Misha Verbitsky  July 20th, 2023
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BrazilChina Joint Mathematical Meeting Я, кстати, тут вот https://sbm.org.br/jointmeetingchina/ сегодня вещаю
Perverse coherent sheaves on hyperkahler manifolds and Weil conjectures
Let (M, I,J,K) be a compact hyperkahler manifold, and L=aI+bJ+cK a general complex structure. All complex subvarieties of $(M,L)$ are evendimensional, which allows one to define the middle perversity, constructing a selfdual tstructure on its category of coherent sheaves. All coherent sheaves on (M,L) are semistable, and the category of coherent sheaves on (M,L) admits a full embedding to the category of coherent sheaves on any its deformation (and is essentially independent on the deformation), hence the notion of the "perverse coherent sheaf" makes sense on algebraic hyperkahler manifolds as well. Just like for the constructible sheaves, the perverse coherent sheaves are extensions of simple perverse coherent sheaves, which are always stable.
In BeilinsonBernsteinDeligne (BBD), the Weil conjectures were interpreted as a theorem about purity of a direct image of a pure perverse sheaf. Instead of fixing the Frobenius action, as in the BBD setup, we fix the lifting of the sheaf to the twistor space; such a lifting is unique for stable sheaves, and exists for all semistable sheaves. The role of the weight filtration is played by the O(i)filtration on the sheaf restricted to the rational curves in the twistor space. The hyperkahler version of "Weil conjectures" predicts that the weights are increased under derived direct images, and the direct images of pure perverse sheaves remain pure, which is actually true, at least in the smooth case.
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