Related papers: Big $q$-ample Line Bundles

The main problem addressed in the paper is the Torelli problem for n-dimensional varieties of general type, more specifically for varieties with ample canonical bundle. It asks under which geometrical condition for a variety the period map…

In this paper we contribute to the construction of families of arithmetically Cohen-Macaulay (aCM) indecomposable vector bundles on a wide range of polarized surfaces $(X,\Oo_X(1))$ for $\Oo_X(1)$ an ample line bundle. In many cases, we…

Let X be an irreducible 2n-dimensional holomorphic symplectic manifold. A reflexive sheaf F is very modular, if its Azumaya algebra End(F) deforms with X to every Kahler deformation of X. We show that if F is a slope-stable reflexive sheaf…

We study integral points on varieties with infinite \'etale fundamental groups. More precisely, for a number field $F$ and $X/F$ a smooth projective variety, we prove that for any geometrically Galois cover $\varphi\colon Y \to X$ of degree…

Let C be a generic non-singular curve of genus g defined over a field of characteristic different from 2. We show that for every line bundle on C of degree at most g+1, the natural product map S^2(H^0(L))\to H^0(C,L^2) is injective. We also…

We show that if $X$ is a smooth uniruled projective variety and $L$ a big and semiample $\mathbb{Q}$-divisor on $X$, then there exists a proper closed subset $W\subset X$ such that every subvariety $Y$ satisfying $a(Y,L)> a(X,L)$ is…

If $X\subset\operatorname{Gr}(2,6)$ is the Fano variety of lines of a smooth cubic fourfold, then we show that the restriction to $X$ of any Schur functor of the tautological quotient bundle is modular and slope polystable. Moreover it is…

In this note we show that given a lc pair $(X, \Delta)$, a large enough multiple of the bundle $K_X+ \Delta$ is effective provided that its Chern class contains an effective $\bQ$-divisor.

Let $X$ be a geometrically irreducible smooth projective curve, of genus at least three, defined over the field of real numbers. Let $G$ be a connected reductive affine algebraic group, defined over $\mathbb R$, such that $G$ is nonabelian…

Let $L$ be a nef and big line bundle on a scheme $X$. It is well known that if $X$ is a projective over a field then the augmented base locus and the exceptional base locus agree. This result is extended to projective schemes over arbitrary…

Let A be an ample line bundle on a projective toric variety X of dimension n. We show that if l>=n-1+p, then A^l satisfies the property N_p. Applying similar methods, we obtain a combinatorial theorem: For a given lattice polytope P we give…

In the holomorphic or algebraic setting we consider a vector bundle E on a smooth subvariety X in a smooth variety Y over a field of characteristic zero. Assuming E extends to the l-th neighborhood of X in Y, we study cohomological…

A Q-homology plane is a normal complex algebraic surface having trivial rational homology. We classify singular Q-homology planes which are C^1- or C*-ruled. We analyze their completions, the number of different rulings, the number of…

The aim of this paper is to study the cohomology theory of monoid schemes in general and apply it to vector and line bundles. We will prove that over separated monoid schemes, any vector bundle is a coproduct of line bundles and then go on…

Let $X$ be an irreducible smooth complex projective variety. Let $G$ be a linear algebraic group over $\mathbb{C}$. We define the notion of Lie algebroid valued connection on holomorphic principal $G$--bundles on $X$, and study their basic…

For a line bundle $L$ on a smooth projective surface $X$ and nonnegative integers $k_1, \ldots, k_N$, Okounkov \cite{Oko} introduced the reduced generating series $\big \langle {\rm ch}_{k_1}^{L} \cdots {\rm ch}_{k_N}^{L} \big \rangle'$ for…

Coman, Guedj and Zeriahi proved that, for an ample line bundle $L$ on a projective manifold $X$, any singular positive metric on the line bundle $L|_{V}$ along a subvariety $V \subset X$ can be extended to a global singular positive metric…

An algebraic variety X is embedded to the order k via a line bundle L if the global sections of L generate all (simultaneous) jets of order k on X or if they separate all zero-dimensional subschemes of length at most k+1. Even though we…

A Q-homology plane is a normal complex algebraic surface having trivial rational homology. We obtain a structure theorem for Q-homology planes with smooth locus of non-general type. We show that if a Q-homology plane contains a non-quotient…

We develop a theory of smooth relative connections over the real path algebra $\mathbb{R}Q$ on smooth twisted quiver bundles. We give obstructions to the existence of a smooth relative connection on twisted quiver bundles. For tree-type…