Differential Geometry and Topology

course: Topology of semismall maps and applications to hyperKahler varieties

speaker: Luca Migliorini (Università di Bologna)

abstract: I will discuss the applications of a recent general result due to M.A. de Cataldo and myself to the topology of (not necessarily compact) HK manifolds. Several families of HK manifolds, such as Hilebrt schemes of points on a surface, or Nakajima quiver varieties, or the recent Hyperkahler toric varieties of Bielawski-Dancer, are endowed with a contraction to a singular variety. Such contractions satisfy an apparently mysterious `semismall estimate' on the dimension of the fibres. Associated to a stratification of the contraction map is a series of bilinear forms, one for each stratum: the intersection form on the fibre of a generic point in the stratum computed inside a transversal slice. The result quoted above states that these forms are definite, with sign depending on the codimension of the stratum. We will discuss several examples of HK contraction, a conceptual explanation of the semismall estimate, and (if time permits) sketch the proof of definiteness of the forms in the compact case, which is much more elementary.

timetable:
Thu 16 Sep, 17:00 - 18:15, Sala Conferenze Centro De Giorgi
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