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# Cohomology of abelian varieties

### Crystalline cohomology of abelian varieties - MathOverflo

To add a bit more to Brian's comment: the crystalline cohomology of an abelian variety (over a finite field of characteristic p, say) is canonically isomorphic to the Dieudonné module of the p-divisible group of the abelian variety (which is a finite free module over the Witt vectors of the field with a semi-linear Frobenius). If you start with an abelian scheme over the Witt vectors of this field then the crystalline cohomology of the special fibre is canonically isomorphic to the. We define a subclass of Hessenberg varieties called abelian Hessenberg varieties, inspired by the theory of abelian ideals in a Lie algebra developed by Kostant and Peterson. We give an inductive formula for the -representation on the cohomology of an abelian regular semisimple Hessenberg variety with respect to the action defined by Tymoczko

### The cohomology of abelian Hessenberg varieties and the

Let κbe a ﬁeld and let X be an abelian variety over κ. One can then associate to X another abelian variety Xp, called the dual of X, with the following features: piqThe dual pX can be identiﬁed with (or deﬁned as) the identity component PicpXq of the Picard variety PicpXq following: after passing to some ﬁnite extension of K, the abelian variety Ais isogenous to a principally polarizedone;moreover,thisconjectureisinsensitivetobasechangeandtheconjecturesfortwoisogenous abelianvarietiesareequivalent. Theorem 2.1.6 ([Del82, Theorem 2.11], [Ogu82, Theorem 4.14], [Bla94]). For any abelian variety, ever The cohomology of abelian Hessenberg varieties and the Stanley-Stembridge conjecture Megumi Harada, Martha Precup We define a subclass of Hessenberg varieties called abelian Hessenberg varieties, inspired by the theory of abelian ideals in a Lie algebra developed by Kostant and Peterson Cohomology of structure sheaf of abelian variety. Let X be an abelian variety over C of dimension n. Consider the structure sheaf O X. It's Euler characteristic is zero, because χ ( O X) = ( O X n) / n!. And the self intersection of O X is 0 of the basic theory of abelian varieties: for ' 6= p, ´et 1 (X)' »= Z2g l, obtained by passing to the limit under the Galois groups of ['n] : A = J(X)! A = J(X), the multiplication by elln maps. 1.2unless ' is the characteristic of k If we look instead at the p-adic cohomology groups Hi(X;Zp) of a (smooth

### Duality for Cohomology of Curves With Coefficients in

• We show that the infinitesimal deformations of Brill-Noether loci Wd attached to a smooth non-hyperelliptic curve C are in one-to-ne correspondence with the deformations of C. As an application, we..
• We give a formula for the Eisenstein cohomology of local systems on the partial compactification of the moduli of principally polarized abelian varieties given by rank 1 degenerations
• such an abelian variety, which acts as identity on a certain quotient of its middle singular cohomology, then it acts as identity on the deepest part of this ltration on the Chow group of 0-cycles of the abelian variety. As an application, we prove that for the generalized Kummer variety associated to a complex abelian surface and the automorphism induced from a symplectic automorphism of the.
• l-Arlie Etale Cohomology of PEL Type Shimura Varieties with N on-Trivial Coefficients Elena Mantovan Mathematics 253-37 Caltech Pasadena, CA 91125 USA mantovan~caltech.edu Abstract. Given a Shimura datum (C, h) of PEL type, let p be an odd prime at which G is unramified. In [13], we established a formula computing the l-adic cohomology of the associated Shimura varieties (regarded as a.

### Étale cohomology and reduction of abelian varieties - NASA/AD

• abelian varieties a fertile testing ground and collection of examples for many im-portant phenomena in algebraic geometry, including the study of line bundles, the Hodge theory on the singular cohomology, questions of embeddings into projective spaces, groups of automorphisms, among numerous other areas of study. Thus, through the study of abelian varieties, many of the fundamental properties.
• Abelian varieties are a natural generalization of elliptic curves to higher dimensions, whose geometry and classification are as rich in elegant results as in the one-dimensional ease. The use of theta functions, particularly since Mumford's work, has been an important tool in the study of abelian varieties and invertible sheaves on them. Also, abelian varieties play a significant role in the.
• Introduction to abelian varieties Rigidity lemma and applications Theorems of the cube and square Isogenies The Tate module Duality, étale cohomology, fundamental group Serre-Tate Néron models Something about local fields References . S. Bosch, W. Lütkebohmert, M. Raynaud. Néron Models. Springer (1990) G. van der Geer, B. J. J. Moonen.
• Rational cohomology tori and the Chow ring. A complex algebraic variety X is called a rational cohomology torus if X is normal and In [3], the authors studied properties of rational cohomology tori. They showed that if X is a rational cohomology torus, then there exists a finite cover to an abelian variety such that ⁎ is an isomorphism
• The cohomology of an abelian variety, or any projective complex manifold, is the -module part of a -Hodge structure for every (I am not sure if this works for homology when it is for non-abelian varieties). I didn't talk about this because it would be harder work to motivate. In the case of abelian varieties, all the information is already in the
• Rham cohomology that works better for singular varieties; the difference, roughly, is the replacement of the cotan-gent sheaf with the cotangent complex. Theorems from [Bha12] show: (a) derived de Rham cohomology agrees with crystalline cohomology for lci varieties, and (b) derived de Rham cohomology is computed by a conjugate spectra
• On the vanishing of weight one Koszul cohomology of abelian varieties MPG-Autoren Aprodu, Marian Max Planck Institute for Mathematics, Max Planck Society; Externe Ressourcen Es sind keine externen Ressourcen hinterlegt. Volltexte (frei zugänglich) Aprodu_On the vanishing_oa_2016.pdf.

### The cohomology of the moduli space of abelian varieties

• This follows from [KrW], [W], and for this assertion we have to assume that the perverse sheaf K is defined over a finitely generated field over the prime field in the case of po
• Fundamentals of (Abelian) Group Cohomology Hard Arithmeti
• Abelian Varieties - William Stein's Homepag
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