Winter School on Poisson Structures in Algebraic Geometry

January 15-17, 2019                   Haeundae, Busan

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◆ Lecture Series (6 hours) by Yoshinori Namikawa (Kyoto)

◇ Title: Introduction to Poisson geometry for algebraic geometers

◇ Abstract: The Poisson geometry has been extensively studied from the view point of differential geometry, dynamical system and quatizations. In this lecture I will talk on the Poisson geometry from algebro-geometric point of view. For example, the simultaneous resolutions of Klein surface singularities can be well explained in terms of Poisson structures, and one can develop Grothendieck-Brieskorn theory in higher dimensional situations. The topics of the lectures will include Poisson deformations, Kaledin's theory on Poisson schemes. I will explain these with many examples.

 

◆ Lecture Series (3 hours) by Brent Pym (Edinburgh/ McGill)

◇ Title: Introduction to symplectic leaves of Poisson varieties

◇ Abstract: One of the key basic facts in Poisson geometry and its applications is that any Poisson variety has a natural decomposition into subspaces, called its "symplectic leaves". In these lectures, we will give a basic introduction to the study of symplectic leaves of projective Poisson manifolds, from an algebraic-geometric point of view. We will see many examples, which will lead us naturally to a discussion of Bondal's conjecture about the dimensions of symplectic leaves on Poisson Fano manifolds.

 

◆ Seminar Talk (1 hour) by Takahiro Nagaoka (Kyoto)

◇ Title: The universal Poisson deformation space of hypertoric varieties and its applications.

◇ Abstract: A hypertoric variety $Y(A, alpha)$ is a (holomorphic) symplectic variety, which is defined as a Hamiltonian reduction of a complex vector space by a torus action. This is an analogue of a toric variety. Actually, its geometric properties can be studied through the associated hyperplane arrangements (instead of polytopes). By definition, there exists a projective morphism $pi:Y(A, alpha) to Y(A, 0)$, and for generic $alpha$, this gives a crepant resolution of each affine hypertoric variety $Y(A, 0)$. In general, for a (conical) symplectic variety and its crepant resolution, Namikawa showed the existence of the universal Poisson deformation space of them. We construct the universal Poisson deformation space of hypertoric varieties $Y(A, alpha)$ and $Y(A, 0)$ explicitly. We will explain this construction. In application, we can classify affine hypertoric varieties by the associated matroids. If time permits, we will also talk about applications to counting crepant resolutions of affine hypertoric varieties.

 

◆ Seminar Talk (1 hour) by Ryo Yamagishi (Kyoto)

◇ Title: Singular cubic fourfolds and their Fano schemes.

◇ Abstract: Beauville and Donagi showed that the moduli space (called the Fano scheme) of lines on a smooth cubic fourfold is a four-dimensional compact symplectic manifold. In this talk, I will discuss what happens if we replace the cubic fourfold by a mildly singular one. In particular, I will show that the corresponding Fano scheme has the same symplectic singularities as the Hilbert scheme of two points on ADE surface singularities.
 

 

 

▶ Organized by Jun-Muk Hwang (KIAS)

 

▶ Supported by National Researcher Program (NRF)

 

▶ Contact: Ms. Da-eun Kim (dekim@kias.re.kr)