Related papers: Rationality in Differential Algebraic Geometry

We construct a cohomology theory controlling the deformations of a general Drinfel'd algebra. The picture presented here has two sides -- the combinatorial one related with the fact of the existence of a graded Lie algebra structure on the…

We prove analogues of several well-known results concerning rational morphisms between quadrics for the class of so-called quasilinear $p$-hypersurfaces. These hypersurfaces are nowhere smooth over the base field, so many of the geometric…

We introduce the notion of a manifold admitting a simple compact Cartan 3-form $\om^3$. We study algebraic types of such manifolds specializing on those having skew-symmetric torsion, or those associated with a closed or coclosed 3-form…

We prove that the set of orthogonal separable coordinates on an arbitrary (pseudo-)Riemannian manifold carries a natural structure of a projective variety, equipped with an action of the isometry group. This leads us to propose a new,…

We show a possibility to apply certain philosophical concepts to the analysis of concrete mathematical structures. Such application gives a clear justification of topological and geometric properties of considered mathematical objects.

A Stein covering of a complex manifold may be used to realise its analytic cohomology in accordance with the Cech theory. If, however, the Stein covering is parameterised by a smooth manifold rather than just a discrete set, then we…

We present alternative postulates for Euclidean geometry whose merit is that they lead to a new class of invariants and associated geometries for real finite-dimensional unital associative algebras.

We give an elementary characterization of rational functions among meromorphic functions in the complex plane.

Jordan algebras were first introduced in an effort to restructure quantum mechanics purely in terms of physical observables. In this paper we explain why, if one attempts to reformulate the internal structure of the standard model of…

In this paper we make an overview of results relating the recent "discoveries" in differential geometry, such as higher structures and differential graded manifolds with some natural problems coming from mechanics. We explain that a lot of…

Some of the well known Fefferman like constructions of parabolic geometries end up with a new structure on the same manifold. In this paper, we classify all such cases with the help of the classical Onishchik's lists \cite{onish1} and we…

In this paper we provide, first, a general symbolic algorithm for computing the symmetries of a given rational surface, based on the classical differential invariants of surfaces, i.e. Gauss curvature and mean curvature. In practice, the…

We consider rationally connected complex projective manifolds M and show that their loop spaces--infinite dimensional complex manifolds--have properties similar to those of M. Furthermore, we give a finite dimensional application concerning…

This is an overview of higher structural constructions in physics. The main motivations of our current attempt are as follows: (i) to provide a brief introduction to derived algebraic geometry, (ii) to understand how derived objects…

We investigate the duality between local (complex analytic) projective structures on surfaces and two dimensional (complex analytic) neighborhoods of rational curves having self-intersection +1. We study the analytic classification,…

Regular algebraic surfaces isogenous to a higher product of curves can be obtained from finite groups with ramification structures. We find unmixed ramification structures for finite groups constructed as p-quotients of particular infinite…

Characteristic class relations in Dolbeault cohomology follow from the existence of a holomorphic Cartan geometry (for example, a holomorphic conformal structure or a holomorphic projective connection). These relations can be calculated…

We prove rationality criteria over algebraically non-closed fields of characteristic $0$ for five out of six types of geometrically rational Fano threefolds of Picard number $1$ and geometric Picard number bigger than $1$. For the last type…

Complexes and cohomology, traditionally central to topology, have emerged as fundamental tools across applied mathematics and the sciences. This survey explores their roles in diverse areas, from partial differential equations and continuum…

This is an introduction to: (1) the enumerative geometry of rational curves in equivariant symplectic resolutions, and (2) its relation to the structures of geometric representation theory. Written for the 2015 Algebraic Geometry Summer…