Related papers: Abundance for varieties with many differential for…

Let $X$ be a smooth projective variety over an algebraically field $k$ with ${\rm char}(k)=p>0$ and $F:X\to X_1$ be the relative Frobenius morphism. When ${\rm dim}(X)=1$, we prove that $F_*W$ is a stable bundle for any stable bundle $W$…

We show that a compact Kahler manifold with nonpositive holomorphic sectional curvature has nef canonical bundle. If the holomorphic sectional curvature is negative then it follows that the canonical bundle is ample, confirming a conjecture…

In this paper we show that any smoothable complex projective variety, smooth in codimension two, with klt singularities and numerically trivial canonical class admits a finite cover, \'etale in codimension one, that decomposes as a product…

Call a projective variety $X$ Calabi-Yau if its canonical divisor is ${\bf Q}$-linearly equivalent to zero. The smallest positive integer $m$ with $mK_X$ linearly equivalent to zero is called the index of $X$. We construct Calabi-Yau…

In this paper, we study an algebraic fiber space in positive characteristic whose generic fiber $F$ has finitely generated canonical ring and sufficiently large Frobenius stable canonical ring. An example of such a case is when $F$ is…

We give the first examples of nef line bundles on smooth projective varieties over finite fields which are not semi-ample. More concretely, we find smooth curves on smooth projective surfaces over finite fields such that the normal bundle…

We study nonsingular branched coverings of a homogeneous space X. There is a vector bundle associated with such a covering which was conjectured by O. Debarre to be ample when the Picard number of X is one. We prove this conjecture, which…

Hartshorne in "Ample vector bundles" proved that $E$ is ample if and only if $\OOO_{P(E)}(1)$ is ample. Here we generalize this result to flag manifolds associated to a vector bundle $E$ on a complex manifold $X$: For a partition $a$ we…

Jet ampleness of line bundles generalizes very ampleness by requiring the existence of enough global sections to separate not just points and tangent vectors, but also their higher order analogues called jets. We give sharp bounds…

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 prove that smooth, projective, $K$-trivial, weakly ordinary varieties over a perfect field of characteristic $p>0$ are not geometrically uniruled. We also show a singular version of our theorem, which is sharp in multiple aspects. Our…

Let $k$ be an algebraically closed field of characteristic zero, and let $X/k$ be a projective variety. The conjectures of Demailly--Green--Griffiths--Lang posit that every integral subvariety of $X$ is of general type if and only if $X$ is…

Given a reflexive sheaf on a mildly singular projective variety, we prove a flatness criterion under certain stability conditions. This implies the algebraicity of leaves for sufficiently stable foliations with numerically trivial canonical…

In this article we show that if $(X, \Delta)$ is a log canonical compact K\"ahler threefold pair such that $K_X+\Delta$ is nef and the numerical dimension $\nu(X, K_X+\Delta)=2$, then $K_X+\Delta$ is semi-ample. This result combined with…

In this short note we will show that every homogeneous strictly nef vector bundle on a complex flag variety is ample. Following this, we consider whether ampleness of a bundle on an abelian variety can be tested on curves.

We prove several results relating the nonvanishing and the existence of good minimal models of different pairs that have the same underlying variety.

For a complex manifold equipped with an anti-holomorphic involution, which is referred to as a real variety, the Smith-Thom inequality states that the total $\mathbb{F}_2$-Betti number of the real locus is not greater than the total…

We find lower bounds on the rank of a "real" vector bundle over an involutive space, such that "real" vector bundles of higher rank have a trivial summand and such that a stable isomorphism for such bundles implies ordinary isomorphism. We…

A $\mathbf{Q}$-Cartier divisor $D$ on a projective variety $M$ is {\it almost nup}, if $(D , C) > 0$ for every very general curve $C$ on $M$. An algebraic variety $X$ is of {\it almost general type}, if there exists a projective variety $M$…

We give a criterion for a nef divisor $D$ to be semiample on a Calabi--Yau threefold $X$ when $D^3=0=c_2(X)\cdot D$ and $c_3(X)\neq 0$. As a direct consequence, we show that on such a variety $X$, if $D$ is strictly nef and $\nu(D)\neq 1$,…