Narrow Results

Let G be a connected reductive linear algebraic group defined over an algebraically closed field k of positive characteristic. Let Z(G) ⊂ G be the center, and , where each G_i is simple with trivial center. For i ∈ [1, m], let ρ_i : G → G_i be the natural projection. Fix a proper parabolic subgroup P of G such that for each i ∈ [1, m], the image ρ_i(G) ⊂ G_i is a proper parabolic subgroup. Fix a strictly anti-dominant character χ of P such that χ is trivial on Z(G). Let M be a smooth projective variety, defined over k, equipped with a very ample line bundle ξ. Let E_G → M be a principal G-bundle. We prove that the following six statements are equivalent:

(1) The line bundle E_G(χ) → E_G/P associated to the principal P-bundle E_G → E_G/P for the character χ is numerically effective.

(2) The sequence of principal G-bundles is bounded, where F_M is the absolute Frobenius morphism of M.

(3) The principal G-bundle E_G is strongly semistable with respect to ξ, and c₂(ad(E_G)) is numerically equivalent to zero.

(4) The principal G-bundle E_G is strongly semistable with respect to ξ, and [c₂(ad(E_G))c₁(ξ)^d-2] = 0.

(5) The adjoint vector bundle ad(E_G) is numerically effective.

(6) For every pair of the form (Y,ψ), where Y is an irreducible smooth projective curve and ψ : Y → M is a morphism, the principal G-bundle ψ*E_G → Y is semistable.

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 $ℝ$, such that $G$ is nonabelian and has one simple factor. We prove that the isomorphism class of the moduli space of principal $G$-bundles on $X$ determine uniquely the isomorphism class of $X$.

In this paper, we compute the pseudo-effective cones of various projective bundles ${\mathbb{P}}_X(E)$ over higher dimensional varieties $X$ under some assumptions on $X$ as well as on the vector bundle $E$. We also compute the volume function on fiber product $\mathbb{P}(E){\times }_C\mathbb{P}(F)$ of two projective bundles over a smooth irreducible complex projective curve $C$. In particular, we show that the volume function on the fiber product of two ruled surfaces is of polynomial type.

We give an algorithm to determine whether a kernel sheaf over a smooth projective curve over an algebraically closed field is semistable. The algorithm uses symmetric powers to make destabilizing subbundles visible as global sections.

After providing a suitable definition of numerical effectiveness for Higgs bundles, and a related notion of numerical flatness, in this paper we prove, together with some side results, that all Chern classes of a Higgs-numerically flat Higgs bundle vanish, and that a Higgs bundle is Higgs-numerically flat if and only if it is has a filtration whose quotients are flat stable Higgs bundles. We also study the relation between these numerical properties of Higgs bundles and (semi)stability.

Let $E$ be a vector bundle of rank $r$ on a smooth complex projective variety $X$. In this paper, we compute the nef and pseudoeffective cones of divisors on the Grassmann bundle Gr${}_X(k,E)$ parametrizing $k$-dimensional subspaces of the fibers of $E$, where $1\le k\le$ rk($E$), under assumptions on $X$ as well as on the vector bundle $E$. In particular, we show that nef cone and the pseudoeffective cone of Gr${}_X(k,E)$ coincide if and only if nef cone and pseudoeffective cone of $X$ coincide under the assumption that $E$ is a slope semistable bundle on $X$ with $c_2$(End($E$))=0. We also discuss about the nefness and ampleness of the universal quotient bundle $Q_k$ on Gr${}_X(k,E)$.

We announce a result about the extension of the Hitchin–Kobayashi correspondence to principal Higgs bundles. A principal Higgs bundle on a compact Kähler manifold, with structure group a connected linear algebraic reductive group, is semistable if and only if it admits an approximate Hermitian–Yang–Mills structure.