Related papers: Abelian varieties, homogeneous spaces and duality
A mixture of an historical article, and of a survey of recent developments, containing also a couple of new results.
The amoebas associated to algebraic varieties are certain concave regions in the Euclidean space whose shape reminds biological amoebas. This term was formally introduced to Mathematics in 1994 by Gelfand, Kapranov and Zelevinski. Some…
In this paper we introduce new affine algebraic varieties whose points correspond to associative algebras. We show that the algebras within a variety share many important homological properties. In particular, any two algebras in the same…
An increasingly important area of interest for mathematicians is the study of Abelian differentials. This growing interest can be attributed to the interdisciplinary role this subject plays in modern mathematics, as various problems of…
In this paper we study abelian varieties which correspond to CM points in the coarse moduli space of principally polarized abelian varieties with multiplication by a maximal order in a quaternion algebra over a totally real number field.…
In three dimensions, an abelian gauge field is related by duality to a free, periodic scalar field. Though usually considered on Euclidean space, this duality can be extended to a general three-manifold M, in which case topological features…
Let $V$ be a finite abelian group of odd order, equipped with a non-degenerate, alternating form $\omega\colon V\times V \to \mathbb{Z}/m\mathbb{Z}$. We give closed formulas for the character values of the Weil representation associated…
This is a survey paper about a selection of results in complex algebraic geometry that appeared in the recent and less recent litterature, and in which rational homogeneous spaces play a prominent r{\^o}le. This selection is largely…
The manuscript is an overview of the motivations and foundations lying behind Voevodsky's ideas of constructing categories similar to the ordinary topological homotopy categories. The objects of these categories are strictly related to…
This is an expository lecture, for the Abel bicentennial (Oslo, 2002), describing some recent work on the (small) quantum cohomology ring of Grassmannians and other homogeneous varieties.
We study derived equivalences of Abelian varieties in terms of their associated symplectic data. For simple Abelian varieties over an algebraically closed field of characteristic zero we prove that the natural correspondence introduced by…
Part I introduced diptych varieties $V_{ABLM}$ and gave a rigorous construction of them in the case $d,e\ge 2$ and $de>4$. Here we prove the existence of $V_{ABLM}$ in all the cases with $de\le4$. At the same time we construct some classes…
This is an expanded version of my talk given at the International Conference ``Algebra and Number Theory'' dedicated to the 80th anniversary of V. E. Voskresenskii, which was held at the Samara State University in May 2007. The goal is to…
We use tools of mathematical logic to analyse the notion of a path on an complex algebraic variety, and are led to formulate a "rigidity" property of fundamental groups specific to algebraic varieties, as well as to define a bona fide…
We provide two new characterizations of bounded orthogonally additive polynomials from a uniformly complete vector lattice into a convex bornological space using harmonic means and completely partitioned weighted geometric means. Our result…
We study projectively flat holomorphic vector bundles over Riemann surfaces. To each such bundle, we naturally assign a Wronskian line bundle. The main idea is a notion of the division of two meromorphic sections. Abel's identity is…
This is the first draft of a set of lecture notes developed for one-half of a seminar on two approaches to the notion of "Abelian", namely those of universal algebra, and of category theory. The half pertaining to the universal-algebraic…
If A and B are abelian varieties over a number field K such that there are non-trivial geometric homomorphisms of abelian varieties between reductions of A and B at most primes of K, then there exists a non-trivial (geometric) homomorphism…
The notion of acyclic matching property was provided by Losonczy and it was proved that torsion-free groups admit this property. In this paper, we introduce a duality of acyclic matching as a tool for classification of some Abelian groups,…
This paper studies a class of Abelian varieties that are of $\GL_2$-type and with isogenous classes defined over a number field $k$. We treat the cases when their endomorphism algebras are either (1) a totally real field $K$ or (2) a…