K(pi,1) via stability conditions

Michael Wemyss (University of Glasgow)

Wednesday 20th November, 2019 16:00-17:00 Maths 311B


Usually in algebraic geometry settings we are interested in (1) autoequivalences, and (2) establishing contractibility of the stability manifold.  Actually, in practice (2) is not strictly true, as it depends on which subcategory and on which component we choose.  Depending on the choices, the difficulty of the problem varies wildly.  I will explain how to approach both (1) and (2) for the situation of flops, which are certain birational surgeries, using contraction algebras, and will motivate this from the viewpoint of mirror symmetry.   The derived category of a contraction algebra is the "source" category of a spherical functor and it behaves exceptionally well.  Very easily we can describe its stability manifold as a covering space, and contractibility comes for free.  There are three main corollaries: K(pi,1) for all intersection arrangements in ADE root systems (which includes ADE braid groups, and the Coxeter groups I_n with n=3,4,5,6,8), plus faithfulness of group actions in various settings (the first avoiding normal forms), plus contractibility of stability manifolds in some 3-CY settings.  This is joint work with Jenny August.

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