The Néron component series of an abelian variety

We introduce the Néron component series of an abelian variety A over a complete discretely valued field. This is a power series in ${\mathbb{Z}}\left[\left[T\right]\right]$ , which measures the behaviour of the number of components of the Néron model of A under tame ramification of the base field. If A is tamely ramified, then we prove that the Néron component series is rational. It has a pole at T = 1, whose order equals one plus the potential toric rank of A. This result is a crucial ingredient of our proof of the motivic monodromy conjecture for abelian varieties. We expect that it extends to the wildly ramified case; we prove this if A is an elliptic curve, and if A has potential purely multiplicative reduction.     

Néron Models and Formal Groups

We show that formal groups can be used to simplify the construction of Néron models. Also we give a new proof of the stable reduction theorem for abelian varieties. Received: September 2007     

Quantitative Néron theory for torsion bundles

Let R be a discrete valuation ring with algebraically closed residue field, and consider a smooth, geometrically connected, and projective curve C _K over the field of fractions K. For any positive integer r prime to the residual characteristic, we consider the finite K-group scheme of r-torsion line bundles on C _K . We determine when there exists a finite R-group scheme, which is a model of over R; in other words, we establish when the Néron model of is finite. The obvious idea would be to study the points of the Néron model over R, but in general these do not represent r-torsion line bundles on a semistable reduction of C _K . Instead, we recast the notion of models on a stack-theoretic base: there, we find finite Néron models, which represent r-torsion line bundles on a stack-theoretic semistable reduction of C _K . This allows us to quantify the lack of finiteness of the classical Néron models and finally to provide an efficient criterion for it. A. Chiodo was financially supported by the Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme, MEIF-CT-2003-501940.     

Rational points of the group of components¶of a Néron model

The structure of component groups of Néron models has been investigated on several occasions. Here we admit non-separably closed residue fields and are interested in the subgroup of rational points or, in other terms, in the subgroup of geometrically connected components of a Néron model. We consider Néron models of abelian varieties and of algebraic tori and give detailed computations in the case of Jacobians of curves. Received: 4 May 1998 / Revised version: 12 October 1998     

Complex analytic Néron models for arbitrary families of intermediate Jacobians

Given a family of intermediate Jacobians (for a polarizable variation of integral Hodge structure of odd weight) on a Zariski-open subset of a complex manifold, we construct an analytic space that naturally extends the family. Its two main properties are: (a) the horizontal and holomorphic sections are precisely the admissible normal functions without singularities; (b) the graph of any admissible normal function has an analytic closure inside our space. As a consequence, we obtain a new proof for the zero locus conjecture of M. Green and P. Griffiths. The construction uses filtered D\mathcal {D}-modules and M. Saito’s theory of mixed Hodge modules; it is functorial, and does not require normal crossing or unipotent monodromy assumptions.     

Fréchet algebras of finite type

The main objects of study in this paper are Fréchet algebras having an Arens Michael representation in which every Banach algebra is finite dimensional. We shall classify these algebras using a theorem which ensures that the image of any continuous linear map of a Fréchet space of finite type (i.e., for which the defining seminorms have a finite dimensional cokernel) into any Fréchet space is in fact closed.This work is part of the research project of the European Research Training Network Analysis and Operators, contract HPRN-CT 2000 00116, funded by the European Commission.     

On the Standard Conjecture for Compactifications of Néron Models of Three-Dimensional Abelian Varieties with Multiplications in an Imaginary Quadratic Field

Mathematical Notes -     

Lie algebras of vector fields on smooth affine varieties

We reprove the results of Jordan [18 Jordan, D. (1986). On the ideals of a Lie algebra of derivations. J. London Math. Soc. 33:33–39.[Crossref], [Web of Science ®] , [Google Scholar]] and Siebert [30 Siebert, T. (1996). Lie algebras of derivations and a?ne algebraic geometry over fields of characteristic 0. Math. Ann. 305:271–286.[Crossref], [Web of Science ®] , [Google Scholar]] and show that the Lie algebra of polynomial vector fields on an irreducible a?ne variety X is simple if and only if X is a smooth variety. Given proof is self-contained and does not depend on papers mentioned above. Besides, the structure of the module of polynomial functions on an irreducible smooth a?ne variety over the Lie algebra of vector fields is studied. Examples of Lie algebras of polynomial vector fields on an N-dimensional sphere, non-singular hyperelliptic curves and linear algebraic groups are considered.     

Cyclic cohomology of certain nuclear Fréchet algebras and DF algebras

We give explicit formulae for the continuous Hochschild and cyclic homology and cohomology of certain \(\hat \otimes\)-algebras. We use well-developed homological techniques together with some niceties of the theory of locally convex spaces to generalize the results known in the case of Banach algebras and their inverse limits to wider classes of topological algebras. To this end we show that, for a continuous morphism ?: x → y of complexes of complete nuclear DF-spaces, the isomorphism of cohomology groups H ⁿ(?): H ⁿ (x) → H ⁿ (y) is automatically topological. The continuous cyclic-type homology and cohomology are described up to topological isomorphism for the following classes of biprojective \(\hat \otimes\)-algebras: the tensor algebra E \(\hat \otimes\) F generated by the duality (E,F,<·,·>) for nuclear Fréchet spaces E and F or for nuclear DF-spaces E and F; nuclear biprojective Köthe algebras λ(P) which are Fréchet spaces or DF-spaces; the algebra of distributions ε^*(G) on a compact Lie group G.     

A variant of N��ron models over curves

We study a variant of the Néron models over curves which has recently been found by the second named author in a more general situation using the theory of Hodge modules. We show that its identity component is a certain open subset of an iterated blow-up along smooth centers of the Zucker extension of the family of intermediate Jacobians and that the total space is a complex Lie group over the base curve and is Hausdorff as a topological space. In the unipotent monodromy case, the image of the map to the Clemens extension coincides with the Néron model defined by Green, Griffiths and Kerr. In the case of families of Abelian varieties over curves, it coincides with the Clemens extension, and hence with the classical Néron model in the algebraic case (even in the non-unipotent monodromy case).     

Modèles semi-factoriels et modèles de Néron

Let S be the spectrum of a discrete valuation ring with function field K. Let X be a scheme over S. We will say that X is semi-factorial over S if any invertible sheaf on the generic fiber X _K can be extended to an invertible sheaf on X. Here we show that any proper geometrically normal scheme over K admits a proper, flat, normal and semi-factorial model over S. We also construct some semi-factorial compactifications of regular S-schemes, such as Néron models of abelian varieties. The semi-factoriality property for a scheme X/S corresponds to the Néron property of its Picard functor. In particular, one can recover the Néron model of the Picard variety ${{\rm Pic}_{X_K/K,{\rm red}}^0}$ of X _K from the Picard functor Pic _X/S , as in the known case of curves. This provides some information on the relative algebraic equivalence on the S-scheme X.     

On simplicial commutative algebras with vanishing André-Quillen homology

In this paper, we study the André-Quillen homology of simplicial commutative ℓ-algebras, ℓ a field, having certain vanishing properties. When ℓ has non-zero characteristic, we obtain an algebraic version of a theorem of J.-P. Serre and Y. Umeda that characterizes such simplicial algebras having bounded homotopy groups. We further discuss how this theorem fails in the rational case and, as an application, indicate how the algebraic Serre theorem can be used to resolve a conjecture of D. Quillen for algebras of finite type over Noetherian rings, having non-zero characteristic. Oblatum 3-III-1999 & 3-V-2000?Published online: 11 October 2000     

The classification of N-dimensional non-associative Jordan algebras with (N − 3)-dimensional annihilator

The paper is devoted to give a complete classification of all n-dimensional non-associative Jordan algebras with (n?3)-dimensional annihilator over an algebraically closed field of characteristic ≠2. We also give a complete classification of all n-dimensional Jordan algebras with (n?1)- and (n?2)-dimensional annihilator.     

The first cohomology of N = 1 super Virasoro algebras

Abstract

In this paper, we determine the first cohomology groups of centerless N?=?1 super Virasoro algebras with coefficients in their tensor density modules. As applications, we classify all central extensions and derivations of the complete spectrum-generating superalgebra.     

Polynomial clones of Mal’cev algebras with small congruence lattices

We show that the polynomials of every finite Mal’cev algebra with congruence lattice of height at most 2 can be described by a finite set of relations.     

Artin–Schelter regular algebras of dimension five with two generators

We study and classify Artin–Schelter regular algebras of dimension five with two generators under an additional Z²${\mathbb{Z}}^2$-grading by Hilbert driven Gröbner basis computations. All the algebras we obtained are strongly noetherian, Auslander regular, and Cohen–Macaulay. One of the results provides an answer to Fløystad–Vatne?s question in the context of Z²${\mathbb{Z}}^2$-grading. Our results also achieve a connection between Lyndon words and Artin–Schelter regular algebras.     

Growth of generalized Temperley–Lieb algebras connected with simple graphs

We prove that the generalized Temperley–Lieb algebras associated with simple graphs Γ have linear growth if and only if the graph Γ coincides with one of the extended Dynkin graphs [(A)\tilde]_n {\tilde A_n} , [(D)\tilde]_n {\tilde D_n} , [(E)\tilde]₆ {\tilde E_6} , or [(E)\tilde]₇ {\tilde E_7} . An algebra TL_{G, t} T{L_{\Gamma, \tau }} has exponential growth if and only if the graph Γ coincides with none of the graphs A_n {A_n} , D_n {D_n} , E_n {E_n} , [(A)\tilde]_n {\tilde A_n} , [(D)\tilde]_n {\tilde D_n} , [(E)\tilde]₆ {\tilde E_6} , and [(E)\tilde]₇ {\tilde E_7} .