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Register a new userInfinitesimal Bloch regulator
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Authors
Sinan Unver
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In this paper, we continue our project of defining and studying the infinitesim
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This text is based on a talk by the first named author at the first congress of the SMF (Tours, 2016). We present Blochs conductor formula, which is a conjectural formula describing the change of topology in a family of algebraic varieties when the parameter specialises to a critical value. The main objective of this paper is to describe a general approach to the resolution of Blochs conjecture based on techniques from both non-commutative geometry and derived geometry.
Consider a Frobenius kernel G in a split semisimple algebraic group, in very good characteristic. We provide an analysis of support for the Drinfeld center Z(rep(G)) of the representation category for G, or equivalently for the representation category of the Drinfeld double of kG. We show that thick ideals in the corresponding stable category are classified by cohomological support, and calculate the Balmer spectrum of the stable category of Z(rep(G)). We also construct a $pi$-point style rank variety for the Drinfeld double, identify $pi$-point support with cohomological support, and show that both support theories satisfy the tensor product property. Our results hold, more generally, for Drinfeld doubles of Frobenius kernels in any smooth algebraic group which admits a quasi-logarithm, such as a Borel subgroup in a split semisimple group in very good characteristic.
The purpose of this article is to show that the Castelnuovo theory for abelian varieties, developed by G. Pareschi and M. Popa, can be infinitesimalized. More precisely, we prove that an irreducible principally polarized abelian variety has a finite scheme in extremal position, in the sense of Castelnuovo theory for abelian varieties, if, and only if, it is a Jacobian and the scheme is contained in a unique Abel-Jacobi curve.
In this paper we describe the infinitesimal deformations of null-filiform Leibniz superalgebras over a field of zero characteristic. It is known that up to isomorphism in each dimension there exist two such superalgebras $NF^{n,m}$. One of them is a Leibniz algebra (that is $m=0$) and the second one is a pure Leibniz superalgebra (that is $m eq 0$) of maximum nilindex. We show that the closure of union of orbits of single-generated Leibniz algebras forms an irreducible component of the variety of Leibniz algebras. We prove that any single-generated Leibniz algebra is a linear integrable deformation of the algebra $NF^{n}$. Similar results for the case of Leibniz superalgebras are obtained.
We describe infinitesimal deformations of complex naturally graded filiform Leibniz algebras. It is known that any $n$-dimensional filiform Lie algebra can be obtained by a linear integrable deformation of the naturally graded algebra $F_n^3(0)$. We establish that in the same way any $n$-dimensional filiform Leibniz algebra can be obtained by an infinitesimal deformation of the filiform Leibniz algebras $F_{n}^1,$ $F_{n}^2$ and $F_{n}^3(alpha)$. Moreover, we describe the linear integrable deformations of above-mentioned algebras with a fixed basis of $HL^2$ in the set of all $n$-dimensional Leibniz algebras. Among these deformations we found one new rigid algebra.
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