Michel Brion
We obtain analogues of classical results on automorphism groups of holomorphic fiber bundles, in the setting of group schemes. Also, we establish a lifting property of the connected automorphism group, for torsors under abelian varieties. These results will be applied to the study of homogeneous bundles over abelian varieties.
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Isomorphisms and automorphisms of multiprojective bundles and symmetric powers of projective bundles
本文研究射影空间上多重射影丛及射影丛相对对称幂的同构分类问题，并进一步刻画了这些对象的自同构群。
Let X be an irreducible smooth complex projective curve of genus at least 3. Fix a line bundle L on X. Let M_{Sp}(L) be the moduli space of symplectic bundles (E, ExE ---> L) on X, with the symplectic form taking values in L. We show that the automorphism group of M_{Sp}(L) is generated by automorphisms sending E to ExM, where M is a 2-torsion line bundle, and automorphisms induced by automorphisms of X.
The automorphism group of a projective bundle P(E) over a simplicial toric variety is described when the bundle E is a direct sum of line bundles. Applications to study of moduli of complete intersections on toric varieties, including Calabi Yau m-folds and Del Pezzo surfaces, are indicated.
Let X be an irreducible smooth complex projective curve of genus g at least 4. Let M(r,Λ) be the moduli space of stable vector bundles over X or rank r and fixed determinant Λ, of degree d. We give a new proof of the fact that the automorphism group of M(r,Λ) is generated by automorphisms of the curve X, tensorization with suitable line bundles, and, if r divides 2d, also dualization of vector bundles.
Let $X$ be a smooth complex projective curve, and let $x\in X$ be a point. We compute the automorphism group of the moduli space of framed vector bundles on $X$ of rank $r \geq 2$ with a framing over $x$. It is shown that this automorphism group is generated by the following three: (1) pullbacks using automorphisms of the curve $X$ that fix the marked point $x$, (2) tensorization with certain line bundles over $X$ and (3) the action of $\operatorname{PGL}_r(\mathbb{C})$ through composition with the framing.
Automorphisms of $\mathbb{P}^1$-bundles over rational surfaces
本文研究了光滑射影理性表面上的$\mathbb{P}^1$-纤维 bundle 的自同构群分类，揭示了其中性成分最大化的条件。
Given a holomorphic conic bundle without sections, we show that finite groups acting by its fiberwise bimeromorphic transformations are bounded. This provides an analog of a similar result obtained by T.Bandman and Yu.Zarhin for quasi-projective conic bundles.
We find the automorphism group of the moduli space of parabolic bundles on a smooth curve (with fixed determinant and system of weights). This group is generated by: automorphisms of the marked curve, tensoring with a line bundle, taking the dual, and Hecke transforms (using the filtrations given by the parabolic structure). A Torelli theorem for parabolic bundles with arbitrary rank and generic weights is also obtained. These results are extended to the classification of birational equivalences which are defined over "big" open subsets (3-birational maps, i.e. birational maps giving an isomor…
In this paper we construct the Poincare line bundle for the stack of Higgs bundles on smooth projective curves and show that it induces a fully-faithful Fourier-Mukai transform on the category of quasi-coherent sheaves.
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…
Parabolic automorphisms of hyperk{ä}hler manifolds: Orbits and Betti maps
本文研究了带有 lagrangian 纤维的不可约 holomorphically symplectic 流形的抛物自同构，证明了相关 Betti 映射具有最大秩，从而得出诱导平移为有限阶的纤维集合是稠密的。
We give a theorem of Leray-Hirsch type for Chow groups and use it to study the Hogde and Grothendieck's standard conjectures for algebraic fiber bundles of Leray-Hirsch type. Morevoer, the Hodge conjecture for product varieties will also be considered.
Automorphism groups and linearizability of rational Fano conic bundle threefolds
本文研究了Fano锥面三维流形的自同构群与线性化问题，证明了No. 2.18类Fano三维流形对于其自同构群的线性化性质。
We find all possible isomorphisms and 3-birational maps (i.e., birational maps which induce an isomorphism between open subsets whose respective complements have codimension at least 3) between moduli spaces of parabolic vector bundles with fixed degree. We prove that every 3-birational map can be described as a composition of tensorization by a fixed line bundle, Hecke transformations, dualization, taking pullback by an isomorphism between the curves and the action of the group of automorphisms of the Jacobian variety of the curve which fix the r-torsion. In particular, we prove a Torelli typ…
We analyze and completely describe the four cases when the Hitchin fibration on a $2$-dimensional moduli space of irregular Higgs bundles over $\mathbb{C}P^{1}$ has a single singular fiber. The case when the fiber at infinity is of type $I_0^*$ is further analyzed, and we give constructions of all the possible configurations of singular curves inelliptic fibrations having this type of singular fiber at infinity.
In this note, we give a simple necessary condition for the Zariski relative tangent space and the Grothendieck relative tangent space to be isomorphic.
Automorphisms of moduli spaces of principal bundles over a smooth curve
本文研究特征零代数闭域上殆单群G在亏格≥4光滑射影曲线上半稳定G-丛模空间的自同构群，通过分析Hitchin纤维化奇异纤维得到结果，并附带刻画了Hitchin基中两类奇异除子的不可约分支。
Bounds for the order of automorphism groups of cyclic covering fibrations of an elliptic surface
本文针对椭圆曲面的有限循环覆盖纤维化，利用纤维亏格、基曲线亏格、覆盖次数和相对典范除子平方，给出了自同构群有限子群阶的上界估计。
Given an automorphism of a smooth complex algebraic curve, there is an induced action on the moduli space of semi-stable rank 2 holomorphic bundles with fixed determinant. We give a complete description of the fixed variety in terms of moduli spaces of parabolic bundles on the quotient curve.
Given a complex projective manifold $X$ and a divisor $D$ with normal crossings, we say that the logarithmic tangent bundle $T_X(-\log D)$ is R-flat if its pull-back to the normalization of any rational curve contained in $X$ is the trivial vector bundle. If moreover $-(K_X+D)$ is nef, then the log canonical divisor $K_X+D$ is torsion and the maximally rationally chain connected fibration turns out to be a smooth locally trivial fibration with typical fiber $F$ being a toric variety with boundary divisor $D_{|F}$.
Fix $n\geq 5$ general points $p_1, \dots, p_n\in\mathbb{P}^1$, and a weight vector $\mathcal{A} = (a_{1}, \dots, a_{n})$ of real numbers $0 \leq a_{i} \leq 1$. Consider the moduli space $\mathcal{M}_{\mathcal{A}}$ parametrizing rank two parabolic vector bundles with trivial determinant on $\big(\mathbb{P}^1, p_1,\dots , p_n\big)$ which are semistable with respect to $\mathcal{A}$. Under some conditions on the weights, we determine and give a modular interpretation for the automorphism group of the moduli space $\mathcal{M}_{\mathcal{A}}$. It is isomorphic to $\left(\frac{\mathbb{Z}}{2\mathbb{Z…
Canonical bundle formula and a conjecture on certain algebraic fiber spaces by Schnell
本文利用典范丛公式，在两种独立假设下证明了Schnell关于代数纤维空间的猜想，其中一种假设涉及刚性流表示的典范类。
Automorphism groups of $\mathbb{P}^1$-bundles over geometrically ruled surfaces
本文研究了在非理几何射影曲面上的$\mathbb{P}^1$-束$(X,π)$的自同构群$\mathrm{Aut}^\circ(X)$，并证明其在包含于$\mathrm{Bir}(X/S)$中的情况下是最大化的。
Brown proved that the I-function of a toric fibration lies on the overruled Lagrangian cone of its genus zero Gromov-Witten theory, introduced by Coates and Givental. In this paper, we prove the theorem for partial flag variety fibrations. To do so, we will construct new moduli spaces generalising the idea of Ciocan-Fontanine, Kim and Maulik.
We study the groups of biholomorphic and bimeromorphic automorphisms of conic bundles over certain compact complex manifolds of algebraic dimension zero.
In this survey we discuss holomorphic $\mathbb{P}^1$-bundles $p: X \to Y$ over a non-uniruled complex compact Kähler manifold $Y$, paying a special attention to the case when $Y$ is a complex torus. We discuss so called Jordan properties of the groups $Aut(X)$ and $Bim(X)$ of its biholomorphic and bimeromorphic automorphisms, respectively.
We equate dynamical properties (e.g., positive entropy, existence of a periodic curve) of complex projective surface automorphisms with properties of the pull-back actions of such automorphisms on line bundles. We use the properties of the cohomological actions to describe the measures of maximal entropy for automorphisms with positive entropy.
Homology fiber bundles of varieties, that are not topological fiber bundles
构造平坦射影态射，它是\(\mathbb{Z}\)-同调纤维丛，全空间光滑但非正则，反驳了第二作者等人的一个猜想。
Let $\mathcal{M}_{n,d}$ be the moduli space of semi-stable rank $n$, trace-free Higgs bundles with fixed determinant of degree $d$ on a Riemann surface of genus at least $3$. We determine the following automorphism groups of $\mathcal{M}_{n,d}$: (i) the group of automorphisms as a complex analytic variety, (ii) the group of holomorphic symplectomorphisms, (iii) the group of Kähler isomorphisms, (iv) the group of automorphisms of the quaternionic structure, (v) the group of hyper-Kähler isomorphisms. When $n$ and $d$ are coprime we show that $\mathcal{M}_{n,d}$ admits an anti-holomorphic isomor…
This paper, the last in a series of three, studies vector bundles on an elliptic surface whose determinant has odd intersection number with a general fiber and uses this study to calculate certain coefficients of Donaldson polynomials.
We study the null fiber of a moment map related to dual pairs. We construct an equivariant resolution of singularities of the null fiber, and get conormal bundles of closed $ K_C $-orbits in the Lagrangian Grassmannian as the categorical quotient. The conormal bundles thus obtained turn out to be a resolution of singularities of the closure of nilpotent $ K_C $-orbits, which is a "quotient" of the resolution of the null fiber.
Let ${\cal S}{\cal U}(r, L_0)$ denote the moduli space of semi stable vector bundles of rank $r$ and fixed determinant $L_0$ of degree $d$ on a smooth curve $C$ of genus $g \geq 3$. In this paper we describe the group of automorphisms of $ {\cal S}{\cal U}(r, L_0) $. The analogue of this result is carried out for the space ${\cal U}(r,d) $ of semi stable vector bundles of rark $r$ and degree $d$. As an application of the technics we use, we give in the appendix at the end of the paper a proof of the Torelli theorem for the moduli spaces ${\cal S}{\cal U}(r, L_0) $ for any rank $r$ and degree $…
We review some basic facts on vector fields, in the complex-analytic setting, thus, obtaining a rationality result and an extension of the Birkhoff-Grothendieck theorem, as follows: (1) Let $Z$ be a compact complex manifold endowed with a very ample line bundle $L$. Denote by $\mathfrak{g}_L$ the extended Lie algebra of infinitesimal automorphisms of $L$. If the representation of $\mathfrak{g}_L$ on the space of holomorphic sections of $L$ is irreducible then $Z$ is rational; (2) Let $P$ be a holomorphic principal bundle over the Riemann sphere, with structural group $G$ whose Lie algebra is n…
Let $R$ be an integral domain over a field $k$, and $G$ a subgroup of the automorphism group of the polynomial ring $R[x_1,..., x_n]$ over $R$. In this paper, we discuss when $G$ is diagonalizable under the assumption that $G$ is diagonalizable over the field of fractions of $R$. We are particularly interested in the case where $G$ is a finite abelian group. Kraft-Russell (2014) implies that every finite abelian subgroup of ${\rm Aut}_R(R[x_1,x_2])$ is diagonalizable if $R$ is an affine PID over $k={\bf C}$. One of the main results of this paper says that the same holds for a PID $R$ over any …
The Automorphism Equivariant Hitchin Index
本文针对一维复环面下黎曼曲面自同构对秩2奇次固定行列式Higgs丛模空间上T等变行列式线丛上同调的作用，定义并研究自同构等变Hitchin指标，证明了其基于商曲面抛物Higgs丛模空间上同调配对的计算公式。
Infinitesimal automorphisms and obstruction theory on the moduli of $L$-valued $G$-Higgs bundles
研究 $L$-valued $G$-Higgs 包的微分自同构及稳定性模堆结构，构造对称完美障碍理论以定义 Vafa-Witten 不变量。
Let $\mathfrak{g}$ be a semisimple Lie algebra, $\mathfrak{t}$ its Cartan subalgebra and $W$ the Weyl group. The goal of this paper is to prove an isomorphism between suitable completions of the equivariant Borel-Moore homology of certain affine Springer fibers for $\mathfrak{g}$ and the global sections of a bundle related to a Procesi bundle on the smooth locus of a partial resolution of $(\mathfrak{t}\oplus \mathfrak{t}^*)/W$. We deduce some applications of our isomorphism including a conditional application to the center of the small quantum group. Our main method is to compare certain bimo…
To each connected component in the space of semisimple representations from the orbifold fundamental group of the base orbifold of a Seifert fibered homology 3-sphere into the Lie group U(2,1), we associate a real number called the "orbifold Toledo invariant." For each such orbifold, there exists an elliptic surface over it, called a Dolgachev surface. Using the theory of Higgs bundles on these Dolgachev surfaces, we explicitly compute all values taken on by the orbifold Toledo invariant.
On the bundles of WZW chiral blocks over the moduli space of a punctured rational curve we construct isomorphisms that implement the action of outer automorphisms of the underlying affine Lie algebra. These bundle-isomorphisms respect the Knizhnik-Zamolodchikov connection and have finite order. When all primary fields are fixed points, the isomorphisms are endomorphisms; in this case, the bundle of chiral blocks is typically a reducible vector bundle. A conjecture for the trace of such endomorphisms is presented; the proposed relation generalizes the Verlinde formula. Our results have applicat…
A geometric characterization of the structure of the group of automorphisms of an arbitrary Birkhoff-Grothendieck bundle splitting $\bigoplus_{i=1}^{r} \mathcal(m_{i})$ over $\mathbb{C}\mathbb{P}^{1}$ is provided, in terms of its action on a suitable space of generalized flags in the fibers over a finite subset $S\subset\mathbb{C}\mathbb{P}^{1}$. The relevance of such characterization derives from the possibility of constructing geometric models for diverse moduli spaces of stable objects in genus 0, such as parabolic bundles, parabolic Higgs bundles, and logarithmic connections, as collection…
Supersymmetry contains initially noninvertible objects, but it is common to deal with the invertible ones only, factorizing former in some extent. We propose to reconsider this ansatz and try to redefine such fundamental notions as supermanifolds, fiber bundles and homotopies using some weakening invertibility conditions. The prefix semi- reflects the fact that the underlying morphisms form corresponding semigroups consisting of a known group part and a new ideal noninvertible part. We found that the absence of invertibility gives us the generalization of the cocycle conditions for transition …
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