We give bounds for various homological invariants (including Castelnuovo-Mumford regularity, degrees of local cohomology, and injective dimension) of finitely generated VI-modules in the non-describing characteristic case. It turns out that the formulas of these bounds for VI-modules are the same as the formulas of corresponding bounds for FI-modules.

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Let$S$be a closed Shimura variety uniformized by the complex$n$-ball associated with a standard unitary group. The Hodge conjecture predicts that every Hodge class in $${H^{2k} (S, \mathbb{Q})}$$ , $${k=0,\dots, n}$$ , is algebraic. We show that this holds for all degrees$k$away from the neighborhood $${\bigl]\tfrac13n,\tfrac23n\bigr[}$$ of the middle degree. We also prove the Tate conjecture for the same degrees as the Hodge conjecture and the generalized form of the Hodge conjecture in degrees away from an interval (depending on the codimension$c$of the subvariety) centered at the middle dimension of S. These results are derived from a general theorem that applies to all Shimura varieties associated with standard unitary groups of any signature. The proofs make use of Arthur’s endoscopic classification of automorphic representations of classical groups. As such our results rely on the stabilization of the trace formula for the (disconnected) groups $${GL (N) \rtimes \langle \theta \rangle}$$ associated with base change.

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Homological Projective duality (HP-duality) theory, introduced by Kuznetsov \cite{Kuz07HPD}, is one of the most powerful frameworks in the homological study of algebraic geometry. The main result (HP-duality theorem) of the theory gives complete descriptions of bounded derived categories of coherent sheaves of (dual) linear sections of HP-dual varieties. We show the theorem also holds for more general intersections beyond linear sections. More explicitly, for a given HP-dual pair $(X,Y)$, then analogue of HP-duality theorem holds for their intersections with another HP-dual pair $(S,T)$, provided that they intersect properly. We also prove a relative version of our main result. Taking $(S,T)$ to be dual linear subspaces (resp. subbundles), our method provides a more direct proof of the original (relative) HP-duality theorem.