Revisiting mixed geometry

Quoc P. Ho Department of Mathematics, Hong Kong University of Science and Technology Penghui Li YMSC, Tsinghua University

Algebraic Geometry mathscidoc:2205.45012

We provide a uniform construction of “mixed versions” or “graded lifts” in the sense of Beilinson-Ginzburg–Soergel which works for arbitrary Artin stacks. In particular, we obtain a general construction of graded lifts of many categories arising in geometric representation theory and categorified knot invariants. Our new theory associates to each Artin stack of finite type Y over F_q a symmetric monoidal DG-category Shv_{gr,c}(Y) of constructible graded sheaves on Y along with the six-functor formalism, a perverse t-structure, and a weight (or co-t-)structure in the sense of Bondarko and Pauksztello. This category sits in between the category Shv_{m,c}(Y_n) of constructible mixed ℓ-adic sheaves in the sense of Beilinson–Bernstein–Deligne--Gabber for any F_{q^n} -form Y_n of Y and the category Shv_c(Y) of constructible ℓ-adic sheaves on Y, compatible with the six-functor formalism, perverse t-structures, and Frobenius weights. Classically, mixed versions were only constructed in very special cases. However, the category Shv_{gr,c}(Y) agrees with those previously constructed when they are available. For example, for any reductive group G with a fixed pair T ⊂ B of a maximal torus and a Borel subgroup, we have an equivalence of monoidal DG weight categories Shv_{gr,c}(B\G/B) ≃ Ch^b(SBim_W), where Ch^b(SBim_W) is the monoidal DG-category of bounded chain complexes of Soergel bimodules and W is the Weyl group of G.
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  title={Revisiting mixed geometry},
  author={Quoc P. Ho, and Penghui Li},
Quoc P. Ho, and Penghui Li. Revisiting mixed geometry. 2022.
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