Benoît Claudon
In this paper, we show an extension type theorem for twisted pluricanonical sections on a family of smooth projective manifolds (the twisting line bundle being pseudo-effective and having a prescribed multiplier ideal on the central fiber).
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In this note we prove that certain twisted flag varieties carry Ulrich bundles.
We count invariants of the moduli spaces of twisted Higgs bundles on a smooth projective curve.
Remarks on Higgs bundles twisted by a vector bundle
本文研究紧 Riemann 曲面上秩二向量丛扭曲的 Higgs 丛，通过 Hecke 变换刻画相伴线丛扭曲对，并建立其模空间与谱覆盖上秩一无挠 Higgs 层正像之间的谱对应。
We prove a canonical identifications of the spaces of generalized theta functions on the moduli spaces of anti-invariant vector bundles in the ramified case and the conformal blocks associated to twisted Kac-Moody affine algebras.
On the Ulrichness of twisted syzygies and dual syzygies bundles
本文研究了扭曲斜坐标系和对偶斜坐标系包在不同极化下的Ulrich性质，并探讨了相关条件和部分结论。
We establish a striking connection between Abramovich's and Vistoli's twisted bundles and Gieseker vector bundles. This note grew out of an attempt to understand a recent draft of Seshadri.
A $\mathrm{U}(p,q)$-Higgs bundle on a Riemann surface (twisted by a line bundle) consists of a pair of holomorphic vector bundles, together with a pair of (twisted) maps between them. Their moduli spaces depend on a real parameter $α$. In this paper we study wall crossing for the moduli spaces of $α$-polystable twisted $\mathrm{U}(p,q)$-Higgs bundles. Our main result is that the moduli spaces are birational for a certain range of the parameter and we deduce irreducibility results using known results on Higgs bundles. Quiver bundles and the Hitchin-Kobayashi correspondence play an essential rol…
We study polystable Higgs bundles twisted by a line bundle over a compact Kähler manifold. These form a Tannakian category when the first and second Chern classes of the bundle are zero. In this paper we identify the corresponding Tannaka group in the case in which the line bundle is of finite order. This group is described in terms of the pro-reductive completion of the fundamental group of the manifold, and the character associated to the line bundle.
Twisted Higgs bundles and coendoscopy
本文研究了被中心gerbe扭曲的G-Higgs束，并展示了扭曲Hitchin纤维的Ngô乘积公式类比。
Let $X$ be a smooth complex projective curve of genus $g$ and let $L$ be a line bundle on $X$ with $\mathrm{deg}\,L>0$. Let $\mathbf{M}$ be the moduli space of semistable rank 2 $L$-twisted Higgs bundles with trivial determinant on $X$. Let $\mathbf{M}_{X}$ be the moduli space of stable rank 2 $L$-twisted Higgs bundles with determinant $\mathcal{O}(-x)$ for some $x\in X$ on $X$. We construct a cycle in the product of a stack of rational maps from nonsingular curves to $\mathbf{M}_{X}$ and $\mathrm{Pic}^{2\mathrm{deg}\,L}(X)$ by using Hecke modifications of a stable $L$-twisted Higgs bundle in …
Let $X$ be a Hyperkähler manifold, and let $H$ be an ample divisor on $X$. We give a lower bound in terms of the Beauville-Bogomolov form $q(H)$ for the twisted cotangent bundle $Ω_X \otimes H$ to be pseudoeffective. If $X$ is deformation equivalent to the Hilbert scheme of a K3 surface the lower bound can be written down explicitly and we study its optimality.
Hausel and Rodriguez-Villegas conjectured that the intersection form on the moduli space of stable PGL_n-Higgs bundles on a curve vanishes if the degree is coprime to n. In this note we prove this conjecture. Along the way we show that moduli spaces of stable chains are irreducible for stability parameters larger than the stability condition induced form stability of Higgs bundles.
Twisted ind-Grassmannians are ind-varieties $\GG$ obtained as direct limits of Grassmannians $G(r_m,V^{r_m})$, for $m\in\ZZ_{>0}$, under embeddings $ϕ_m:G(r_m,V^{r_m})\to G(r_{m+1}, V^{r_{m+1}})$ of degree greater than one. It has been conjectured in \cite{PT} and \cite{DP} that any vector bundle of finite rank on a twisted ind-Grassmannian is trivial. We prove this conjecture under the assumption that the ind-Grassmannian $\GG$ is sufficiently twisted, i.e. that $\lim_{m\to\infty}\frac{r_m}{°ϕ_1...\degϕ_m}=0$.
Our main goal in this article is to establish a quantitative version of the positivity properties of twisted relative pluricanonical bundles and their direct images. The notion of "singular Hermitian metric" on vector bundles (together with an appropriate definition of positivity of the associated curvature) plays a major role here, and its properties are studied via a version of Ohsawa-Takegoshi extension theorem. Part of this article is based on the joint work of the first named author with Bo Berndtsson, and it can be seen as an expanded and updated version of it.
We characterize smooth irreducible varieties with Ulrich twisted normal bundle.
Spectral coverings without embeddings
本文研究扭曲希格斯丛谱对应的一种弱化版本，通过有限覆盖及其上的向量丛构造扭曲希格斯丛，并分析其稳定性及与传统谱覆盖数据的关系。
In this paper, we provide two different resolutions of structural sheaves of projectivized tangent bundles of smooth complete intersections. These resolutions allow in particular to obtain convenient (and completely explicit) descriptions of cohomology of twisted symmetric powers of cotangent bundles of complete intersections, which are easily implemented on computer. We then provide several applications. First, we recover the known vanishing theorems on the subject, and show that they are optimal via some non-vanishing theorems. Then, we study the symmetric algebra of global sections of symme…
We prove a closed formula counting semistable twisted Higgs bundles of fixed rank and degree over a smooth projective curve defined over a finite field. We also prove a formula for the Donaldson-Thomas invariants of the moduli spaces of twisted Higgs bundles over a curve.
Several important cases of vector bundles with extra structure (such as Higgs bundles and triples) may be regarded as examples of twisted representations of a finite quiver in the category of sheaves of modules on a variety/manifold/ringed space. We show that the category of such representations is an abelian category with enough injectives by constructing an explicit injective resolution. Using this explicit resolution, we find a long exact sequence that computes the Ext groups in this new category in terms of the Ext groups in the old category. The quiver formulation is directly reflected in…
On varieties with Ulrich twisted conormal bundles
本文研究射影空间中余法丛的某次扭曲为Ulrich向量丛的簇，证明此类簇必为曲线，并构造了若干满足该性质的曲线例子。
Let C be a generic curve, E a generic vector bundle on C. Then, for every line bundle on C the twisted Petri map P:H^0(C,L\otimes E)\otimes H^0(C, K\otimes L^*\otimes E^{*})--> H^0(C, K) is injective.
Let $X$ be a smooth projective variety over $\mathbb C$. We prove that a twisted Higgs vector bundle $(\calE\, ,θ)$ on $X$ admits an Einstein--Hermitian connection if and only if $(\calE\, ,θ)$ is polystable. A similar result for twisted vector bundles (no Higgs fields) was proved by S. Wang in \cite{Wa}. Our approach is simpler.
Let $S \subset \mathbb{P}^g$ be a smooth $K3$ surface of degree $2g-2$, $g \geq 3$. We classify all the cases for which $h^0(\mathcal{N}_{S/\mathbb{P}^g}(-2)) \neq 0$ and the cases for which $h^0(\mathcal{N}_{S/\mathbb{P}^g}(-2)) < h^0(\mathcal{N}_{C/\mathbb{P}^{g-1}}(-2))$ for $C \subset \mathbb{P}^{g-1}$ a general canonical curve section of $S$.
In this paper, we consider the preservation of stability by using the notion of Twisted stability. As applications, (1) we show that moduli spaces of vector bundles on K3 and abelian surfaces are irreducible, (2) we compute Hodge polynomials of some moduli spaces of vector bundles on Enriques surfaces and on elliptic surfaces with a section.
Moduli spaces of twisted equivariant G-bundles over a curve
本文构造了紧黎曼曲面上多稳定(θ,c)-扭曲Γ-等变G-丛的粗模空间，推广了Ramanathan的G-丛模空间构造，并给出非连通群Ĝ-丛模空间的GIT构造。
The cherry on top of this stacky paper is the following: for any g>1 we give a finite group G such that the moduli space of connected admissible G-covers of genus g is a smooth, fine moduli space, which is a Galois cover of the moduli space of stable curves. The proof relies on methods introduced by Looijenga and Pikaart-De Jong, and on the theory of twisted G-covers, a theory announced without proofs in math.AG/9811059, section 3, and developed in the first few sections of this paper. This includes a moduli description of the normalization of the Harris-Mumford space of admissible covers, and…
Picard bundles and Twisted Picard bundles on the Jacobian of a curve
研究了曲线雅可比上皮卡束的限制及其稳定性，揭示了稳定ACM束的构造与theta divisor的关系。
On the vanishing of Twisted negative K-theory and homotopy invariance
本文重新探讨扭曲K-理论的Weibel猜想，研究Prüfer整环的扭曲负K-群消失问题，并发现有限维Prüfer整环的扭曲K-理论具有同伦不变性。
Semihomogenous vector bundles, $\mathbb Q$-twisted sheaves, duality, and linear systems on abelian varieties
本文研究了abelian varieties上的半同调向量束与$\mathbb Q$-扭曲sheaves之间的关系，并探讨了其在上同调秩函数和线性系统中的应用。
Let $V$ be a vector bundle over a smooth curve $C$. In this paper, we study twisted Brill--Noether loci parametrising stable bundles $E$ of rank $n$ and degree $e$ with the property that $h^0 (C, V \otimes E) \ge k$. We prove that, under conditions similar to those of Teixidor i Bigas and of Mercat, the Brill-Noether loci are nonempty, and in many cases have a component which is generically smooth and of the expected dimension. Along the way, we prove the irreducibility of certain components of both twisted and "nontwisted" Brill--Noether loci. We describe the tangent cones to the twisted Bril…
Torelli Theorem for moduli stacks of vector bundles on tame Deligne Mumford curves
本文证明了向量丛模空间的两个Torelli定理：从固定行列式模栈恢复曲线，及从扭曲模空间恢复曲线和Brauer类，方法涉及特殊线性线性化、Hitchin构造和Kummer覆盖等。
In this note, we obtain various formulas for the curvature of the $L^2$ metric on the direct image of the relative canonical bundle twisted by a holomorphic line bundle endowed with a positively curved metric with analytic singularities, generalizing some of Berndtsson's seminal results in the smooth case. When the twist is assumed to be relatively big, we further provide a very explicit lower bound for the curvature of the $L^2$ metric.
In this note we prove a new explicit formula for the invariants of moduli spaces of twisted Higgs bundles over P^1 and we relate these invariants to the invariants of moduli spaces of representations of some infinite symmetric quiver. The formula can be easily implemented for computations and it is in agreement with the existing conjectures for the invariants of twisted Higgs bundles over curves. Results of this note were presented in Oberwolfach in May 2014.
We generalize the classical Beauville-Narasimhan-Ramanan correspondence to the case of parabolic Higgs bundles with regular singularities and Higgs $V$-bundles. Using this correspondence along with Bott-Morse theoretic techniques we provide an exact component count for moduli spaces of maximal parabolic $\text{Sp}\left( 2n,\mathbb{R} \right)$-Higgs bundles with fixed parabolic structure.
We prove the Kobayashi-Hitchin correspondence and the approximate Kobayashi-Hitchin correspondence for twisted holomorphic vector bundles on compact Kähler manifolds. More precisely, if $X$ is a compact manifold and $g$ is a Gauduchon metric on $X$, a twisted holomorphic vector bundle on $X$ is $g-$polystable if and only if it is $g-$Hermite-Einstein, and if $X$ is a compact Kähler manifold and $g$ is a Kähler metric on $X$, then a twisted holomorphic vector bundle on $X$ is $g-$semistable if and only if it is approximate $g-$Hermite-Einstein.
Given a vector bundle $E$ on a smooth projective curve or surface $X$ carrying the structure of a $V$-twisted Hitchin pair for some vector bundle $V$, we observe that the associated tautological bundle $E^{[n]}$ on the punctual Hilbert scheme of points $X^{[n]}$ has an induced structure of a $((V^\vee)^{[n]})^\vee$-twisted Hitchin pair, where $(V^\vee)^{[n]}$ is a vector bundle on $X^{[n]}$ constructed using the dual $V^\vee$ of $V$. In particular, a Higgs bundle on $X$ induces a logarithmic Higgs bundle on the Hilbert scheme $X^{[n]}$. We then show that the known results on stability of tauto…
On varieties with Ulrich twisted tangent bundles
本文研究使得扭曲切丛 $T_X(k)$ 成为 Ulrich 向量丛的射影簇 $X$，给出了 $k$ 的精确界，分类了 $k=1$ 的情形，并证明了当 $n \ge 4$ 时 $k=2$ 的情形不存在。
In this paper, we study holomorphic rank-2 vector bundles on non-K\" ahler elliptic surfaces. Our main tool for analysing these bundles is of course the spectral cover. However, given the non-Kähler condition, the elliptic surfaces we are considering do not have sections and gerbes naturally arise in this context. The spectral construction presented in this paper is a modification of the Fourier-Mukai transform for elliptic fibrations without a section. After examining some of the properties of this Fourier-Mukai transform, we give a complete classification of vector bundles on these surfaces.
We study twisted vector bundles of infinite rank on gerbes, giving a new spin on Grothendieck's famous problem on the equality of the Brauer group and cohomological Brauer group. We show that the relaxed version of the question has an affirmative answer in many, but not all, cases, including for any algebraic space with the resolution property and any algebraic space obtained by pinching two closed subschemes of a projective scheme. We also discuss some possible theories of infinite rank Azumaya algebras, consider a new class of "very positive" infinite rank vector bundles on projective variet…
The paper is Part III of our ongoing project to study a case of Crepant Transformation Conjecture: K-equivalence Conjecture for ordinary flops. In this paper we prove the invariance of quantum rings for general ordinary flops, whose local models are certain non-split toric bundles over arbitrary smooth base. An essential ingredient in the proof is a quantum splitting principle, which reduces a statement in Gromov--Witten theory on non-split bundles to the case of split bundles.
Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces
该研究证明亏格≥2的光滑射影曲线上，秩与次数互素的李代数胚联络模空间的动机类仅依赖线丛次数，与李代数胚结构及线丛本身无关，且等于对应扭曲希格斯丛模空间的动机类，并推广至E多项式。
Higgs bundles twisted by a vector bundle
本文推广光滑复射影曲线上的Higgs丛理论，用秩2向量丛替代典范丛扭曲希格斯场，定义了Hitchin映射、给出谱对应并阐述对应Hitchin方程的Hitchin-Kobayashi对应，填补了曲线与高维簇Higgs丛理论的中间空白。
Morita theory of twisted sheaves on $μ_{n}$-gerbes of line bundles
本文研究线丛 $μ_{n}$-gerbes 上扭曲层的 Morita 理论，给出 Azumaya 代数 Morita 等价的显式判据，并以反例说明 Căldăraru 猜想对一般 Deligne–Mumford 栈不成立。
Flow lines on the moduli space of rank $2$ twisted Higgs bundles
本文研究紧黎曼曲面上秩为2扭曲希格斯丛模空间中希格斯场L^2范数平方的梯度流线，证明其空间可用割线簇进行代数几何分类，并给出莫尔斯理论紧化的代数解释。
We study the algebraic geometry of twisted Higgs bundles of cyclic type along complex curves. These objects, which generalize ordinary cyclic Higgs bundles, can be identified with representations of a cyclic quiver in a twisted category of coherent sheaves. Referring to the Hitchin fibration, we produce a fibre-wise geometric description of the locus of such representations within the ambient twisted Higgs moduli space. When the genus is 0, we produce a concrete geometric identification of the moduli space as a vector bundle over an associated (twisted) A-type quiver variety; we count the numb…
We show that the residual categories of quadric surface bundles are equivalent to the (twisted) derived categories of some scheme under the following hypotheses. Case 1: The quadric surface bundle has a smooth section. Case 2: The total space of the quadric surface bundle is smooth and the base is a smooth surface. We provide two proofs in Case 1 describing the scheme as the hyperbolic reduction and as a subscheme of the relative Hilbert scheme of lines, respectively. In Case 2, the twisted scheme is obtained by performing birational transformations to the relative Hilbert scheme of lines. Fin…
The enumerative geometry of r-th roots of line bundles is the subject of Witten's conjecture and occurs in the calculation of Gromov-Witten invariants of orbifolds. It requires the definition of the suitable compact moduli stack and the generalization of the standard techniques from the theory of moduli of stable curves. In math.AG/0603687, we construct a compact stack by describing the notion of stability in the context of twisted curves. In this paper, by working with stable twisted curves, we extend Mumford's formula for the Chern character of the Hodge bundle to the direct image of the uni…
We prove a closed formula counting semistable twisted (or meromorphic) Higgs bundles of fixed rank and degree over a smooth projective curve of genus g, defined over a finite field, when the degree of the twisting line bundle is at least 2g-2; this includes the case of usual Higgs bundles. We obtain at the same time a closed expression for the Donaldson-Thomas invariants of the moduli spaces of twisted Higgs bundles. We similarly deal with twisted quiver bundles of type A (affine or finite), obtaining in particular a Harder-Narasimhan-type formula counting semistable U(p,q)-bundles over any sm…
This is a review article on some applications of generalised parabolic structures to the study of torsion free sheaves and $L$-twisted Hitchin pairs on nodal curves. In particular, we survey on the relation between representations of the fundamental group of a nodal curve and the moduli spaces of generalised parabolic bundles and generalised parabolic $L$-twisted Hitchin pairs on its normalisation as well as on an analogue of the Hitchin map for generalised parabolic $L$-twisted Hitchin pairs.
Logarithmic spectral correspondence for $V$--twisted Higgs bundles on punctured curves
研究了穿孔曲线上的$V$-扭曲Higgs包的对数性质，通过Hecke紧致化方法分类了此类包，并扩展了ABK的谱对应理论到穿孔情况。
Ordinarily, quiver varieties are constructed as moduli spaces of quiver representations in the category of vector spaces. It is also natural to consider quiver representations in a richer category, namely that of vector bundles on some complex variety equipped with a fixed sheaf that twists the morphisms. Representations of A-type quivers in this twisted category --- known in the literature as "holomorphic chains" --- have practical use in questions concerning the topology of the moduli space of Higgs bundles. In that problem, the variety is a Riemann surface of genus at least 2, and the twist…
KW-Euler Classes via Twisted Symplectic Bundles
本文计算分类栈上秩2向量丛的KW-Euler类，通过新策略恢复Levine的Witt-层上同调公式，并发展扭曲辛丛及Künneth公式等基础理论。
Given a double quiver, we study homological algebra of twisted quiver sheaves with the moment map relation using the short exact sequence of Crawley-Boevey, Holland, Gothen, and King. Then in a certain one-parameter space of the stability conditions, we obtain a wall-crossing formula for the generalized Donaldson-Thomas invariants of the abelian category of framed twisted quiver sheaves on a smooth projective curve. To do so, we closely follow the approach of Chuang, Diaconescu, and Pan in the ADHM quiver case, which makes use of the theory of Joyce and Song. The invariants virtually count fra…
Very stable and wobbly loci for elliptic curves
本文研究亏格1复代数曲线上由线丛扭曲的非常稳定与摆动丛，证明扭曲稳定丛非非常稳定，并利用半稳定性证明摆动轨迹为除数，进而验证了Drinfeld关于其闭性与维数的猜想。
Non-connected Lie groups, twisted equivariant bundles and coverings
本文研究由有限群作用和中心值2-上循环定义的群扩张，建立Ĝ-丛与Galois覆盖上扭曲Γ-等变丛的对应，并用非阿贝尔上同调加以描述。
On the behavior of analytic torsion for twisted canonical bundles under degenerations
研究在退化情况下扭曲canonical包的分析扭量行为，证明其对数在判别位形附近具有渐近展开，并提供非等价情况下的系数公式。
A twisted Higgs bundle on a Kähler manifold $X$ is a pair $(E,ϕ)$ consisting of a holomorphic vector bundle $E$ and a holomorphic bundle morphism $ϕ\colon M\otimes E \to E$ for some holomorphic vector bundle $M$. Such objects were first considered by Hitchin when $X$ is a curve and $M$ is the tangent bundle of $X$, and also by Simpson for higher dimensional base. The Hitchin-Kobayashi correspondence for such pairs states that $(E,ϕ)$ is polystable if and only if $E$ admits a hermitian metric solving the Hitchin equation. This correspondence is a powerful tool to decide whether there exists a s…
Duality invariance of Faltings heights, Hodge line bundles and global periods
本文证明整体域上 Abel 簇与其对偶的 Faltings 高度、带自然度量的 Hodge 线丛，以及 BSD 猜想中的实周期和整体周期均具有对偶不变性。
We prove an extension of the nonabelian Hodge theorem in which the underlying objects are twisted torsors over a smooth complex projective variety. In the prototypical case of $GL_n$-torsors, one side of this correspondence consists of vector bundles equipped with an action of a sheaf of twisted differential operators in the sense of Beilinson and Bernstein; on the other side, we endow them with appropriately defined twisted Higgs data. The proof we present here is formal, in the sense that we do not delve into the analysis involved in the classical nonabelian Hodge correspondence. Instead, we…
Topological recursion and variations of spectral curves for twisted Higgs bundles
本文针对扭 Higgs 丛提出保留亚纯奇性的扭拓扑递归，并证明零亏格 Eynard-Orantin 微分给出谱曲线周期矩阵的 Taylor 展开。
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