Related papers: Stability of Arakelov bundles and tensor products …

Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field $k$ with ${\rm char}(k)=p>0$ and $F:X\to X_1$ be the relative Frobenius morphism. For any vector bundle $W$ on $X$, we prove that instability of…

We investigate the flat holomorphic vector bundles over compact complex parallelizable manifolds $G / \Gamma$, where $G$ is a complex connected Lie group and $\Gamma$ is a cocompact lattice in it. The main result proved here is a structure…

Let (X,H) be a polarized smooth projective surface satisfying H^1(X,O_X)=0 and let F be either a rank one torsion-free sheaf or a rank two {\mu}H-stable vector bundle on X. Assume that c_1(F)/=0. In this article it is shown that the rank…

Let $X$ be a smooth complex projective curve of genus $g\geq 2$ and let $K$ be its canonical bundle. In this note we show that a stable vector bundle $E$ on $X$ is very stable, i.e. $E$ has no non-zero nilpotent Higgs field, if and only if…

Let $\alpha : X \to Y$ be a general degree $r$ primitive map of nonsingular, irreducible, projective curves over an algebraically closed field of characteristic zero or larger than $r$. We prove that the Tschirnhausen bundle of $\alpha$ is…

Let $X$ be a smooth projective curve over the complex numbers. To every representation $\rho\colon \GL(r)\lra \GL(V)$ of the complex general linear group on the finite dimensional complex vector space $V$ which satisfies the assumption that…

Let $\pi:Y\to X$ be a surjective morphism between two irreducible, smooth complex projective varieties with ${\rm dim}Y>{\rm dim}X >0$. We consider polarizations of the form $L_c=L+c\cdot\pi^*A$ on $Y$, with $c>0$, where $L,A$ are ample…

We show the stability of certain syzygies of line bundles on curves, which we call transforms, and are kernels of the evaluation map on subspaces of the space of global sections. For the transforms constructed, we prove the existence of…

Let $E$ be a vector bundle on a smooth complex projective variety $X$. We study the family of sections $s_t\in H^0(E\otimes L_t)$ where $L_t\in Pic^0(X)$ is a family of topologically trivial line bundle and $L_0=\mathcal O_X,$ that is, we…

We prove quantitative versions for several results from geometric partial differential equations. Firstly, we obtain a double stability theorem for Serrin's overdetermined problem in spaceforms. Secondly, we prove stability theorems for…

Let f:X-->Y be a semi-stable family of complex abelian varieties over a curve Y of genus q, and smooth over the complement of s points. If F(1,0) denotes the non-flat (1,0) part of the corresponding variation of Hodge structures, the…

Let $M=P(E)$ be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle $E \to \Sigma$ over a compact complex curve $\Sigma$ of genus $\ge 2$. Building on ideas of Fujiki, we prove that $M$…

The existence of stable ACM vector bundles of high rank on algebraic varieties is a challenging problem. In this paper, we study stable Ulrich bundles (that is, stable ACM bundles whose corresponding module has the maximum number of…

The spectral side of the (conjectural) Betti geometric Langlands correspondence concerns sheaves on the character stack of an algebraic curve; in particular, the categories in question are manifestly invariant under deformations of the…

Let $X$ be a compact complex manifold of dimension $n$ and let $m$ be a positive integer with $m\leq n$. Assume that $X$ admits a K\"ahler metric $\omega$ and a weakly positive, $\partial\bar\partial$-closed, smooth $(n-m,\,n-m)$-form…

Let $U\subset K$ be an open and dense subset of a compact metric space and let $\{\Phi_t\}_{t\ge0}$ be a Markov semigroup on the space of bounded Borel measurable functions on $U$ with the strong Feller property. Suppose that for each…

Let $X$ be a smooth projective curve over an algebraically closed field $k$. Let $\mathcal{G}$ be a parahoric group scheme on $X$ as in \cite{pr}. Via the principle of Hecke correspondences, we set-up relationships between the cohomology of…

Let f:X-->Y be a semi-stable family of complex abelian varieties over a curve Y of genus q, and smooth over the complement of s points. If F(1,0) denotes the non-flat (1,0) part of the corresponding variation of Hodge structures, the…

We construct an explicit example of a stable bundle on the twistor space $\mathrm{Tw}(M)$ of a hyperk\"ahler manifold $M$ whose restrictions to all the fibres of the natural twistor projection $\pi : \mathrm{Tw}(M) \to \mathbb{CP}^1$ are…

Let $R$ be an excellent Henselian discrete valuation ring with algebraically closed residue field $k$ of any characteristic. Fix integers $r,d$ with $r\ge 2$. Let $X_R$ be a regular fibred surface over Spec($R$) with special fibre denoted…