Related papers: Horospherical varieties with quotient singularitie…

We prove that a representation of the fundamental group of a quasi-projective manifold into the group of formal diffeomorphisms of one variable either is virtually abelian or, after taking the quotient by its center, factors through an…

For each strongly connected finite-dimensional (pure) simplicial complex we construct a finite group, the group of projectivities of the complex, which is a combinatorial but not a topological invariant. This group is studied for…

In this paper we obtain a criterion of flexibility for an affine complexity-zero horospherical variety. This result generalizes previously known results on flexibility of normal horospherical varieties, horospherical varieties with an…

An algebraic variety is said to have the $A_k$-property if any $k$ points are contained in some common affine open neighbourhood. A theorem of W{\l}odarczyk states that a normal variety has the $A_2$-property if and only if it admits a…

Let $A$ be an abelian scheme of dimension at least four over a $\mathbb{Z}$-finitely generated integral domain $R$ of characteristic zero, and let $L$ be an ample line bundle on $A$. We prove that the set of smooth hypersurfaces $D$ in $A$…

We give a characterization of smooth quadrics in terms of the existence of full exceptional collections of certain type, which generalizes a result of C.Vial for projective spaces.

We consider subtorus actions on divisorial toric varieties. Here divisoriality means that the variety has many Cartier divisors like quasiprojective and smooth ones. We characterize when a subtorus action on such a toric variety admits a…

We describe the generic singularity of a Schubert variety of type A on each irreducible component of its singular locus. This singularity is given either by a cone of rank one matrices, or a quadratic cone.

We study the equivariant real structures on complex horospherical varieties, generalizing classical results known for toric varieties and flag varieties. In particular, we obtain a necessary and sufficient condition for the existence of…

We characterise the virtually abelian groups which are fundamental groups of compact K\"ahler manifolds and of smooth projective varieties. We show that a virtually abelian group is K\"ahler if and only if it is projective. In particular,…

A Q-homology plane is a normal complex algebraic surface having trivial rational homology. We obtain a structure theorem for Q-homology planes with smooth locus of non-general type. We show that if a Q-homology plane contains a non-quotient…

Quasifolds are singular spaces that generalize manifolds and orbifolds. They are locally modeled by manifolds modulo the smooth action of countable groups and they are typically not Hausdorff. If the countable groups happen to be all…

By the fundamental work of Griffiths one knows that, under suitable assumption, homological and algebraic equivalence do not coincide for a general hypersurface section of a smooth projective variety $Y$. In the present paper we prove the…

We give a complete classification of Riemannian and Lorentzian surfaces of arbitrary codimension in a pseudo-sphere whose pseudo-spherical Gauss maps are of 1-type or, in particular, harmonic. In some cases a concrete global classification…

We describe complete simplicial toric varieties on which a unipotent group acts with a finite number of orbits. We also provide a complete list of such varieties in the case where the dimension is equal to 2.

A plurisubharmonic singularity is extreme if it cannot be represented as the sum of non-homothetic singularities. A complete characterization of such singularities is given for the case of homogeneous singularities (in particular, those…

Homology Hirzebruch characteristic classes for singular varieties have been recently defined by Brasselet-Schuermann-Yokura as an attempt to unify previously known characteristic class theories for singular spaces (e.g., MacPherson-Chern…

Let $G$ be a connected reductive algebraic group. In this note we prove that for a quasi-affine $G$-spherical variety the weight monoid is determined by the weights of its non-trivial $\mathbb{G}_a$-actions that are homogeneous with respect…

Let G < SL(V) be a finite group, V is finite dimensional over a field F, p=char F and S(V) is the symmetric algebra of V. We determine when the subring of G-invariants S(V)^G is a polynomial ring. As a consequence, we classify, if F is…

In this article we study the cohomological and homological (due to Jannsen) Hodge conjecture for singular varieties. The motivation for studying singular varieties comes from the fact that any smooth projective variety X is birational to a…