Related papers: The Nonvanishing problem for varieties with nef an…

A nilmanifold is a quotient of a nilpotent group $G$ by a co-compact discrete subgroup. A complex nilmanifold is one which is equipped with a $G$-invariant complex structure. We prove that a complex nilmanifold has trivial canonical bundle.…

The paper consists of three parts. In the first of them different kinds stability are discussed. In particular, the stability concept with respect to nef divisor is introduced. A structure of rigid and superrigid vector bundles on smooth…

Let X be a projective manifold of dimension n. Beltrametti and Sommese conjectured that if A is an ample divisor such that $K_X+(n-1)A$ is nef, then $K_X+(n-1)A$ has non-zero global sections. We prove a weak version of this conjecture in…

The goal of this work is to study positivity of subvarieties with nef normal bundle in the sense of intersection theory. After Ottem's work on ample subschemes, we introduce the notion of a nef subscheme, which generalizes the notion of a…

Let $k$ be an $F$-finite field containing an infinite perfect field of positive characteristic. Let $(X, \Delta)$ be a projective log canonical pair over $k$. In this note we show that, for a semi-ample divisor $D$ on $X$, there exists an…

We prove that the non-nef locus and the restricted base locus of a pseudoeffective divisor coincide on KLT pairs. We also extend to KLT pairs F. Russo's characterization of nef and abundant divisors by means of asymptotic multiplier ideals.

We prove the rationality of the descendent partition function for stable pairs on nonsingular toric 3-folds. The method uses a geometric reduction of the 2- and 3-leg descendent vertices to the 1-leg case. As a consequence, we prove the…

Let $(X,L)$ be an $n$-dimensional polarized variety. Fujita's conjecture says that if $L^n>1$ then the adjoint bundle $K_X+nL$ is spanned and $K_X+(n+1)L$ is very ample. There are some examples such that $K_X+nL$ is not spanned or…

In this paper, we show an extension type theorem for twisted pluricanonical sections on a family of smooth projective manifolds (the twisting line bundle being pseudo-effective and having a prescribed multiplier ideal on the central fiber).

We classify non-algebraic compact K\"ahler threefolds admitting an endomorphism $f: X \to X$ of degree at least two.

We show the invariance of plurigenera for generalized polarized pairs with abundant nef parts and generalized canonical singularities. This is obtained by investigating a type of newly introduced multiplier ideal sheaf which is of…

An affine algebraic variety $X$ is rigid if the algebra of regular functions ${\mathbb K}[X]$ admits no nonzero locally nilpotent derivation. We prove that a factorial trinomial hypersurface is rigid if and only if every exponent in the…

We determine the 6-dimensional solvmanifolds admitting an invariant complex structure with holomorphically trivial canonical bundle. Such complex structures are classified up to isomorphism, and the existence of strong K\"ahler with torsion…

In this work we consider compact K\"ahler manifolds with non-positive mixed curvature which is a "convex combination" of Ricci curvature and holomorphic sectional curvature. We show that in this case, the canonical line bundle is nef.…

We prove that the linear system $|-1/3K_X| on a non-singular Fano fivefold $X$ of index 3 contains an irreducible divisor with only canonical singularities.

In this article, we study the existence of non-negative solutions of the class of non-local problem of $n$-Kirchhoff type $$ \left\{ \begin{array}{lr} \quad - m(\int_{\Omega}|\nabla u|^n)\Delta_n u = f(x,u) \; \text{in}\; \Omega,\quad u…

Let X be a projective 3-fold with at most Q-factorial terminal singularities on which K_X is nef and big. Suppose the canonical index r(X)>1. For any positive integer m, it is interesting to consider the base point freeness and…

This paper has three objectives. First to recall the link between the classical Legendre-Fenschel transformation and a useful isomorphism between 1-jets of functions on a vector bundle and on its dual. As a particular consequence we obtain…

For a $\mathbb{Q}$-Fano 3-fold $X$ on which $K_X$ is a canonical divisor, we investigate the geometry induced from the linear system $|-mK_X|$ in this paper and prove that the anti-$m$-canonical map $\varphi_{-m}$ is birational onto its…

This paper concerns the construction of minimal varieties with small canonical volumes. The first part devotes to establishing an effective nefness criterion for the canonical divisor of a weighted blow-up over a weighted hypersurface, from…