In characteristic zero, semistable principal bundles on a nonsingular projective curve with a semisimple structure group form a bounded family, as shown by Ramanathan in 1970's using the Narasimhan-Seshadri theorem. This was the first step in his construction of moduli for principal bundles. In this paper we prove boundedness in finite characteristics (other than characteristic 2), when the structure group is a semisimple, simply connected algebraic group of classical type. The main ingredient is an analogue of the Mukai-Sakai theorem (which says that any vector bundle admits a proper subbundle whose degree is `not too small') in the present situation.
In this expository article, we follow the work of Langer to prove the boundedness of the moduli space of semistable torsion-free sheaves over a projective variety, in any characteristic.
We give a new simple proof of boundedness of the family of semistable sheaves with fixed numerical invariants on a fixed smooth projective variety. In characteristic zero our method gives a quick proof of Bogomolov's inequality for semistable sheaves on a smooth projective variety of any dimension $\ge 2$ without using any restriction theorems.
Let $X$ be a normal projective variety defined over an algebraically closed field $k$ of positive characteristic. Let $G$ be a connected reductive group defined over $k$. We prove that some Frobenius pull back of a principal $G$-bundle admits the canonical reduction $E_P$ such that its extension by $P\to P/R_u(P)$ is strongly semistable. Then we show that there is only a small difference between semistability of a principal $G$-bundle and semistability of its Frobenius pull back. This and the boundedness of the family of semistable torsion free sheaves imply the boundedness of semistable princ…
Let H be a semisimple algebaric group and let X be a smooth projective curve defined over an algebraically closed field k. In the first part of this paper we show that the moduli of semistable principal H-bundles exists once given a "low-height" representation of H. We also show the projectivity of the moduli space if p > ψ, where ψis a representation theoritic index. The projectivity is a consequence of a semistable reduction theorem. The irreducibility of the moduli space of semistable H-bundles for simple and simply connected group is obtained as a consequence.
In this paper we construct the moduli of semistable principal bundles with structure group a semisimple group, over nonsingular curves in positive characteristic, with good bounds on the prime.
We show boundedness for PT-semistable objects of any Chern classes on a smooth projective three-fold $X$. Then we show that the stack of objects in the heart $\langle \Coh_{\leq 1}(X), \Coh_{\geq 2}(X)[1] \rangle$ satisfies a version of the valuative criterion for completeness. In the remainder of the paper, we give a series of results on how to compute cohomology with respect to this heart.
In this article we study the behaviour of semistable principal $G$-bundles over a smooth projective variety $X$ under the extension of structure groups in positive characteristic. We extend some results of Ramanan-Ramanathan \cite{Ramanan-Ramanathan} on rationality of instability flags and show that the associated vector bundles via representations of $G$ are not too unstable and the instability can be bounded by a constant independent of semistable bundles. As a consequence of this the boundedness of the set of isomorphism classes of semistable $G$-bundles with fixed degree and Chern classes …
Let $H$ be a semisimple algebraic group. We prove the semistable reduction theorem for $μ$--semistable principal $H$--bundles over a {\it smooth projective variety $X$} defined over the field $\bc$. When $X$ is a {\it smooth projective surface} and $H$ is simple, we construct the algebro--geometric Donaldson--Uhlenbeck compactification of the moduli space of $μ$--semistable principal $H$--bundles with fixed characteristic classes and describe its points. For large characteristic classes we show that the moduli space of $μ$--stable principal $H$--bundles is non--empty.
Let $X$ be a smooth variety defined over an algebraically closed field of arbitrary characteristic and $Ø_X(H)$ be a very ample line bundle on $X$. We show that for a semistable $X$-bundle $E$ of rank two, there exists an integer $m$ depending only on $Δ(E).H^{\dim(X)-2}$ and $H^{\dim(X)}$ such that the restriction of $E$ to a general divisor in $|mH|$ is again semistable. As corollaries we obtain boundedness results, and weak versions of Bogomolov's theorem and Kodaira's vanishing theorem for surfaces in arbitrary characteristic.
Let $G$ be a semisimple linear algebraic group over the field $\mathbb C$, and let $C$ be an irreducible smooth complex projective curve of genus at least three. We compute the Brauer group of the smooth locus of the moduli space of semistable principal $G$--bundles over $C$. We also compute the Brauer group of the moduli stack of principal $G$--bundles over $C$.
We study moduli of semistable twisted sheaves on smooth proper morphisms of algebraic spaces. In the case of a relative curve or surface, we prove results on the structure of these spaces. For curves, they are essentially isomorphic to spaces of semistable vector bundles. In the case of surfaces, we show (under a mild hypothesis on the twisting class) that the spaces are asympotically geometrically irreducible, normal, generically smooth, and l.c.i. over the base. We also develop general tools necessary for these results: the theory of associated points and purity of sheaves on Artin stacks, t…
Let $X$ be a smooth projective variety over a perfect field $k$ of characteristic $p>0$, and $V$ be a vector bundle over $X$. It is well known that if $X$ is a curve and $V$ is not strongly semistable, then some Frobenius pullback $(F^t)^*V$ is a direct sum of strongly semistable bundles. A natural question to ask is whether this still holds in higher dimension. Indranil Biswas, Yogish I. Holla, A.J. Parameswaran, and S. Subramanian showed that there is always a counterexample to this over any algebraically closed field of positive characteristic which is uncountable. However, we will produce …
Semisimple representations of the free product Z_p*Z_q determine θ-semistable representations of a specific quiver Q_pq. The dimension vectors of θ-stable representations of this quiver were classified by Le Bruyn and Adriaenssens. In this paper we classify the moduli spaces M_α^ss(Q_pq,θ) which are smooth projective varieties.
Let $X$ be an irreducible smooth projective algebraic curve of genus $g \geq 2$ over the ground field $\bc$ and let $G$ be a semisimple simply connected algebraic group. The aim of this paper is to introduce the notion of semistable and stable parahoric torsors under a certain Bruhat-Tits group scheme $\mathcal G$ and construct the moduli space of semistable parahoric $\mathcal G$--torsors; we also identify the underlying topological space of this moduli space with certain spaces of homomorphisms of Fuchsian groups into a maximal compact subgroup of $G$. The results give a generalization of th…
We give a moduli-theoretic treatment of the existence and properties of moduli spaces of semistable quiver representations, avoiding methods from geometric invariant theory. Using the existence criteria of Alper--Halpern-Leistner--Heinloth, we show that for many stability functions, the stack of semistable representations admits an adequate moduli space, and prove that this moduli space is proper over the moduli space of semisimple representations. We construct a natural determinantal line bundle that descends to a semiample line bundle on the moduli space and provide new effective bounds for …
Let $X$ be a compact connected Riemann surface of genus at least two, and let ${G}$ be a connected semisimple affine algebraic group defined over $\mathbb C$. For any $δ\in π_1({G})$, we prove that the moduli space of semistable principal ${G}$--bundles over $X$ of topological type $δ$ is simply connected. In contrast, the fundamental group of the moduli stack of principal ${G}$--bundles over $X$ of topological type $δ$ is shown to be isomorphic to $H^1(X, π_1({G}))$.
A finite quiver $Q$ without loops or 2-cycles defines a 3CY triangulated category $D(Q)$ and a finite heart $A(Q)$. We show that if $Q$ satisfies some (strong) conditions then the space of stability conditions $Stab(A(Q))$ supported on this heart admits a natural family of semisimple Frobenius manifold structures, constructed using the invariants counting semistable objects in $D(Q)$. In the case of $A_n$ evaluating the family at a special point we recover a branch of the Saito Frobenius structure of the $A_n$ singularity $y^2 = x^{n+1}$. We give examples where applying the construction to eac…
Let M be a geometrically irreducible smooth projective variety, defined over a finite field k, such that M admits a k-rational point x_0. Let \varpi(M,x_0) denote the corresponding fundamental group--scheme introduced by Nori. Let E_G be a principal G-bundle over M, where G is a reduced reductive linear algebraic group defined over the field k. Fix a polarization ξon M. We prove that the following three statements are equivalent: The principal G-bundle E_G over M is given by a homomorphism \varpi(M,x_0) --> G. There are integers b > a > 0 such that the principal G-bundle (F^b_M)^*E_G is isomor…
