold X as the universal covering space π : X → E[n] of degree 2. I will show that for n ≥ 3, the number of distinct Enriques quotients of X is one. Furthermore X admits no Enriques quotient of X other than the original.
Speaker: Naoki Koseki (Tokyo)
Title: A functorial construction of moduli of sheaves.
Abstract: I will explain the paper of L.A-Consul and A.King, "A functorial construction of moduli of sheaves". We will show that the moduli space of coherent sheaves can be constructed as a closed subscheme of the moduli space of quiver representations. In the first talk, we recall two moduli problems; moduli of sheaves and moduli of quiver representations. Stability conditions are important notion in these moduli theory. After that, we recall the construction of the moduli space of quiver representations. In the second talk, we construct the morphism from the moduli functor of sheaves to the moduli functor of quiver representations. The key point is to show that the morphism preserves stability.
Speaker: Genki Ouchi (IPMU)
Title: Derived automorphisms of K3 surfaces of Picard number one inspired by complex dynamics
Abstract: The notion of categorical entropy of endofunctors of triangulated categories is introduced by Dimitrov-Haiden-Katzarkov-Kontsevich. Recently, Kikuta and Takahashi proved that if an endofunctor is a derived pullback of surjective endomorphism of a smooth projective variety, then the categorical entropy is equal to the topological entropy. In this talk, I would like to talk about derived automorphisms of K3 surfaces of Picard number one from point of view of complex dynamics. It is well known that K3 surfaces of Picard number one have only trivial automorphisms. However, we can show that there are derived automorphisms of positive entropy for any algebraic K3 surface. We will see the relation between entropy (or logarithm of spectral radius) of derived automorphisms and dynamical property of induced autmorphisms of the space of stability conditions for K3 surfaces of Picard number one. As an application, we will discuss the existence problem of Gepner type stability conditions on triangulated categories of matrix factorizations for defining polynomials of special cubic fourfolds with associated K3 surfaces. Moreover, we will discuss relation between derived automorphisms of K3 surfaces and (birational) automorphisms of moduli spaces of stable objects on them via some examples. This talk is based on work in progress.
Speaker: Xun Yu (POSTECH)
Title: McKay correspondence and new Calabi-Yau threefolds
Abstract: Classification of topological types of Calabi-Yau threefolds is still an open problem. It turns out that we can obtain some new Calabi-Yau threefolds by considering crepant resolutions of the quotient varieties of smooth quintic threefolds by Gorenstein group actions. In the first lecture, we will discuss the classical McKay correspondences (which relate the topology of crepant resolutions to representation theory of finite groups) and classification of automorphism groups of smooth quintic threefolds. In the second lecture, we will explain how to determine all possible Hodge numbers of crepant resolutions of quotients of smooth quintic threefolds and will relate such crepant resolutions to known Calabi-Yau threefolds constructed by other methods.