arXiv subjects

Rodion N. Déev

Publications and source records attributed to Rodion N. Déev.

3 recordsLinked to original sources

arXiv2021

Haupt--Kapovich theorem revisited

A theorem of O. Haupt, rediscovered by M. Kapovich and celebrated by his proof invoking Ratner theory, describes the set of de Rham cohomology classes on a topological orientable surface, which can be realized by an abelian differential in some respective complex structure, in purely topological terms. We make an attempt to describe similarly pairs and triples of cohomology classes, which can be realized by abelian differentials in some complex structure. This leads us to some interesting problems in algebraic geometry of curves, and gives an unexpected local description of the Teichmüller space.

math.AG↗

arXiv2016

Cousin groups and Hodge structures

We consider the geometric properties of Hodge Cousin groups, introduced in an unpublished paper \cite{OVV}, emphasizing the case of Hodge Cousin groups corresponding to polarized $\mathbb{Q}$-Hodge structures. Basing on this consideration, we introduce the class of abelian Cousin groups and prove an analogue of Poincaré complete reducibility theorem for them.

math.DG↗

arXiv2016

Compact fibrations with hyperkähler fibers

Essential dimension of a family of complex manifolds is the dimension of the image of its base in the Kuranishi space of the fiber. We prove that any family of hyperkähler manifolds over a compact simply connected base has essential dimension not greater than $1$. A similar result about families of complex tori is also obtained.

math.DG↗