Related papers: Gromov elliptic resolutions of quartic double soli…

We prove that every nontrivial principal $G_m$-bundle over a complete uniformly rational variety is algebraically elliptic in the sense of Gromov.

We survey on algebraically elliptic varieties in the sense of Gromov.

We show that every smooth cubic hypersurface X in P^{n+1}, n> 1 is algebraically elliptic in Gromov's sense. This gives the first examples of non-rational projective manifolds elliptic in Gromov's sense. We also deduce that the punctured…

We study quartic double solids admitting icosahedral symmetry.

We establish the equivalence of Gromov ellipticity and subellipticity in the algebraic category.

We investigate basic properties of uniformly rational varieties, i.e. those smooth varieties for which every point has a Zariski open neighborhood isomorphic to an open subset of A^n. It is an open question of Gromov whether all smooth…

We study the rationality problem for nodal quartic double solids. In particular, we prove that nodal quartic double solids with at most six singular points are irrational, and nodal quartic double solids with at least eleven singular points…

We study equivariant birational geometry of (rational) quartic double solids ramified over (singular) Kummer surfaces.

It was shown in [S. Kaliman, M. Zaidenberg, Gromov ellipticity of cones over projective manifolds, Math. Res. Lett. (to appear), arXiv:2303.02036 (2023)] that the affine cones over flag manifolds and rational smooth projective surfaces are…

In the present paper is devoted to the study of elliptic quadratic operator equations over the finite dimensional Euclidean space. We provide necessary and sufficient conditions for the existence of solutions of elliptic quadratic operator…

A rational perfect cuboid is a rectangular parallelepiped whose edges and face diagonals are given by rational numbers and whose space diagonal is equal to unity. Finding such a cuboid is equivalent to finding a perfect cuboid with all…

We classify rational, irreducible quartic symmetroids in projective 3-space. They are either singular along a line or a smooth conic section, or they have a triple point or a tacnode.

We construct singular quartic double fivefolds whose Kuznetsov component admits a crepant categorical resolution of singularities by a twisted Calabi--Yau threefold. We also construct rational specializations of these fivefolds where such a…

We prove that nonsingular retract rational algebraic varieties over any infinite field are uniformly retract rational. As a consequence, every rational, projective, nonsingular complex variety is algebraically elliptic.

We establish the existence of strong solutions to a class of nonlinear strongly coupled and uniform elliptic systems consisting of more than two equations. The existence of of nontrivial and non constant solutions (or pattern formations)…

New embeddings of weighted Sobolev spaces are established. Using such embeddings, we obtain the existence and regularity of positive solutions with Navier boundary value problems for a weighted fourth order elliptic equation. We also obtain…

We find classes of projective manifolds that are elliptic in the sense of Gromov and such that the affine cones over these manifolds also are elliptic off their vertices. For example, the latter holds for any generalized flag manifold of…

We consider a semilinear elliptic equation on a smooth bounded domain $\Om$ in $\R^2$, assuming that both the domain and the equation are invariant under reflections about one of the coordinate axes, say the y-axis. It is known that…

Using a method developped in [1] and [2], we prove the existence of weak non trivial solutions to fourth order elliptic equations with singularities and with critical Sobolev growth.

This article discusses two versions of elliptic equations obtained from a system of equations describing a rational cuboid. Analysis of elliptic equations shows that they are equivalent, and that there are rational points on the elliptic…