Introduction to Middle Intersection Cohomology and Perverse Sheaves

In this article we give a geometric construction of a tilting perverse sheaf on Drinfeld’s compactification, by applying the nearby cycles functor to a family of nondegenerate Whittaker sheaves. Its restrictions along the defect stratification are shown to be certain perverse sheaves attached to the nilpotent radical of the Langlands dual Lie algebra. We also describe the subquotients of the monodromy filtration using the Picard–Lefschetz oscillators introduced by Schieder. We give an argument that the subquotients are semisimple based on the action, constructed by Feigin, Finkelberg, Kuznetsov, and Mirković, of the Langlands dual Lie algebra on the global intersection cohomology of quasimaps into flag varieties.

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Polynomial identities related to special Schubert varieties

Applicable Algebra in Engineering Communication and Computing ◽

10.1007/s00200-021-00496-6 ◽

2021 ◽

Author(s):

Francesca Cioffi ◽

Davide Franco ◽

Carmine Sessa

Keyword(s):

Arbitrary Number ◽

Schubert Variety ◽

Decomposition Theorem ◽

Point Of View ◽

Explicit Description ◽

Intersection Cohomology ◽

Schubert Varieties ◽

Polynomial Identities ◽

Poincaré Polynomial

AbstractLet $$\mathcal S$$ S be a single condition Schubert variety with an arbitrary number of strata. Recently, an explicit description of the summands involved in the decomposition theorem applied to such a variety has been obtained in a paper of the second author. Starting from this result, we provide an explicit description of the Poincaré polynomial of the intersection cohomology of $$\mathcal S$$ S by means of the Poincaré polynomials of its strata, obtaining interesting polynomial identities relating Poincaré polynomials of several Grassmannians, both by a local and by a global point of view. We also present a symbolic study of a particular case of these identities.

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Explicit Models for Perverse Sheaves, II

Algebras and Representation Theory ◽

10.1007/s10468-007-9058-1 ◽

2007 ◽

Vol 11 (2) ◽

pp. 149-178 ◽

Cited By ~ 1

Author(s):

F. Gudiel-Rodríguez ◽

L. Narváez-Macarro

Keyword(s):

Perverse Sheaves

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Perverse sheaves on rank stratifications

Duke Mathematical Journal ◽

10.1215/s0012-7094-99-09609-6 ◽

1999 ◽

Vol 96 (2) ◽

pp. 317-362 ◽

Cited By ~ 18

Author(s):

Tom Braden ◽

Mikhail Grinberg

Keyword(s):

Perverse Sheaves

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Donaldson–Thomas invariants versus intersection cohomology of quiver moduli

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2017-0010 ◽

2019 ◽

Vol 2019 (754) ◽

pp. 143-178 ◽

Cited By ~ 5

Author(s):

Sven Meinhardt ◽

Markus Reineke

Keyword(s):

Moduli Space ◽

Stability Condition ◽

Intersection Cohomology ◽

Quiver Representations ◽

Compactly Supported ◽

Generic Stability ◽

Relative Version ◽

Zero Potential ◽

Stable Locus

Abstract The main result of this paper is the statement that the Hodge theoretic Donaldson–Thomas invariant for a quiver with zero potential and a generic stability condition agrees with the compactly supported intersection cohomology of the closure of the stable locus inside the associated coarse moduli space of semistable quiver representations. In fact, we prove an even stronger result relating the Donaldson–Thomas “function” to the intersection complex. The proof of our main result relies on a relative version of the integrality conjecture in Donaldson–Thomas theory. This will be the topic of the second part of the paper, where the relative integrality conjecture will be proven in the motivic context.

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