Restriction of Tangent Bundle and Semistability

Let G be a simple linear algebraic group defined over ? and P ? G a maximal proper parabolic subgroup such that m: = dim _? G/P ≥ 5. Let ι: Z ₁ ∩ Z ₂?G/P be a smooth complete intersection such that degree(Z _i ) ≥ (m ? 1)·index(G/P)/m, i = 1, 2. Then the vector bundle ι*T(G/P) → Z ₁ ∩ Z ₂ is semistable.     

Semi-self-decomposable distributions on Z<Subscript>+</Subscript>

We present a notion of semi-self-decomposability for distributions with support in Z ₊. We show that discrete semi-self-decomposable distributions are infinitely divisible and are characterized by the absolute monotonicity of a specific function. The class of discrete semi-self-decomposable distributions is shown to contain the discrete semistable distributions and the discrete geometric semistable distributions. We identify a proper subclass of semi-self-decomposable distributions that arise as weak limits of subsequences of binomially thinned sums of independent Z ₊-valued random variables. Multiple semi-self-decomposability on Z ₊ is also discussed.     

Moduli spaces for principal bundles in arbitrary characteristic

In this article, we solve the problem of constructing moduli spaces of semistable principal bundles (and singular versions of them) over smooth projective varieties over algebraically closed ground fields of positive characteristic.     

Semistability of Frobenius Direct Image of Representations of Cotangent Bundles

Let k be an algebraically closed field of characteristic p > 0, X a smooth projective variety over k with a fixed ample divisor H, F_X : X → X the absolute Frobenius morphism on X. Let E be a rational GL_n(k)-bundle on X, and ρ : GL_n(k) → GL_m(k) a rational GL_n(k)-representation of degree at most d such that ρ maps the radical RGL_n(k)) of GL_n(k) into the radical R(GL_m(k)) of GL_m(k). We show that if \(F_X^{N*}(E)\) is semistable for some integer \(N \ge {\max {_{0 < r < m}}}(_r^m) \cdot {\log _p}(dr)\), then the induced rational GL_m(k)-bundle E(GL_m(k)) is semistable. As an application, if dimX = n, we get a sufficient condition for the semistability of Frobenius direct image \(F_{X*}(\rho*(\Omega_X^1))\), where \(\rho*(\Omega_X^1)\) is the vector bundle obtained from \(\Omega_X^1\) via the rational representation ρ.     

Semistability of Logarithmic Cotangent Bundle on Some Projective Manifolds

In this article, we investigate the semistability of logarithmic de Rham sheaves on a smooth projective variety (X, D), under suitable conditions. This is related to existence of Kähler–Einstein metric on the open variety. We investigate this problem when the Picard number is one. Fix a normal crossing divisor D on X and consider the logarithmic de Rham sheaf Ω _X (log D) on X. We prove semistability of this sheaf, when the log canonical sheaf K _X  + D is ample or trivial, or when ?K _X  ? D is ample, i.e., when X is a log Fano n-fold of dimension n ≤ 6. We also extend the semistability result for Kawamata coverings, and this gives examples whose Picard number can be greater than one.