Related papers: Counting $h^0(D)$ on primary Burniat surfaces

Khovanov defined graded homology groups for links L in R^3 and showed that their polynomial Euler characteristic is the Jones polynomial of L. Khovanov's construction does not extend in a straightforward way to links in I-bundles M over…

The classification of algebraic vector bundles of rank 2 over smooth affine fourfolds is a notoriously difficult problem. Isomorphism classes of such vector bundles are not uniquely determined by their Chern classes, in contrast to the…

We study varieties $X \subseteq \mathbb P^N$ of dimension $n$ such that $T_X(k)$ is an Ulrich vector bundle for some $k \in \mathbb Z$. First we give a sharp bound for $k$ in the case of curves. Then we show that $k \le n+1$ if $2 \le n \le…

Here we study vector bundles $E$ on the Hirzebruch surface $F_e$ such that their twists by a spanned, but not ample, line bundle $M = \mathcal {O}_{F_e}(h+ef)$ have natural cohomology, i.e. $h^0(F_e,E(tM)) >0$ implies $h^1(F_e,E(tM)) = 0$.

We prove that any rank two arithmetically Cohen-Macaulay vector bundle on a general hypersurface of degree at least three in $\mathbb{P}^5$ must be split.

We study the projective normality of the projective bundle of an Ulrich vector bundle embedded through the complete linear system of its tautological line bundle. The focus will be on Ulrich bundles defined over curves, surfaces with…

Fujita's second theorem for K\"ahler fibre spaces over a curve asserts that the direct image $V$ of the relative dualizing sheaf splits as the direct sum $ V = A \oplus Q$, where $A$ is ample and $Q$ is unitary flat. We focus on our…

Let $X$ be a complex surface obtained as the quotient of the complex Euclidean space $\mathbb{C}^2$ by a discrete subgroup of rank $3$. We investigate the cohomology group $H_0^1(X, E)$ with compact support for a unitary flat line bundle…

In this paper, we investigate the existence of Ulrich bundles on a smooth complete intersection of two $4$-dimensional quadrics in $\mathbb P^5$ by two completely different methods. First, we find good ACM curves and use Serre…

We study the resolution of an Ulrich bundle of arbitrary rank on the Segre fourfold $\PP^2\times\PP^2$. We characterize the Ulrich bundles $\Vv$ of arbitrary rank on $\PP^2\times\PP^2$ with $h^1(\Vv\otimes\Omega\boxtimes\Omega)=0$ or with…

The cohomology groups of line bundles over complex tori (or abelian varieties) are classically studied invariants of these spaces. In this article, we compute the cohomology groups of line bundles over various holomorphic, non-commutative…

Let M be a moduli space of stable vector bundles on a curve with rank and degree fixed and coprime. We give a simple proof that the rational cohomology of M is generated by the Kunneth components of the Chern classes of the universal…

A strong version of the quantization conjecture of Guillemin and Sternberg is proved. For a reductive group action on a smooth, compact, polarized variety (X,L), the cohomologies of L over the GIT quotient X // G equal the invariant part of…

We show that the zeroth cohomology of effective line bundles on del Pezzo and Hirzebruch surfaces can always be computed in terms of a topological index.

This work provides a curve-based approach to Ulrich bundles on surfaces, establishing a correspondence that characterizes their existence, with a focus on applications to surfaces in $\mathbb{P}^3$.

In these notes a recently developed technique for the computation of line bundle-valued sheaf cohomology group dimensions on toric varieties is reviewed. The key result is a vanishing theorem for the contributing components which depends on…

In this paper we generalise the theory of real vector bundles to a certain class of non-Hausdorff manifolds. In particular, it is shown that every vector bundle fibred over these non-Hausdorff manifolds can be constructed as a colimit of…

Smooth complex surfaces polarized with an ample and globally generated line bundle of degree three and four, such that the adjoint bundle is not globally generated, are considered. Scrolls of a vector bundle over a smooth curve are shown to…

We calculate the ordinary $C_2$-cohomology, with Burnside ring coefficients, of $BU(2)$, the classifying space for $C_2$-equivariant complex 2-plane bundles, using an extended grading that allows us to capture a more natural set of…

Let $X$ be a smooth variety defined over an algebraically closed field of arbitrary characteristic and $\O_X(H)$ be a very ample line bundle on $X$. We show that for a semistable $X$-bundle $E$ of rank two, there exists an integer $m$…