Related papers: A note on non-vanishing and applications

We describe non-locally connected planar continua via the concepts of fiber and numerical scale. Given a continuum $X\subset\mathbb{C}$ and $x\in\partial X$, we show that the set of points $y\in \partial X$ that cannot be separated from $x$…

We study non-Kaehler manifolds with trivial logarithmic tangent bundle. We show that each such manifold arises as a fiber bundle with a compact complex parallelizable manifold as basis and a toric variety as fiber.

Let $f:X\to Y$ be a proper, dominant morphism of smooth varieties over a number field $k$. When is it true that for almost all places $v$ of $k$, the fibre $X_P$ over any point $P\in Y(k_v)$ contains a zero-cycle of degree $1$? We develop a…

We study real linear spaces in projective space that avoid the real points of a non-degenerate projective variety. For a variety $X \subset \mathbb{P}^{n-1}$ with a real smooth point, we define the avoidance locus $\mathcal{A}_k(X)$ as the…

Let $X$ be a normal variety. Assume that for some reduced divisor $D \subset X$, logarithmic 1-forms defined on the snc locus of $(X, D)$ extend to a log resolution $\tilde X \to X$ as logarithmic differential forms. We prove that then the…

We study a useful numerical invariant of normal surface singularities, introduced recently by T. Kawachi. Using this invariant, we give a quick proof of the (well-known) fact that all log-canonical surface singularities are either elliptic…

In this short note, we will show the following weak evidence of S. Lang conjecture over function fields. Let f : X ---> Y be a projective and surjective morphism of algebraic varieties over an algebraically closed field k of characteristic…

For a projective variety $X$ defined over a non-Archimedean complete non-trivially valued field $k$, and a semipositive metrized line bundle $(L, \phi)$ over it, we establish a metric extension result for sections of $L^{\otimes n}$ from a…

Let $X$ be a projective variety over an algebraically closed field $k$ of arbitrary characteristic $p \ge 0$. A surjective endomorphism $f$ of $X$ is $q$-polarized if $f^\ast H \sim qH$ for some ample Cartier divisor $H$ and integer $q >…

Let $C$ be a smooth projective irreducible curve defined over a finite field $\mathbb{F}_q$ and $K=\mathbb{F}_q(C)$. Let $A\subset K$ be the ring of functions regular outside a fixed place $\infty$ of $K$. Let…

Let f: X \to Z be a surjective morphism of smooth complex projective varieties with connected fibers. Suppose that L is a pseudo-effective divisor on X that is f-numerically trivial. We show that there is a divisor D on Z such that L is…

Let $X$ be a smooth complex projective variety of dimension three and let $L$ be an ample line bundle on $X$. In this paper, we provide a lower bound of the dimension of the global sections of $m(K_{X}+L)$ under the assumption that…

Let $\pi\,:\, X \,\longrightarrow\, Y$ be a finite morphism of smooth projective varieties defined over an algebraically closed field of characteristic zero. We study the necessary and sufficient criteria for $\pi$ such that there exists a…

In the present paper Mori extremal rays of a smooth projective manifold X are divided into two classes: L-supported and L-negligible (where ``L'' stands for ``Lefschetz'' since the division comes from Hard Lefschetz Theorem). Roughly…

Given a (meromorphic) fibration $f:X\to Y$ where $X$ and $Y$ are compact complex manifolds of dimensions $n$ and $m$, we define $L_f$ to be the invertible subsheaf of the sheaf of holomorphic $m$-forms of $X$ given by the saturation of…

We give some explicit upper bounds on the effective birationality of the canonical or anti-canonical system for a singular surface. In particular, we show that for any surface $X$ with $\epsilon$-lc singularity and the canonical divisor…

We study the moduli functor of flat bundles on smooth, possibly non-proper, algebraic variety $X$ (over a field of characteristic zero). For this we introduce the notion of \emph{formal boundary} of $X$, denoted by $\partial X$, which is a…

For each pseudo-Anosov map $\phi$ on surface $S$, we will associate it with a $\mathbb{Q}$-submodule of $\mathbb{R}$, denoted by $A(S,\phi)$. $A(S,\phi)$ is defined by an interaction between the Thurston norm and dilatation of pseudo-Anosov…

Let $(X,\Delta)$ be a projective, $\mathbb{Q}$-factorial log canonical pair and let $L$ be a pseudoeffective $\mathbb{Q}$-divisor on $X$ such that $K_X + \Delta + L$ is pseudoeffective. Is there an effective $\mathbb{Q}$-divisor $M$ on $X$…

We give explicit, highly symmetric equations for the versal deformation of the singularity $L_{n+1}^n$ consisting of n+1 lines through the origin in n-dimensional affine space in generic position. These make evident that the base space of…