Related papers: Properties of legendrian subvarieties of projectiv…

We propose a geometric and categorical approach to the Hodge Conjecture for all smooth projective complex varieties. By embedding any such variety into a flat family with general fibers smooth complete intersections, we prove the conjecture…

We study braid varieties and their relation to open positroid varieties. We discuss four different types of braids associated to open positroid strata and show that their associated Legendrian links are all Legendrian isotopic. In…

We prove, in any positive characteristic, Parseval-Rayleigh identities for the residue map of a homogeneous complete intersection. As an application, we give a conceptual proof of the folklore fact that generic homogeneous complete…

This paper proves that every projective toric variety is the fine moduli space for stable representations of an appropriate bound quiver. To accomplish this, we study the quiver $Q$ with relations $R$ corresponding to the finite-dimensional…

We consider when a smooth vector bundle endowed with a connection possesses non-trivial, local parallel sections. This is accomplished by means of a derived flag of subsets of the bundle. The procedure is algebraic and rests upon the…

We prove that all complex analytic subvarieties of a generic compact hyperkaehler manifold are even-dimensional. Moreover, these subvarieties are holomorphically symplectic.

In some varieties of algebras one can reduce the question of finding most general unifiers (mgus) to the problem of the existence of unifiers that fulfill the additional condition called projectivity. In this paper we study this problem for…

We exhibit a family of homogeneous hypersurfaces in affine space, one in each dimension, generalising the Cayley surface.

We give a short proof of the Zariski-Lipman conjecture for toric varieties: any complex toric variety with locally free tangent sheaf is smooth.

Firstly we show a generalization of the (1,1)-Lefschetz theorem for projective toric orbifolds and secondly we prove that on 2k-dimensional quasi-smooth hypersurfaces coming from quasi-smooth intersection surfaces, under the Cayley trick,…

We present a smooth, complete toric threefold with no nontrivial nef line bundles. This is a counterexample to a recent conjecture of Fujino.

We study the global structure of Lorentzian manifolds with partial sectional curvature bounds. In particular, we prove completeness theorems for homogeneous and isotropic cosmologies as well as static spherically symmetric spacetimes. The…

In this note we gather and review some facts about existence of toric spaces over 3-dimensional simple polytopes. First, over every combinatorial 3-polytope there exists a quasitoric manifold. Second, there exist combinatorial 3-polytopes,…

We study the representability of motivic spheres by smooth varieties. We show that certain explicit "split" quadric hypersurfaces have the $\mathbb A^1$-homotopy type of motivic spheres over the integers and that the $\mathbb A^1$-homotopy…

Let $X$ be a smooth complex projective variety. A recent conjecture of S. Kov\'acs states that if t\ he $p^{\text{th}}$-exterior power of the tangent bundle $T_X$ contains the $p^{\text{th}}$-exterior power of an ample vector bundle, then…

We prove that every endomorphism of an infinite-dimensional vector space splits as the sum of four idempotents and as the sum of four square-zero endomorphisms, a result that is optimal in general.

We show that for a complete complex algebraic variety the pure component of homology coincides with the image of intersection homology. Therefore pure homology is topologically invariant. To obtain slightly more general results we introduce…

We prove that every irreducible component of a fibre of a complex Lagrangian fibration is Lagrangian subvariety. Especially, complex Lagrangian fibations are equidimensional.

Every convex homogeneous polynomial (or form) is nonnegative. Blekherman has shown that there exist convex forms that are not sums of squares via a nonconstructive argument. We provide an explicit example of a convex form of degree four in…

The aim of this work is to construct examples of pairs whose logarithmic cotangent bundles have strong positivity properties. These examples are constructed from any smooth n-dimensional complex projective varieties by considering the sum…