Serge Cantat, Abdelghani Zeghib
We classify all holomorphic actions of higher rank lattices on compact Kaehler manifolds of dimension 3. This provides a complete answer to Zimmer's program for holomorphic actions on compact Kaehler manifolds of dimension at most 3.
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DHYM connections on higher rank holomorphic vector bundles over ${\mathbb{P}}(T_{{\mathbb{P}^{2}}})$
本文构造了首个非平凡dHYM连接的例子，并提供了存在dHYM连接的代数条件，拓展了相关研究的边界。
We classify compact Kähler manifolds $M$ of dimension $n\geq 3$ on which acts a lattice of an almost simple real Lie group of rank $\geq n-1$. This provides a new line in the so-called Zimmer program, and characterizes certain type of complex tori by a property of their automorphisms groups.
A holomorphic chain on a compact Riemann surface is a tuple of vector bundles together with homomorphisms between them. We show that the moduli space of holomorphic chains of rank one is identified with a fiber product of projective space bundles. We compute the Euler characteristic of the moduli space. The stability of chains involves real vector parameters. We also show that the variation of parameters corresponds to the characters of Gm.
There is an isomorphism between the moduli spaces of $σ$-stable holomorphic triples and some of the critical submanifolds of the moduli space of $k$-Higgs bundles of rank three, whose elements $(E,φ^k)$ correspond to variations of Hodge structure, VHS. There are special embeddings on the moduli spaces of $k$-Higgs bundles of rank three. The main objective here is to study the cohomology of the critical submanifolds of such moduli spaces, extending those embeddings to moduli spaces of holomorphic triples.
In this paper we study holomorphic rank two vector bundles on the blow up of $ {\bf C}^2$ at the origin. A classical theorem of Birchoff and Grothendieck says that any holomorphic vector bundle on the projective plane ${\bf P}^1$ splits into a sum of line bundles. If $E$ is a holomorphic vector bundle over the blow up of $ {\bf C}^2$ at the origin, then the restriction of $E$ to the exceptional divisor is a vector bundle over ${\bf P}^1$ and therefore splits. Moreover we assume that $E$ is a rank two bundle that has zero first Chern class. Hence its restriction to the exceptional divisor is of…
A numerically flat rank-two bundle without a holomorphic connection on a $\partial\bar\partial$-threefold
在满足$\partial\bar\partial$-引理的紧致复三维流形上构造秩二数值平坦全纯向量丛，证明其无全纯联络，否定回答了Cao--Deng--Matsumura的问题。
The interest in rigid vector bundles (with respect to determinant preserving deformations) stems from various sources. From a geometric point of view, non-Kähler manifolds are of particular interest with respect to this problem. In this article, a description of the various possible cases of rigid rank 2 bundles on non-Kähler surfaces is presented. The largest gap of knowledge is for minimal class VII surfaces with $b_{2}(X)\ge 4$.
We prove formulas (found by Witten in 1992 using physical methods) for intersection pairings in the cohomology of the moduli space M(n,d) of stable holomorphic vector bundles of rank n and degree d (assumed coprime) on a Riemann surface of genus g greater than or equal to 2. We also use these formulas for intersection numbers to obtain a proof of the Verlinde formula for the dimension of the space of holomorphic sections of a line bundle over M(n,d).
The existence problem for vector bundles on a smooth compact complex surface consists in determining which topological complex vector bundles admit holomorphic structures. For projective surfaces, Schwarzenberger proved that a topological complex vector bundle admits a holomorphic (algebraic) structure if and only if its first Chern class belongs to the Neron-Severi group of the surface. In contrast, for non-projective surfaces there is only a necessary condition for the existence problem (the discriminant of the vector bundles must be positive) and the difficulty of the problem resides in the…
We prove that actions of complex reductive Lie groups on a holomorphic vector bundle over a complex compact manifold are locally extendable to its local moduli space.
We extend the equivariant holomorphic Morse inequalities of circle actions to cases with torus and non-Abelian group actions on holomorphic vector bundles over Kahler manifolds and show the necessity of the Kahler condition. For torus actions, there is a set of inequalities for each choice of action chambers specifying directions in the Lie algebra of the torus. We apply the results to invariant line bundles over toric manifolds. If the group is non-Abelian, there is in addition an action of the Weyl group on the fixed-point set of its maximal torus. The sum over the fixed points can be rearra…
The Serre construction of rank two holomorphic bundles with a section is adapted to construct generalized holomorphic bundles on a generalized complex 4-manifold from the data of a set of points on an elliptic curve. The motivation is the special case of rank two Poisson modules on a complex surface with a holomorphic Poisson structure.
Given a complex manifold endowed with a $\mathbb{C}^\times$-action and a DQ-algebra equipped with a compatible holomorphic Frobenius action (F-action), we prove that if the $\mathbb{C}^\times$-action is free and proper, then the category of F-equivariant DQ-modules is equivalent to the category of modules over the sheaf of invariant sections of the DQ-algebra. As an application, we deduce the codimension three conjecture for formal microdifferential modules from the one for DQ-modules on a symplectic manifold.
This paper attempts to provide an analogue of the Novikov conjecture for algebraic (or Kähler) manifolds. Inter alia, we prove a conjecture of Rosenberg's on the birational invariance of higher Todd genera. We argue that in the algebraic geometric setting the Novikov philosophy naturally includes non-birational mappings.
Let Y be a hypersurface in a 2n-dimensional holomorphic symplectic manifold X. The restriction $σ|_Y$ of the holomorphic symplectic form induces a rank one foliation on Y. We investigate situations where this foliation has compact leaves; in such cases we obtain a space of leaves Y/F which has dimension 2n-2 and admits a holomorphic symplectic form.
Let $M$ be an irreducible smooth complex projective variety equipped with an action of a compact Lie group $G$, and let $({\mathcal L},h)$ be a $G$-equivariant holomorphic Hermitian line bundle on $M$. Given a compact connected Riemann surface $X$, we construct a $G$-equivariant holomorphic Hermitian line bundle $(L\,,H)$ on $X\times M$ (the action of $G$ on $X$ is trivial), such that the corresponding Quillen determinant line bundle $({\mathcal Q}, h_Q)$, which is a $G$--equivariant holomorphic Hermitian line bundle on $M$, is isomorphic to the given $G$--equivariant holomorphic Hermitian lin…
For every integer $g \,\geq\, 2$ we show the existence of a compact Riemann surface $Σ$ of genus $g$ such that the rank two trivial holomorphic vector bundle ${\mathcal O}^{\oplus 2}_Σ$ admits holomorphic connections with $\text{SL}(2,{\mathbb R})$ monodromy and maximal Euler class. Such a monodromy representation is known to coincide with the Fuchsian uniformizing representation for some Riemann surface of genus $g$. The construction carries over to all very stable and compatible real holomorphic structures for the topologically trivial rank two bundle over $Σ$ and gives the existence of holo…
Motivated by the Green-Griffiths conjecture, we study maximal rank holomorphic maps from $\C^p$ into complex manifolds. When $p>1$ such maps should in principle be more tractable than entire curves. We extend to this setting the jet-bundles techniques introduced by Semple, Green-Griffiths and Demailly. Our main application is the non-existence of maximal rank holomorphic maps from $\C^2$ into the very general degree $d$ hypersurface in $\bP^4$, as soon as $d\geq 93.$
In this note, we continue the investigation of a projective Kähler manifold $M$ of semi-negative holomorphic sectional curvature $H$. We introduce a new differential geometric numerical rank invariant which measures the number of linearly independent {\it truly flat} directions of $H$ in the tangent spaces. We prove that this invariant is bounded above by the nef dimension and bounded below by the numerical Kodaira dimension of $M$. We also prove a splitting theorem for $M$ in terms of the nef dimension and, under some additional hypotheses, in terms of the new rank invariant.
Using mainly tools from [B.13] and [B.15] we give a necessary and sufficient condition in order that a holomorphic action of a connected complex Lie group $G$ on a reduced complex space $X$ admits a strongly quasi-proper meromorphic quotient. We apply this characterization to obtain a result which assert that, when $G = K.B$ \ with $B$ a closed complex subgroup of $G$ and $K$ a real compact subgroup of $G$, the existence of a strongly quasi-proper meromorphic quotient for the $B-$action implies, assuming moreover that there exists a $G-$invariant Zariski open dense subset in $X$ which is good …
High rank solutions to the 2D Toda Lattice System are constructed simultaneously with the effective calculation of coefficients of the high rank commuting ordinary difference operators. Our technic is based on the study of discrete dynamics of Tyurin Parameters characterizing the stable holomorphic vector bundles over the algebraic curves (Riemann Surfaces).
We study $3$-folds with an action of a algebraic torus $T$ and finite fixed point set. In particular, assuming the torus action has (exactly) $6$ fixed points we show that aside from Mori fibre spaces, the topology of such spaces is strongly restricted. For $T= \mathbb{C}^{*}$ we prove that there are two explicit infinite families plus a finite number of exceptional cases. For $T = \mathbb{C}^{*} \times \mathbb{C}^{*}$ there are $2$ exceptional cases which are described explicitly.
The cohomology of the moduli spaces of stable bundles M(n,d), of coprime rank n and degree d, over a Riemann surface (of genus g > 1) have been intensely studied over the past three decades. We prove in this paper that the Pontryagin ring of M(n,d) vanishes in degrees above 2n(n-1)(g-1) and that this bound is strict (i.e. there exists a non-zero element of degree 2n(n-1)(g-1) in Pont(M(n,d)).) This result is a generalisation of a 1967 Newstead conjecture that Pont(M(2,1)) vanished above 4(g-1) (or equivalently that β^g =0.) These results have been independently proved by Lisa Jeffrey and Jonat…
We prove that rational and 1-rational singularities of complex spaces are stable under taking quotients by holomorphic actions of reductive and compact Lie groups. This extends a result of Boutot to the analytic category and yields a refinement of his result in the algebraic category. As one of the main technical tools vanishing theorems for cohomology groups with support on fibres of resolutions are proven.
Given a holomorphic line bundle $L$ on a compact complex torus $A$, there are two naturally associated holomorphic $Ω_A$--torsors over $A$: one is constructed from the Atiyah exact sequence for $L$, and the other is constructed using the line bundle $(p^*_1 L^*)\otimes (α^*L)$, where $α$ is the addition map on $A\times A$, and $p_1$ is the projection of $A\times A$ to the first factor. In \cite{BHR}, it was shown that these two torsors are isomorphic. The aim here is to produce a canonical isomorphism between them through an explicit construction.
For simple and simply-connected complex algebraic group G, we conjecture the existence of a functor eta_G from the category of 2-bordisms to the category of holomorphic symplectic varieties with Hamiltonian action, such that gluing of boundaries corresponds to the holomorphic symplectic quotient with respect to the diagonal action of G. We describe various properties of eta_G obtained via string-theoretic analysis. Mathematicians are urged to construct eta_G rigorously.
Proper holomorphic maps between bounded symmetric domains with small rank differences
本文研究秩差较小时不可约有界对称域间逆紧全纯映射的刚性，刻画了映射的标准嵌入分解形式，证明了部分域型组合下映射不存在，通过边界值构造模空间映射并结合几何子结构刚性完成证明。
Slices for reductive group actions in algebraic and holomorphic symplectic geometry
本文在复形几何和辛几何中推广了辛切片定理，研究复可约代数群作用的局部结构，并利用Slodowy切片和分解类进行深入分析。
$Z$-critical equations for holomorphic vector bundles on Kähler surfaces
本文证明了在紧 Kähler 曲面上，秩 2 全纯向量丛上 $Z$-正且 $Z$-临界 Hermite 度量的存在性蕴含该丛的 $Z$-稳定性，并给出了变形 Hermitian Yang-Mills 方程等特例及远离大体积极限的实例。
We describe the basic cohomology ring of the canonical holomorphic foliation on a moment-angle manifold, LVMB-manifold or any complex manifold with a maximal holomorphic torus action. Namely, we show that the basic cohomology has a description similar to the cohomology ring of a complete simplicial toric variety due to Danilov and Jurkiewicz. This settles a question of Battaglia and Zaffran, who previously computed the basic Betti numbers for the canonical holomorphic foliation in the case of a shellable fan. Our proof uses an Eilenberg-Moore spectral sequence argument; the key ingredient is t…
A locally conformally Kähler (LCK) manifold is a complex manifold whose universal cover is Kähler with monodromy group acting on the universal cover by holomorphic homotheties. A Vaisman manifold $M$ is a compact non-Kähler LCK manifold admitting an action of a holomorphic conformal flow lifting to an action on a Kähler cover by nontrivial homotheties. When the orbits of the action on $M$ are compact, it is known that every stable holomorphic vector bundle over $M$, $\dim(M) \geq 3$, is equivariant and filtrable. In the present paper we generalize this result to irregular Vaisman manifolds.
In this work we study the topology of holomorphic rank two bundles over complex surfaces. We consider bundles that are constructed by glueing and show that under certain conditions the topology of the bundle does not depend on the glueing. We present a simple classification of bundles on blown-up surfaces.
Let $X$ be a smooth projective curve of genus $g\geq 2$ over the complex numbers. A holomorphic triple $(E_1,E_2,ϕ)$ on $X$ consists of two holomorphic vector bundles $E_{1}$ and $E_{2}$ over $X$ and a holomorphic map $ϕ\colon E_{2} \to E_{1}$. There is a concept of stability for triples which depends on a real parameter $σ$. In this paper, we determine the Hodge polynomials of the moduli spaces of $σ$-stable triples with $\mathrm{rk}(E_1)=\mathrm{rk}(E_2)=2$, using the theory of mixed Hodge structures (in the cases that they are smooth and compact). This gives in particular the Poincar{é} pol…
Metrics of constant negative curvature on a compact Riemann surface are critical points of the Liouville action functional, which in recent constructions is rigorously defined as a class in a Cech-de Rham complex with respect to a suitable covering of the surface. We show that this class is the square of the metrized holomorphic tangent bundle in hermitian-holomorphic Deligne cohomology. We achieve this by introducing a different version of the hermitian-holomorphic Deligne complex which is nevertheless quasi-isomorphic to the one introduced by Brylinski in his construction of Quillen line bun…
We give a brief survey on the entropy of holomorphic self maps $f$ of compact Kähler manifolds, and rational dominating self maps $f$ of smooth projective varieties. We emphasize the connection between the entropy and the spectral radii of the induced action of $f$ on the homology of the compact manifold. The main conjecture for the rational maps states that modulo birational isomorphism all various notions of entropy and the spectral radii are equal.
Let $X$ be a smooth projective curve of genus $g\geq 2$ over the complex numbers. A holomorphic triple $(E_1,E_2,ϕ)$ on $X$ consists of two holomorphic vector bundles $E_1$ and $E_2$ over $X$ and a holomorphic map $ϕ:E_2 \to E_1$. There is a concept of stability for triples which depends on a real parameter $σ$. In this paper, we determine the Hodge polynomials of the moduli spaces of $σ$-stable triples with $\rk(E_1)=3$, $\rk(E_2)=1$, using the theory of mixed Hodge structures. This gives in particular the Poincaré polynomials of these moduli spaces. As a byproduct, we recover the Hodge polyn…
The paper considers the Dirac operator on a Riemann surface coupled to a symplectic holomorphic vector bundle W. Each spinor in the null-space generates through the moment map a Higgs bundle, and varying W one obtains a holomorphic Lagrangian subvariety in the moduli space of Higgs bundles. Applying this to the irreducible symplectic representations of SL(2) we obtain Lagrangian submanifolds of the rank 2 moduli space which link up with m-period points on the Prym variety of the spectral curve as well as Brill-Noether loci on the moduli space of semistable bundles. The case of genus 2 is inves…
Let $G$ be a reductive group. We prove that a family of polynomial actions of $G$ on $\mathbb{C}^2$, holomorphically parametrized by an open Riemann surface, is linearizable. As an application, we show that a particular class of reductive group actions on $\mathbb{C}^3$ is linearizable. The main step of our proof is to establish a certain restrictive Oka property for groups of equivariant algebraic automorphisms of $\mathbb{C}^2$.
Let $X$ be a smooth projective curve of genus $g\geq 2$ over the complex numbers. A holomorphic pair on $X$ is a couple $(E,ϕ)$, where $E$ is a holomorphic bundle over $X$ of rank $n$ and degree $d$, and $ϕ\in H^0(E)$ is a holomorphic section. In this paper, we determine the Hodge polynomials of the moduli spaces of rank 2 pairs, using the theory of mixed Hodge structures. We also deal with the case in which $E$ has fixed determinant.
We give a new proof that every holomorphic one-form on a smooth complex projective variety of general type must vanish at some point, first proven by Popa and Schnell using generic vanishing theorems for Hodge modules. Our proof relies on Simpson's results on the relation between rank one Higgs bundles and local systems of one-dimensional complex vectors spaces, and the structure of the cohomology jump loci in their moduli spaces.
Dolbeault Cohomology Growth under Curvature Rank Bounds
本文证明在紧复流形上，线丛幂次张量向量丛的 Dolbeault 上同调维数以曲率最大秩为指数增长，无需正性等假设，并推出 Kodaira--Iitaka 维数上界。
Gorenstein singularities with $\mathbb{G}_m$-action and moduli spaces of holomorphic differentials
本文通过测试配置构造Gorenstein奇点，分解全纯微分的strata，并研究其变形空间性质，同时探讨非变strata的紧致化及$K(π,1)$性质。
Irreducible Holomorphic Symplectic manifolds with an action of $\mathbb Z^4_3 : \mathcal A_6$
本文刻画了同时具有最大阶辛群 $\mathbb Z_3^4 : \mathcal A_6$ 作用和非辛自同构的 K3$^{[2]}$ 型不可约全纯辛流形，并证明 Fermat 三次四维流形的直线 Fano 簇达到自同构群最大阶 174960。
This short note considers varieties of the form $G\times S_{\text{reg}}$, where $G$ is a complex semisimple group and $S_{\text{reg}}$ is a regular Slodowy slice in the Lie algebra of $G$. Such varieties arise naturally in hyperkähler geometry, theoretical physics, and in the theory of abstract integrable systems developed by Fernandes, Laurent-Gengoux, and Vanhaecke. In particular, previous work of the author and Rayan uses a Hamiltonian $G$-action to endow $G\times S_{\text{reg}}$ with a canonical abstract integrable system. One might therefore wish to understand, in some sense, all examples…
Algebraically trivial automorphisms of irreducible holomorphic symplectic manifolds
本文扩展 Brandhorst–Cattaneo 的格论方法，对已知 IHS 流形上的代数平凡作用及偶数阶非辛自同构进行分类，并证明非平凡判别式作用情形仅有有限多种可能。
Consider a holomorphic torus action on vector bundles over a complex manifold which lifts to a holomorphic vector bundle. When the connected components of the fixed-point set are partially ordered, we construct, using sheaf-theoretical techniques, two spectral sequences that converges to the twisted Dolbeault cohomology groups and those with compact support, respectively. These spectral sequences are the holomorphic counterparts of the instanton complex in standard Morse theory. The results proved imply holomorphic Morse inequalities and fixed-point formulas on a possibly non-compact manifold.…
A holomorphic toric Poisson manifold is a nonsingular toric variety equipped with a holomorphic Poisson structure, which is invariant under the torus action. In this paper, we computed the Poisson cohomology groups for all holomorphic toric Poisson structures on $CP^n$, with the stand Poisson structure on $CP^n$ as a special case. We also computed the algebraic and the formal Poisson cohomology groups of holomorphic toric Poisson structures on $C^n$.
I prove the existence of slices for an action of a reductive complex Lie group on a Kähler manifold at certain orbits, namely those orbits that intersect the zero level set of a momentum map for the action of a compact real form of the group. I give applications of this result to symplectic reduction and geometric quantization at singular levels of the momentum map. In particular, I obtain a formula for the multiplicities of the irreducible representations occurring in the quantization in terms of symplectic invariants of reduced spaces, generalizing a result of Guillemin and Sternberg.
Let M be a symplectic manifold equipped with a Hamiltonian circle action and let L be an invariant Lagrangian submanifold of M. We study the problem of counting holomorphic disc sections of the trivial M-bundle over a disc with boundary in L through degeneration. We obtain a conjectural relationship between the potential function of L and the Seidel element associated to the circle action. When applied to a Lagrangian torus fibre of a semi-positive toric manifold, this degeneration argument reproduces a conjecture (now a theorem) of Chan-Lau-Leung-Tseng relating certain correction terms appear…
Stable vector bundles on a hyper-Kahler manifold with a rank 1 obstruction map are modular
本文证明不可约全纯辛流形上正秩斜率稳定自反层在障碍映射秩为1时高度模，并构造LLV格向量及三类导出等价来源的例子。
The paper contains a fundamental defect in its framework of using the gauge action to study the rigidity problem. As a result, the calculations leading to the main formula is also incorrect.
We conjecture that the relative Gromov-Witten potentials of elliptic fibrations are (cycle-valued) lattice quasi-Jacobi forms and satisfy a holomorphic anomaly equation. We prove the conjecture for the rational elliptic surface in all genera and curve classes numerically. The generating series are quasi-Jacobi forms for the lattice $E_8$. We also show the compatibility of the conjecture with the degeneration formula. As Corollary we deduce that the Gromov-Witten potentials of the Schoen Calabi-Yau threefold (relative to $\mathbb{P}^1$) are $E_8 \times E_8$ quasi-bi-Jacobi forms and satisfy a h…
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