Hélène Esnault, Claus Hertling
Bolibruch's examples of representations of pi_1(P^1-finitely many points) which are not realizable by Fuchsian differential systems are adapted to curves of higher genus.
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Virtual Poincare polynomial of moduli space of semistable sheaves of rank two on reducible curves
本文研究由两条光滑曲线在一点粘合而成的稳定曲线上秩二半稳定层的模空间，证明其不可约且与分量曲线上稳定向量丛模空间的射影丛双有理，并计算其虚拟庞加莱多项式。
We constructed a projective moduli space of semistable torsion free sheaves with `fixed determinant' on a reducible curve. When a family of smooth curves degenerates to the reducible curve, our moduli space is a degeneration of the moduli spaces of semistable vector bundles with a fixed determinant on the smooth curves. Moreover, the moduli space is reduced and seminormal.
Exploring the interplay of semistable vector bundles and their restrictions on reducible curves
本文研究梳状曲线上半稳定向量丛限制到各分量时的半稳定性判据，并推广核丛半稳定性结果至一般模空间分量情形。
Studying degenerations of moduli spaces of semistable principal bundles on smooth curves leads to the problem of constructing and studying moduli spaces on singular curves. In this note, we will see that the moduli spaces of $δ$-semistable pseudo bundles on a nodal curve constructed by the first author become, for large values of $δ$, the moduli spaces for semistable singular principal bundles. The latter are reasonable candidates for degenerations and a potential basis of further developments as on irreducible nodal curves. In particular, we find a notion of semistability for principal bundle…
We give a proof of the conjecture that a semistable Higgs bundle is strongly Higgs semistable in the case of small ranks, based upon the fact that there exists a gr-semistable Griffiths-transverse filtration on a $\nabla$-invariant semistable vector bundle. The latter is a generalization of a result of Carlos Simpson.
We provide a partial classification of semistable Higgs bundles over a simply connected Calabi-Yau manifolds. Applications to a conjecture about a special class of semistable Higgs bundles are given. In particular, the conjecture is proved for K3 and Enriques surfaces, and some related classes of surfaces.
We extend the notion of the spectrum of semistable rank two bundles (initially constructed on ${\bf P}^3$ by Barth and Elencwajg) to a certain class of compact complex algebraic threefolds.
We prove a semistable reduction theorem for principal bundles on curves in almost arbitrary characteristics. For exceptional groups we need some small explicit restrictions on the characteristic.
We prove a rigidity result for automorphisms of points of certain stacks admitting adequate moduli spaces. It encompasses as special cases variations of the moduli of $G$-bundles on a smooth projective curve for a reductive algebraic group $G$. For example, our result applies to the stack of semistable $G$-bundles, stacks of semistable Hitchin pairs, and stacks of semistable parabolic $G$-bundles. Similar arguments apply to Gieseker semistable $G$-bundles in higher dimensions. We present two applications of the main result. First, we show that in characteristic $0$ every stack of semistable de…
Let $X$ be a smooth projective curve of genus $g$ over an algebraically closed field $k$ of characteristic $p>2$. We prove that any rank $3$ nilpotent semistable Higgs bundle $(E,θ)$ on $X$ is a strongly semistable Higgs bundle. This gives a partially affirmative answer to a conjecture of Lan-Sheng-Zuo \cite{LanShengZuo12ii}\footnotemark[1]. In addition, we prove a tensor product theorem for strongly semistable Higgs bundles with $p$ satisfying some bounds (Theorem \ref{TensorTheorem}). From this we reprove a tensor theorem for semistable Higgs bundles on the condition that the Lan-Sheng-Zuo c…
We show that a principal G bundle on a smooth projective curve over a finite field is strongly semistable if and only if it is defined by a representation of the fundamental group scheme of the curve into G.
Let $k$ be an algebraic closure of finite fields with odd characteristic $p$ and a smooth projective scheme $\mathbf{X}/W(k)$. Let $\mathbf{X}^0$ be its generic fiber and $X$ the closed fiber. For $\mathbf{X}^0$ a curve Faltings conjectured that semistable Higgs bundles of slope zero over $\mathbf{X}^0_{\mathbb{C}_p}$ correspond to genuine representations of the algebraic fundamental group of $\mathbf{X}^0_{\mathbb{C}_p}$ in his $p$-adic Simpson correspondence. This paper intends to study the conjecture in the characteristic $p$ setting. Among other results, we show that isomorphism classes of…
In this paper, our aim is to find the relations amongst the cohomology classes of Brill-Noether subvarieties of the moduli space of semistable bundles over an elliptic curve. We obtain results similar to the Poincaré relations on a Jacobian variety.
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.
We give a Miyaoka-type semistability criterion for principal Higgs G-bundles E on complex projective manifolds of any dimension, i.e., we prove that E is semistable and the second Chern class of its adjoint bundle vanishes if and only if certain line bundles, obtained from the characters of the parabolic subgroups of G, are numerically effective. We also give alternative characterizations in terms of a notion of numerical effectiveness of Higgs vector bundles we have recently introduced.
For any projective curve $X$ let $\bar{M}^d(X)$ be the Simpson moduli space of pure dimension one rank 1 degree $d$ sheaves that are semistable with respect to a fixed polarization $H$ on $X$. When $X$ is a reduced curve the connected component of $\bar{M}^d(X)$ that contains semistable line bundles can be considered as the compactified Jacobian of $X$. In this paper we give explicitly the structure of this compactified Simpson Jacobian for the following projective curves: tree-like curves and all reduced and reducible curves that can appear as Kodaira singular fibers of an elliptic fibration,…
Approximation of semistable bundles on smooth algebraic varieties
本文证明了具有消失数值Chern类的强半稳定丛可由商为相似斜率线丛的过滤逼近，推广了曲线和曲面情形的早期结果并证实了相关猜想。
It is known that a vector bundle E on a smooth projective curve Y defined over an algebraically closed field is semistable if and only if there is a vector bundle F on Y such that the cohomologies of E\otimes F vanish. We extend this criterion for semistability to vector bundles on curves defined over perfect fields. Let X be a geometrically irreducible smooth projective curve defined over a perfect field k, and let E be a vector bundle on X. We prove that E is semistable if and only if there is a vector bundle F on $X$ such that the cohomologies of E\otimes F vanish. We also give an explicit …
We give a class of examples of reducible (d-semistable) threefolds of CY type with two irreducible components for which (it is reasonably easy to prove that) no family of admissible genus zero stable maps sweeps out a surface, yet such stable maps occur in infinitely many degrees.
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…
We construct the Hilbert compactification of the universal moduli space of semistable vector bundles over smooth curves. The Hilbert compactification is the GIT quotient of some open part of an appropriate Hilbert scheme of curves in a Grassmannian. It has all the properties asked for by Teixidor.
Let $C$ be a curve with two smooth components and a single node. Let $\mathcal{U}_C(r,w,χ)$ be the moduli space of $w$-semistable classes of depth one sheaves on $C$ having rank $r$ on both components and Euler characteristic $χ$. In this paper, under suitable assumptions, we produce a projective bundle over the product of the moduli spaces of semistable vector bundles of rank $r$ on each components and we show that it is birational to an irreducible component of $\mathcal{U}_C(r,w,χ)$. Then we prove the rationality of the closed subset containing vector bundles with given fixed determinant.
Let X be a smooth complex projective curve of genus g bigger or equal to 1. If g is bigger than 1 assume further that X is either bielliptic or with general moduli. Under a natural condition on slopes, we prove that there exists a short exact sequence of semistable vector bundles with given ranks and degrees. The importance of the paper consists of the new method adopted for the proves.
This note is an attempt to generalize Bolibruch's theorem from the projective line to curves of higher genus. We show that an irreducible representation of the fundamental group of an open in a curve of higher genus has always a representation as a regular system of differential equations on a semistable bundle of degree 0. Vice-versa, we show that given such a bundle and 3 points on the curve, one can construct an irreducible representation of the curve minus the 3 points such that an associated regular system of differential equations lives on this bundle.
We show that the moduli space of semistable G-bundles on an elliptic curve for a reductive group G is isomorphic to a power of the elliptic curve modulo a certain Weyl group which depend on the topological type of the bundle. This generalises a result of Laszlo to arbitrary connected components and recovers the global description of the moduli space due to Friedman--Morgan--Witten and Schweigert. The proof is entirely in the realm of algebraic geometry and works in arbitrary characteristic.
In this paper, we provide an $L^2$ fine resolution of the prolongation of a nilpotent harmonic bundle in the sense of Simpson-Mochizuki (an analytic analogue of the Kashiwara-Malgrange filtrations). This is the logarithmic analogue of Cattani-Kaplan-Schmid's and Kashiwara-Kawai's results on the $L^2$ interpretation of the intersection complex. As an application, we give an $L^2$-theoretic proof to the Nadel-Kawamata-Viehweg vanishing theorem with coefficients in a nilpotent Higgs bundle.
The normal bundle of a general canonical curve of genus at least 7 is semistable
本文研究任意特征代数闭域上一般典范曲线的法丛稳定性，证明亏格不为4、6时法丛半稳定，且亏格模6余1或3时法丛稳定，将亏格下界推进至7。
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 subbundl…
We study the restrictions of rank 2 semistable vector bundles E on P^2 to conics. A Grauert-Mulich type theorem on the generic splitting is proven. The jumping conics are shown to have the scheme structure of a hypersurface J_{2} in P^5 of degree c_{2}(E) when c_{1}(E)=0 and of degree c_{2}(E)-1 when c_{1}(E)=-1. Some examples of jumping conics and jumping lines are studied in detail.
In this paper we show that semistable vector bundles on a Castelnuovo curve of genus g >= 2 have theta divisors. As a corollary, we deduce that semistable vector bundles on a smooth, general curve of genus g >= 2 which extend to semistable vector bundles on any Castelnuovo degeneration of the general curve admit a theta divisor.
A stacky approach to identifying the semistable locus of bundles
本文利用栈方法刻画曲线上主丛模栈，证明半稳定轨迹是唯一具有概形模空间的最大开子栈，并指出秩2与更高秩情形在分离模空间上的差异。
We give necessary and sufficient conditions for moduli spaces of semistable chains on a curve to be irreducible and non-empty. This gives information on the irreducible components of the nilpotent cone of GL_n-Higgs bundles and the irreducible components of moduli of systems of Hodge bundles on curves. As we do not impose coprimality restrictions, we can apply this to prove connectedness for moduli spaces of U(p,q)-Higgs bundles.
Filtrations of numerically flat Higgs bundles and curve semistable Higgs bundles on Calabi-Yau manifolds
研究卡拉比-丘流形上的希格斯丛，证明数值平坦希格斯丛存在商为稳定局部自由H-nflat的Jordan-Hölder滤过，进而推出单连通卡拉比-丘流形上曲线半稳定希格斯丛判别式为零。
It has been observed, by S. Rayan, that the complex projective surfaces that potentially admit non-trivial examples of semistable co-Higgs bundles must be found at the lower end of the Enriques-Kodaira classification. Motivated by this remark, it is natural to study the geometry of these objects over $\mathbb{P}^1 \times \mathbb{P}^1$. In this paper, we present necessary and sufficient conditions on the Chern classes $c_1,c_2$ of a bundle that guarantee the non-emptiness of the moduli space $\mathcal{M^{\text{co}}}(c_1,c_2)$ of rank 2 semistable co-Higgs bundles over $\mathbb{P}^1 \times \math…
We determine some classes of varieties X - that include the varieties with numerically effective tangent bundle - satisfying the following property: if E is a Higgs bundle such that f*E is semistable for any morphism f from a smooth projective curve to X, then E is slope semistable and the discriminant of E is zero in H4(X,R). We also characterize some classes of varieties such that the underlying vector bundle of a slope semistable Higgs bundle is always slope semistable.
An overview on curve semistable and numerically flat Higgs bundles
本文通过Higgs-Grassmannian方案定义数值平坦等正性条件，证明满足条件的Higgs丛的性质，并探讨其与曲线半稳定Higgs丛的关系。
In a recent paper Ben-Zvi and Nadler proved that the induction map from $B$-bundles of degree 0 to semistable $G$-bundles of degree 0 over an elliptic curve is a small map with Galois group isomorphic to the Weyl group of $G$. We generalize their result to all connected components of $\Bun_G$ for an arbitrary reductive group $G$. We prove that for every degree (i.e. topological type) there exists a unique parabolic subgroup such that any semistable $G$\nobreakdash-bundle of this degree has a reduction to it and moreover the induction map is small with Galois group the relative Weyl group of th…
Lefschetz principle-type theorems for curve semistable Higgs sheaves and applications to elliptic surfaces
本文证明了光滑项目ive簇上半稳定和曲线半稳定Higgs sheaves的Lefschetz原理型定理，并将其应用于椭圆面中，证明了关于曲线半稳定Higgs bundles的猜想。
Strong semistability of Higgs bundles over curves
本文研究曲线上的Higgs束强半稳定性，证明 genus ≤1 时半稳定Higgs束必强半稳定，而 genus ≥2 时构造了大秩非强半稳定的例子，补充了Lan-Sheng-Yang-Zuo和Langer的定理。
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 $X_k$, a generalised tree-like curve of genus $g \ge 2$. Let ${L}_R$ be a line bundle on $X_R$ of degree $d$ such that the degree of the restriction of ${L}_R$ on the rational components of $X_k$ is a multiple of $r$. In this article we prove the existence of a rank $r$ locally free sheaf on $X_R$ of determinant ${L}_R$ such that it is semistable on the fibre…
Let C be a curve of genus g and L a line bundle of degree 2g on C . Let M be the kernel of the evaluation map from the trivial bundle with fibre H^0(C,L) into L . We show that when L is general enough, the rank g bundle M and its exterior powers are stable and admit a reducible theta divisor.
Curve semistable Higgs bundles and smooth projective varieties whose canonical bundle is ample
本文研究典范丛丰富的光滑射影簇上的 Simpson 系统，证明其 Higgs 丛稳定性及 Guggenheimer–Yau 不等式，并在等号时得到判别类消失与曲线半稳定性，同时给出代数双曲性及正实常数下界。
Let $C$ be a smooth complex irreducible projective curve of genus $g$ with general moduli, and let $(L,H^0(L))$ be a generated complete linear series of type $(d,r+1)$ over $C$. The syzygy bundle, denoted by $M_L$, is the kernel of the evaluation map $H^0(L)\otimes\mathcal O_C\to L$. In this work we have a double purpose. The first one is to give new examples of stable syzygy bundles admitting theta divisor over general curves. We prove that if $M_L$ is strictly semistable then $M_L$ admits reducible theta divisor. The second purpose is to study the cohomological semistability of $M_L$, and in…
Semistable Higgs bundles on elliptic surfaces
本文分析了椭圆曲面上的一类 Higgs bundles，证明了在谱曲线既约等特定条件下 Higgs 场的垂直性或标量性，并将其应用于通过曲线拉回半稳定性刻画斜率半稳定 Higgs bundles 的问题。
We compute the Picard group of the moduli space $U'$ of semistable vector bundles of rank $n$ and degree $d$ on an irreducible nodal curve $Y$ and show that $U'$ is locally factorial. We determine the canonical line bundles of $U'$ and $U'_L$, the subvariety consisting of vector bundles with a fixed determinant. For rank 2, we compute the Picard group of other strata in the compactification of $U'$.
A morphism of the moduli functor of admissible semistable pairs to the Gieseker -- Maruyama moduli functor (of semistable coherent torsion-free sheaves) with the same Hilbert polynomial on the surface, is constructed. It is shown that these functors are isomorphic, and main components of moduli scheme for semistable admissible pairs are isomorphic to main components of the Gieseker -- Maruyama moduli scheme
We study the existence of canonical Kähler metrics on the projectivisation of strictly Mumford semistable holomorphic vector bundles over a complex curve. We also provide an algebro-geometric characterization of these metrics.
We compare two constructions that associate to a semistable vector bundle on a Mumford curve a representation of the Schottky group and the algebraic fundamental group respectively.
We construct a compactification of the universal moduli space of semistable principal $G$-bundles over $\overline{\textrm{M}}_{g}$, the fibers of which over singular curves are the moduli spaces of $δ$-semistable singular principal G-bundles.
Let $k $ be the algebraic closure of a finite field of odd characteristic $p$ and $X$ a smooth projective scheme over the Witt ring $W(k)$ which is geometrically connected in characteristic zero. We introduce the notion of Higgs-de Rham flow and prove that the category of periodic Higgs-de Rham flows over $X/W(k)$ is equivalent to the category of Fontaine modules, hence further equivalent to the category of crystalline representations of the étale fundamental group $π_1(X_K)$ of the generic fiber of $X$, after Fontaine-Laffaille and Faltings. Moreover, we prove that every semistable Higgs bund…
Rational Chow ring of the universal moduli space of semistable rank two bundles over genus two curves
本文确定亏格 2 曲线上秩 2 半稳定丛泛模空间的有理 Chow 环，证明其由重言类生成，并顺带得到相关泛 Jacobian 的 Chow 环。
Let $X$ be a geometrically irreducible smooth projective curve defined over a field $k$. Assume that $X$ has a $k$-rational point; fix a $k$-rational point $x\in X$. From these data we construct an affine group scheme ${\mathcal G}_X$ defined over the field $k$ as well as a principal ${\mathcal G}_X$-bundle $E_{{\mathcal G}_X}$ over the curve $X$. The group scheme ${\mathcal G}_X$ is given by a ${\mathbb Q}$--graded neutral Tannakian category built out of all strongly semistable vector bundles over $X$. The principal bundle $E_{{\mathcal G}_X}$ is tautological. Let $G$ be a linear algebraic gr…
$\mathbb{A}^1$-connectedness of moduli of semistable bundles and symplectic bundles on a curve
本文证明无限域上光滑射影曲线半稳定向量丛（固定行列式）及辛向量丛模栈的 $\mathbb{A}^1$-连通性，条件依赖于有理点存在，并推广至拟抛物情形。
Let (X, ω) be a compact connected Kaehler manifold of complex dimension d and E_G a holomorphic principal G-bundle on X, where G is a connected reductive linear algebraic group defined over C. Let Z (G) denote the center of G. We prove that the following three statements are equivalent: (1) There is a parabolic subgroup P of G and a holomorphic reduction of the structure group of E_G to P (say, E_P) such that the bundle obtained by extending the structure group of E_P to L(P)/Z(G) (where L(P) is the Levi quotient of P) admits a flat connection; (2) The adjoint vector bundle ad(E_G) is numerica…
Unirationality of the universal moduli space of semistable bundles over smooth curves
本文通过构造从有理簇上的射影丛到模空间的显式支配有理态射，证明了秩、次数任意且亏格在2到9之间的半稳定向量丛万有模空间 $U_{r,d,g}$ 的单有理性，并将结论推广到 $U_{r,\mathcal{L},g}$。
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 …
Semistable Non Abelian Hodge theorem in positive characteristic
本文证明对于任意约化群G，曲线上半稳定G-Higgs丛的模空间是半稳定平坦G-联络模空间的扭曲形式，并由此推出Hitchin态射分解定理在两种情形下形式相同。
Let E_G be a principal G-bundle over a compact connected Kähler manifold, where G is a connected reductive complex linear algebraic group. We show that E_G is semistable if and only if it admits approximate Hermitian-Einstein structures.
Let $X$ be a compact Riemann surface of genus $g\geq 3$ and let $\mathcal{M}_{\mathrm{par}}$ be the moduli space of semistable parabolic bundles over $X$. Let $\mathcal{M}_{\mathrm{parH}}$ denote the moduli space of semistable parabolic Higgs bundles over $X$. In this article, we study the Picard group of $\mathcal{M}_{\mathrm{par}}$ and $\mathcal{M}_{\mathrm{parH}}$.
Results due to Druel and Beauville show that the blowup of the intermediate Jacobian of a smooth cubic threefold X in the Fano surface of lines can be identified with a moduli space of semistable sheaves of Chern classes c_1=0, c_2=2, c_3=0 on X. Here we further identify this space with a space of matrix factorizations. This has the advantage that this description naturally generalizes to singular and even reducible cubic threefolds. In this way, given a degeneration of X to a reducible cubic threefold X_0, we obtain an associated degeneration of the above moduli spaces of semistable sheaves.
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