scholarly journals ON VANISHING THEOREMS FOR LOCAL SYSTEMS ASSOCIATED TO LAURENT POLYNOMIALS

2017 ◽  
Vol 231 ◽  
pp. 1-22 ◽  
Author(s):  
ALEXANDER ESTEROV ◽  
KIYOSHI TAKEUCHI
Keyword(s):  
Morse Theory ◽  
Hypergeometric Functions ◽  
Vanishing Theorems ◽  
Cohomology Groups ◽  
Laurent Polynomials ◽  
Perverse Sheaves ◽  
Local Systems ◽  
Vanishing Cycles

We prove some vanishing theorems for the cohomology groups of local systems associated to Laurent polynomials. In particular, we extend one of the results of Gelfand et al.[Generalized Euler integrals and$A$-hypergeometric functions, Adv. Math.84(1990), 255–271] to various directions. In the course of the proof, some properties of vanishing cycles of perverse sheaves and twisted Morse theory are used.

scholarly journals Conjecture de monodromie-poids pour quelques variétés de Shimura unitaires

2010 ◽  
Vol 146 (2) ◽  
pp. 367-403 ◽  
Author(s):  
Pascal Boyer
Keyword(s):  
Explicit Description ◽  
Shimura Varieties ◽  
Perverse Sheaves ◽  
Automorphic Representations ◽  
Local Systems ◽  
Langlands Correspondence ◽  
Princeton University ◽  
Vanishing Cycles ◽  
Local Langlands Correspondence ◽  
Invent Math

AbstractIn Boyer [Monodromy of perverse sheaves on vanishing cycles on some Shimura varieties, Invent. Math. 177 (2009), 239–280 (in French)], a sheaf version of the monodromy-weight conjecture for some unitary Shimura varieties was proved by giving explicitly the monodromy filtration of the complex of vanishing cycles in terms of local systems introduced in Harris and Taylor [The geometry and cohomology of some simple Shimura varieties (Princeton University Press, Princeton, NJ, 2001)]. The main result of this paper is the cohomological version of the monodromy-weight conjecture for these Shimura varieties, which we prove by means of an explicit description of the groups of cohomology in terms of automorphic representations and the local Langlands correspondence.

scholarly journals Topology of Subvarieties of Complex Semi-abelian Varieties

2020 ◽  
Author(s):  
Yongqiang Liu ◽  
Laurentiu Maxim ◽  
Botong Wang
Keyword(s):  
Euler Characteristic ◽  
Morse Theory ◽  
Characteristic Property ◽  
Abelian Varieties ◽  
Betti Numbers ◽  
Local Systems ◽  
Affine Manifolds ◽  
Cyclic Covers ◽  
Rank One ◽  
Generic Vanishing

Abstract We use the non-proper Morse theory of Palais–Smale to investigate the topology of smooth closed subvarieties of complex semi-abelian varieties and that of their infinite cyclic covers. As main applications, we obtain the finite generation (except in the middle degree) of the corresponding integral Alexander modules as well as the signed Euler characteristic property and generic vanishing for rank-one local systems on such subvarieties. Furthermore, we give a more conceptual (topological) interpretation of the signed Euler characteristic property in terms of vanishing of Novikov homology. As a byproduct, we prove a generic vanishing result for the $L^2$-Betti numbers of very affine manifolds. Our methods also recast June Huh’s extension of Varchenko’s conjecture to very affine manifolds and provide a generalization of this result in the context of smooth closed sub-varieties of semi-abelian varieties.

scholarly journals Vanishing theorems for perverse sheaves on abelian varieties, revisited

2017 ◽  
Vol 24 (1) ◽  
pp. 63-84 ◽  
Author(s):  
Bhargav Bhatt ◽  
Christian Schnell ◽  
Peter Scholze
Keyword(s):  
Abelian Varieties ◽  
Vanishing Theorems ◽  
Perverse Sheaves

scholarly journals A generalization of Kita and Noumi’s vanishing theorems of cohomology groups of local system

1997 ◽  
Vol 147 ◽  
pp. 63-69 ◽  
Author(s):  
Koji Cho
Keyword(s):  
Algebraic Geometry ◽  
Spectral Sequence ◽  
Local System ◽  
Vanishing Theorems ◽  
Vanishing Theorem ◽  
Cohomology Groups ◽  
De Rham

AbstractWe prove vanishing theorems of cohomology groups of local system, which generalize Kita and Noumi’s result and partially Aomoto’s result. Main ingredients of our proof are the Hodge to de Rham spectral sequence and Serre’s vanishing theorem in algebraic geometry.

scholarly journals Basic hypergeometric functions and orthogonal Laurent polynomials

2012 ◽  
Vol 140 (6) ◽  
pp. 2075-2089 ◽  
Author(s):  
Marisa S. Costa ◽  
Eduardo Godoy ◽  
Regina L. Lamblém ◽  
A. Sri Ranga
Keyword(s):  
Hypergeometric Functions ◽  
Laurent Polynomials ◽  
Orthogonal Laurent Polynomials ◽  
Basic Hypergeometric Functions

WHEN ARE THERE ENOUGH PROJECTIVE PERVERSE SHEAVES?

2021 ◽  
pp. 1-12
Author(s):  
ALESSIO CIPRIANI ◽  
JON WOOLF
Keyword(s):  
Finite Dimensional Algebra ◽  
Perverse Sheaves ◽  
Dimensional Algebra ◽  
Local Systems ◽  
Finite Dimensional ◽  
Stratified Space ◽  
Main Component ◽  
Projective Covers

Abstract Let X be a topologically stratified space, p be any perversity on X and k be a field. We show that the category of p-perverse sheaves on X, constructible with respect to the stratification and with coefficients in k, is equivalent to the category of finite-dimensional modules over a finite-dimensional algebra if and only if X has finitely many strata and the same holds for the category of local systems on each of these. The main component in the proof is a construction of projective covers for simple perverse sheaves.

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