Misha Verbitsky - July 20th, 2023

July 20th, 2023

July 20th, 2023
09:06 am


Brazil-China Joint Mathematical Meeting
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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
even-dimensional, which allows one to define the middle perversity,
constructing a self-dual t-structure 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 Beilinson-Bernstein-Deligne (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 semi-stable
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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