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Register a new userCoherent sheaves on the stack of Langlands parameters
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Xinwen Zhu
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We formulate a few conjectures on some hypothetical coherent sheaves on the stacks of arithmetic local Langlands parameters, including their roles played in the local-global compatibility in the Langlands program. We survey some known results as evidences of these conjectures.
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The central aim of this monograph is to provide decomposition results for quasi-coherent sheaves on the moduli stack of one-dimensional formal groups. These results will be based on the geometry of the stack itself, particularly the height filtration and an analysis of the formal neighborhoods of the geometric points. The main theorems are algebraic chromatic convergence results and fracture square decompositions. There is a major technical hurdle in this story, as the moduli stack of formal groups does not have the finitness properties required of an algebraic stack as usually defined. This is not a conceptual problem, but in order to be clear on this point and to write down a self-contained narrative, I have included a great deal of discussion of the geometry of the stack itself, giving various equivalent descriptions.
The goal of this paper is to start a study of aCM and Ulrich sheaves on non-integral projective varieties. We show that any aCM vector bundle of rank two on the double plane is a direct sum of line bundles. As a by-product, any aCM vector bundle of rank two on a sufficiently high dimensional quadric hypersurface also splits. We consider aCM and Ulrich vector bundles on a multiple hyperplanes and prove the existence of such bundles that do not split, if the multiple hyperplane is linearly embedded into a sufficiently high dimensional projective space. Then we restrict our attention to the double plane and give a classification of aCM sheaves of rank at most $3/2$ on the double plane and describe the family of isomorphism classes of them.
Let $G=Sp_{2n}(mathbb{C})$, and $mathfrak{N}$ be Katos exotic nilpotent cone. Following techniques used by Bezrukavnikov in [5] to establish a bijection between $Lambda^+$, the dominant weights for a simple algebraic group $H$, and $textbf{O}$, the set of pairs consisting of a nilpotent orbit and a finite-dimensional irreducible representation of the isotropy group of the orbit, we prove an analogous statement for the exotic nilpotent cone. First we prove that dominant line bundles on the exotic Springer resolution $widetilde{mathfrak{N}}$ have vanishing higher cohomology, and compute their global sections using techniques of Broer. This allows to show that the direct images of these dominant line bundles constitute a quasi-exceptional set generating the category $D^b(Coh^G(mathfrak{N}))$, and deduce that the resulting $t$-structure on $D^b(Coh^G(mathfrak{N}))$ coincides with the perverse coherent $t$-structure. The desired result now follows from the bijection between costandard objects and simple objects in the heart of this $t$-structure on $D^b(Coh^G(mathfrak{N}))$.
We state a conjecture that relates the derived category of smooth representations of a p-adic split reductive group with the derived category of (quasi-)coherent sheaves on a stack of L-parameters. We investigate the conjecture in the case of the principal block of GL_n by showing that the functor should be given by the derived tensor product with the family of representations interpolating the modified Langlands correspondence over the stack of L-parameters that is suggested by the work of Helm and Emerton-Helm.
More than four decades ago, Eisenbud, Khimv{s}iav{s}vili, and Levine introduced an analogue in the algebro-geometric setting of the notion of local degree from differential topology. Their notion of degree, which we call the EKL-degree, can be thought of as a refinement of the usual notion of local degree in algebraic geometry that works over non-algebraically closed base fields, taking values in the Grothendieck-Witt ring. In this note, we compute the EKL-degree at the origin of certain finite covers $fcolon mathbb{A}^nto mathbb{A}^n$ induced by quotients under actions of Weyl groups. We use knowledge of the cohomology ring of partial flag varieties as a key input in our proofs, and our computations give interesting explicit examples in the field of $mathbb{A}^1$-enumerative geometry.
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