Related papers: Frobenius pull backs of vector bundles in higher d…

Let $\mathcal{V}$ be a mixed characteristic complete discrete valuation ring, $\mathcal{P}$ a separated smooth formal scheme over $\mathcal{V}$, $P$ its special fiber, $X$ a smooth closed subscheme of $P$, $T$ a divisor in $P$ such that…

We investigate when the filtration induced by Beilinson's spectral sequence splits non-canonically into a direct sum decomposition. We conclude that for any vector bundle $\mathcal{E}$ on a projective space over an algebraically closed…

The thesis studies Frobenius-type theorems in non-smooth settings. We extend the definition of involutivity to non-Lipschitz subbundles using generalized functions. We prove the real Frobenius Theorem with sharp regularity on log-Lipschitz…

Let $k$ be an algebraically closed field with characteristic $2$, and let $X$ be a smooth projective algebraic curve of genus $g \geqslant 2$ over $k$. Let $\mathcal{M}^s_X(2,\mathcal{L})$ be the moduli space of rank $2$ stable vector…

We adapt the notions of stability of holomorphic vector bundles in the sense of Mumford-Takemoto and Hermitian-Einstein metrics in holomorphic vector bundles for canonically polarized framed manifolds, i.e. compact complex manifolds X…

In this article, we study the behavior of the stability of pullback of a vector bundle under a finite morphism from a (not necessarily smooth) stacky curve to an orbifold curve. We establish a categorical equivalence between proper formal…

We use the theory of p-curvature of connections to analyze stable vector bundles of rank 2 on curves of genus 2 which pull back to unstable bundles under the Frobenius morphism. We take two approaches, first using explicit formulas for…

Let E be a stable rank 2 vector bundle on a smooth quadric threefold Q in the projective 4-space P. We show that the hyperplanes H in P for which the restriction of E to the hyperplane section of Q by H is not stable form, in general, a…

Let $X$ be a smooth complex projective curve of genus $g\geq 2$. We prove that a parabolic vector bundle $\mathcal{E}$ on $X$ on $X$ is (strongly) wobbly, i.e. $\mathcal{E}$ has a non-zero (strongly) parabolic nilpotent Higgs field, if and…

The so-called Hitchin-Kobayashi correspondence, proved by Donaldson, Uhlenbeck and Yau, establishes that an indecomposable holomorphic vector bundle over a compact Kahler manifold admits a Hermitian-Einstein metric if and only if the bundle…

Consider $E$ a vector bundle over a smooth curve $C$. We compute the $\delta$-invariant of all ample ($\mathbb{Q}$-) line bundles on $\mathbb{P}(E)$ when $E$ is strictly Mumford semistable. We also investigate the case when one assumes that…

Let X be a projective smooth irreducible polarized variety over the field of complex numbers. Typical examples of wide extensions are vector bundles E that have a subsheaf F whose slope is much bigger than the slope of E/F, and such that F…

Let E be a torsion-free sheaf on P2. We give an effective method which uses the Hilbert function of E to construct a weak version of the Harder-Narasimhan filtration of a torsion-free sheaf on P2 subject only to the condition that E be…

We give an example of an orthogonal bundle where the Harder-Narasimhan filtration, with respect to Gieseker semistability, of its underlying vector bundle does not correspond to any parabolic reduction of the orthogonal bundle. A similar…

Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field $k$ of characteristic $p>0$. Let $F_X:X\rightarrow X$ be the absolute Frobenius morphism, and $\E$ a torsion free sheaf on $X$. We give a upper bound…

We prove an elementary but somewhat unexpected result about projective embeddings of smooth varieties X whose cotangent bundles are numerically effective. Specifically, we show that the degree of X in any projective embedding must grow…

In this article we study a special class of vector bundles, called tensors. A tensor consists of a vector bundle $E$ over a smooth irreducible projective variety and a morphism of vector bundles $\varphi$. As for classical vector bundles,…

Given a smooth complex projective variety X and a smooth divisor D on X, we prove the existence of Hermitian-Einstein connections, with respect to a Poincar\'e-type metric on X - D, on polystable parabolic principal Higgs bundles with…

Given two arbitrary vector bundles on the Fargues-Fontaine curve, we give an explicit criterion in terms of Harder-Narasimhan polygons on whether they realize a semistable vector bundle as their extensions. Our argument is largely…

We present bounds for the geometric degree of the tangent bundle and the tangential variety of a smooth affine algebraic variety $V$ in terms of the geometric degree of $V$. We first analyze the case of curves, showing an explicit relation…