Related papers: The openness conjecture for projective manifolds

Using deformation theory of rational curves, we prove a conjecture of Sommese on the extendability of morphisms from ample subvarieties when the morphism is a smooth (or mildly singular) fibration with rationally connected fibers. We apply…

In the present paper we obtain a new homological version of the implicit function theorem and some versions of the Darboux theorem. Such results are proved for continuous maps on topological manifolds. As a consequence, some versions of…

In this paper we provide a positive answer to a conjecture due to A. J. Di Scala, A. Loi, H. Hishi (see [3, Conjecture 1]) claiming that a simply-connected homogeneous K\"ahler manifold M endowed with an integral K\"ahler form $\mu\omega$,…

In this paper, we consider a compact Kahler manifold with extremal Kahler metric and a Mumford stable holomorphic bundle over it. We proved that, if the holomorphic vector field defining the extremal Kahler metric is liftable to the bundle…

We describe explicit open books on arbitrary plumbings of oriented circle bundles over closed oriented surfaces. We show that, for a non-positive plumbing, the open book we construct is horizontal and the corresponding compatible contact…

We study the dynamics of surjective endomorphisms of projective bundles on elliptic curves and relate their dynamical properties to the geometry of the bundle. As an application we prove the Kawaguchi--Silverman conjecture for projective…

A complex manifold $X$ of dimension $n$ together with an ample vector bundle $E$ on it will be called a {\sf generalized polarized variety}. The adjoint bundle of the pair $(X,E)$ is the line bundle $K_X + det(E)$. We study the positivity…

Classification theory and the study of projective varieties which are covered by rational curves of minimal degrees naturally leads to the study of families of singular rational curves. Since families of arbitrarily singular curves are hard…

Let $X$ be a compact K\"ahler manifold. We study plurisupported currents on $X$, i.e. closed, positive $(1,1)$-currents which are supported on a pluripolar set. In particular, we are able present a technical generalization of…

We prove some results on effective very ampleness and projective normality for some varieties with trivial canonical bundle. In the first part we prove an effective projective normality result for an ample line bundle on regular smooth…

The gonality conjecture predicts that the gonality of a curve can be read off Koszul cohomology of line bundles of sufficiently large degree. We verify this conjecture for generic curves of odd genus. The even-genus case was previously…

We first provide an approach to the recent conjecture of Bierstone-Milman-Pawlucki on Whitney's old problem on smooth extendability of functions defined on a closed subset of a Euclidean space, using higher order paratangent bundle they…

Let C be an integral projective curve with surficial singularities. We prove that topologically trivial line bundles on the compactified Jacobian of C are in one-to-one correspondence with line bundles on C (the autoduality conjecture), and…

In this note, we introduce a new curvature condition called the $2-$positive bisectional curvature on compact K\"{a}hler manifolds. We then deduce a characterization theorem for manifolds with $2-$positive bisectional curvature, which can…

Demailly's conjecture, which is a consequence of the Green-Griffiths-Lang conjecture on varieties of general type, states that an algebraically hyperbolic complex projective variety is Kobayashi hyperbolic. Our aim is to provide evidence…

Let $(M,g)$ be a closed oriented negatively curved surface. A unitary connection on a Hermitian vector bundle over $M$ is said to be transparent if its parallel transport along the closed geodesics of $g$ is the identity. We study the space…

We prove the Arnold chord conjecture on cotangent bundles of open manifold by Gromov's nonlinear Fredholm alternative for $J-$holomorphic curves.

We show by an example that the (equivalence class of) singularity of a plurisubharmonic function cannot be determined by the data of its Lelong numbers, in a nontrivial sense. Such an example is provided by Siu-type singular hermitian…

We establish the Hodge conjecture for the top dimensional cohomology group with integer coefficients of any $q$-complete complex manifold $X$ with $q<\dim X$. This holds in particular for the complement $X=\mathbb{C}\mathbb{P}^n\setminus A$…

Let $(M,\omega)$ be a compact K\"ahler manifold with negative holomorphic sectional curvature. It was proved by Wu-Yau and Tosatti-Yang that $M$ is necessarily projective and has ample canonical bundle. In this paper, we show that any…