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We describe when two multiprojective bundles (fibre products of projective bundles over the same base) over projective spaces are isomorphic as abstract varieties. We also describe when two relative symmetric powers of projective bundles…
A linear system of real quadratic forms defines a real projective variety. The real non-singular locus of this variety (more precisely of the underlying scheme) has a highly connected double cover as long as each non-zero form in the system…
We define a birational version of the stability of cotangent sheaves for complex projective manifolds, and more generally for smooth orbifolds. We then show, using standard conjectures in birational classification, that these cotangent…
Given a smooth and projective curve C and a smooth and projective toric variety X, we first describe a compactification of the space of morphisms from C to X representing a fixed homology class, and after we study the intersection theory on…
We develop an intersection theory for a singular hemitian line bundle with positive curvature current on a smooth projective variety and irreducible curves on the variety. And we prove the existence of a natural rational fibration structure…
In this paper, we prove that a smooth projective globally $F$-split variety with numerically flat tangent bundle is an \'etale quotient of an ordinary abelian variety. We also show its logarithmic analog, which contains a characterization…
We show that smooth well formed weighted complete intersections have finite automorphism groups, with several obvious exceptions.
We construct smooth rational real algebraic varieties of every dimension $\ge$ 4 which admit infinitely many pairwise non-isomorphic real forms.
For a given complex projective variety, the existence of entire curves is strongly constrained by the positivity properties of the cotangent bundle. The Green-Griffiths-Lang conjecture stipulates that entire curves drawn on a variety of…
We classify codimension 2 well-formed and quasi-smooth weighted complete intersection del Pezzo surfaces.
In this note we study two features of submanifolds (subvarieties) with ample normal bundles in a compact K\"ahler manifold X. First, we study how algebraic X can be, i.e. we investigate the algebraic dimension. Second, we study curves with…
In this paper we give a new and simplified proof of the variational Hodge conjecture for complete intersection cycles on a hypersurface in projective space.
We give a systematic approach to constructing non-reduced, locally Cohen-Macaulay schemes with reduced support a smooth projective variety. The hierarchy of such structures includes a lot of information about the underlying variety, its…
The goal of this work is to prove an embedding theorem for compact almost complex manifolds into complex algebraic varieties. It is shown that every almost complex structure can be realized by the transverse structure to an algebraic…
We prove that a general hyperplane section of a smooth Legendrian subvariety in a projective space admits Legendrian embedding into another projective space. This gives numerous new examples of smooth Legendrian subvarieties, some of which…
Chiriv\`{\i} and Maffei \cite{CM II} have proved that the multiplication of sections of any two ample spherical line bundles on the wonderful symmetric variety $X=\bar{G/H}$ is surjective. We have proved two criterions that allows ourselves…
We discuss the flatness property of some fiber type contractions of complex smooth projective varieties of arbitrary dimensions. We relate the flatness of some morphisms having one-dimensional fibers with their conic bundles structures,…
Let G be a semisimple connected linear algebraic group over C, and X a wonderful G-variety. We study the possibility of realizing X as a closed subvariety of the projective space of a simple G-module. We describe the wonderful varieties…
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…
The family of varieties of minimal rational tangents on a quasi-homogeneous projective manifold is isotrivial. Conversely, are projective manifolds with isotrivial varieties of minimal rational tangents quasi-homogenous? We will show that…