Related papers: The stable converse soul question for positively c…

Let $G/H$ be a closed, simply connected homogeneous manifold. Suppose every stable class of real vector bundles over $G/H$ contains a homogeneous bundle. Then, for any closed, simply connected smooth manifold $M$ homotopy equivalent to…

We extend two known existence results to simply connected manifolds with positive sectional curvature: we show that there exist pairs of simply connected positively-curved manifolds that are tangentially homotopy equivalent but not…

We provide several results on the existence of metrics of non-negative sectional curvature on vector bundles over certain cohomogeneity one manifolds and homogeneous spaces up to suitable stabilization. Beside explicit constructions of the…

This paper treats the strict semi-stability of the symmetric powers $S^k E$ of a stable vector bundle $E$ of rank $2$ with even degree on a smooth projective curve $C$ of genus $g \geq 2$. The strict semi-stability of $S^2 E$ is equivalent…

We prove that, if $C$ is a smooth projective curve over the complex numbers, and $E$ is a stable vector bundle on $C$ whose slope does not lie in the interval $[-1,n-1]$, then the associated tautological bundle $E^{[n]}$ on the symmetric…

We study the stability of some homogeneous bundles on P^3 by using their representations of the quiver associated to the homgeneous bundles on P^3. In particular we show that homogeneous bundles on P^3 whose support of the quiver…

We establish a link between rational homotopy theory and the problem which vector bundles admit complete Riemannian metric of nonnegative sectional curvature. As an application, we show for a large class of simply-connected nonnegatively…

We give a necessary and sufficient condition for the projectivisation of a slope semistable vector bundle to admit constant scalar curvature K\"ahler (cscK) metrics in adiabatic classes, when the base admits a constant scalar curvature…

Let $E$ be a vector bundle over a smooth curve $C$, and $V$ a generating space of sections of $E$. We characterise Mumford linear stability of the associated projective model of $\mathbb{P} E^\vee$ in $\mathbb{P} V^\vee$ in terms of…

Let $f : X \rightarrow Y$ be a separable finite surjective map between irreducible normal projective varieties defined over an algebraically closed field, such that the corresponding homomorphism between \'etale fundamental groups $f_* :…

This paper addresses Cheeger and Gromoll's question of which vector bundles admit a complete metric of nonnegative curvature, and relates their question to the issue of which sphere bundles admit a metric of positive curvature. We show that…

J. Rosenberg's $\mathbb{S}^1$-stability conjecture states that a closed oriented manifold $X$ admits a positive scalar curvature metric iff $X\times \mathbb{S}^1$ admits a positive scalar curvature metric $h$. As pointed out by J. Rosenberg…

Let $(V,q)$ be a vector bundle on a smooth projective curve $X$ together with a quadratic form $q: \mathrm{Sym}^2(V) \ra \mathcal{O}_X$ (respectively symplectic form $q: \Lambda^2V \ra \mathcal{O}_X$). Fixing the degeneracy locus of the…

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…

The concept of RC-positivity and uniform RC-positivity is introduced by Xiaokui Yang to solve a conjecture of Yau on projectivity and rational connectedness of a compact K\"ahler manifold with positive holomorphic sectional curvature. Some…

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 $M$…

Let $E$ be a vector bundle on a curve $C$ of compact type and $V \subseteq \mathrm{H}^0(C, E)$ be a linear subspace that generates $E$. In this note, we study the stability of the syzygy bundle $M_{E,V}$ associated to $(E, V )$ over certain…

In the previous article (\cite{S}), we proved that slope stability of a holomorphic vector bundle $E$ over a polarized manifold $(X,L)$ implies Chow stability of $(\mathbb{P}E^*,\mathcal{O}_{\mathbb{P}E^*}(1)\otimes \pi^* L^k)$ for $k \gg…

The problem of the stabilization of moduli is discussed within the context of compactified strongly coupled heterotic string theory. It is shown that all geometric, vector bundle and five-brane moduli are completely fixed, within a…

Let $X$ be a smooth, connected complex projective curve of genus at least $2$. A Higgs coherent system is an augmented bundle $(E,V)$, where $E$ is a holomorphic vector bundle, and $V$ is a linear subspace of the spaces of Higgs bundles of…