Potentials for hyper-Kahler metrics with torsion

Geometry of Hyper-Kähler Connections with Torsion

The internal space of a N=4 supersymmetric model with Wess-Zumino term has a connection with totally skew-symmetric torsion and holonomy in $\SP(n)$. We study the mathematical background of this type of connections. In particular, we relate it to classical Hermitian geometry construct homogeneous as well as inhomogeneous examples, characterize it in terms of holomorphic data, develop its potential theory and reduction theory.

Blow-ups and resolutions of strong Kähler with torsion metrics

On a compact complex manifold we study the behaviour of strong Kähler with torsion (strong KT) structures under small deformations of the complex structure and the problem of extension of a strong KT metric. In this context we obtain the analogous result of Miyaoka extension theorem. Studying the blow-up of a strong KT manifold at a point or along a complex submanifold, we prove that a complex orbifold endowed with a strong KT metric admits a strong KT resolution. In this way we obtain new examples of compact simply-connected strong KT manifolds.

On Strominger Kähler-like manifolds with degenerate torsion

In this paper, we study a special type of compact Hermitian manifolds that are Strominger Kähler-like, or SKL for short. This condition means that the Strominger connection (also known as Bismut connection) is Kähler-like, in the sense that its curvature tensor obeys all the symmetries of the curvature of a Kähler manifold. Previously, we have shown that any SKL manifold $(M^n,g)$ is always pluriclosed, and when the manifold is compact and $g$ is not Kähler, it can not admit any balanced or strongly Gauduchon (in the sense of Popovici) metric. Also, when $n=2$, the SKL condition is equivalent …

Bott-Chern 形式性与强 Kähler 带挠率和 Kähler 可解流形上的 Massey 积

Bott-Chern formality and Massey products on strong Kähler with torsion and Kähler solvmanifolds

研究几何 Bott-Chern 形式性与 SKT 度量及 Kähler 可解流形的关系，证明 6 维幂零流形存在 SKT 度量等价于几何 Bott-Chern 形式性，并构造非零 ABC-Massey 积。

Geometry of Quaternionic Kähler connections with torsion

The target space of a (4,0) supersymmetric two-dimensional sigma model with Wess-Zumino term has a connection with totally skew-symmetric torsion and holonomy contained in Sp(n).Sp(1), QKT-connection. We study the geometry of QKT-connections. We find conditions to the existence of a QKT-connection and prove that if it exists it is unique. Studying conformal transformations we obtain a lot of (compact) examples of QKT manifolds. We present a (local) description of 4-dimensional homogeneous QKT structures relying on the known result of naturally reductive homogeneous Riemannian manifolds. We con…

Symmetries of the Dirac operator with skew-symmetric torsion

In this paper, we consider the symmetries of the Dirac operator derived from a connection with skew-symmetric torsion. We find that the generalized conformal Killing-Yano tensors give rise to symmetry operators of the massless Dirac equation, provided an explicitly given anomaly vanishes. We show that this gives rise to symmetries of the Dirac operator in the case of strong Kahler with torsion (KT) and strong hyper-Kahler with torsion (HKT) manifolds.

Pseudoholomorphic Curves in Nearly Kahler $\mathbf{CP}^3$

We study pseudoholomorphic curves in the nearly Kalher $\mathbf{CP}^3$. It is shown that a class of curves called null-torsion are in one to one correspondence with the integrals of a holomorphic contact system on the usual Kahler $\mathbb{CP}^3$ studied by Bryant. Browing Bryant's result we get plenty of such curves. Rational curves are shown to be either vertical or horizontal or null-torsion. Both horizontal and null-torsion curves can be understood by mentioned Bryant's results while vertical curves are just fibers of the twistor fibration.

Torsion in almost Kaehler geometry

We study almost Kähler manifolds whose curvature tensor satisfies the second curvature condition of Gray (shortly ${\cal{AK}}_2$). This condition is interpreted in terms of the first canonical Hermitian connection. It turns out that this condition forces the torsion of this connection to be parallel in directions orthogonal to the Kähler nullity of the almost complex structure. We prove a local structure result for ${\cal{AK}}_2$ manifolds, showing that the basic pieces are manifolds with parallel torsion and special almost Kähler manifolds, a class generalizing, to some algebraic extent, the …

Conification of Kähler and hyper-Kähler manifolds

Given a Kähler manifold $M$ endowed with a Hamiltonian Killing vector field $Z$, we construct a conical Kähler manifold $\hat{M}$ such that $M$ is recovered as a Kähler quotient of $\hat{M}$. Similarly, given a hyper-Kähler manifold $(M,g,J_1,J_2,J_3)$ endowed with a Killing vector field $Z$, Hamiltonian with respect to the Kähler form of $J_1$ and satisfying $\mathcal{L}_ZJ_2= -2J_3$, we construct a hyper-Kähler cone $\hat{M}$ such that $M$ is a certain hyper-Kähler quotient of $\hat{M}$. In this way, we recover a theorem by Haydys. Our work is motivated by the problem of relating the supergr…

Analytic torsion, vortices and positive Ricci curvature

We characterize the global maximizers of a certain non-local functional defined on the space of all positively curved metrics on an ample line bundle L over a Kahler manifold X. This functional is an adjoint version, introduced by Berndtsson, of Donaldson's L-functional and generalizes the Ding-Tian functional whose critical points are Kahler-Einstein metrics of positive Ricci curvature. Applications to (1) analytic torsions on Fano manifolds (2) Chern-Simons-Higgs vortices on tori and (3) Kahler geometry are given. In particular, proofs of conjectures of (1) Gillet-Soulé and Fang (concerning …

Hermitian non-Kähler structures on products of principal $S^{1}$-bundles over complex flag manifolds and applications in Hermitian geometry with torsion

In this paper we provide an explicit description of normal almost contact structures obtained from Cartan-Ehresmann connections (gauge fields) on principal $S^{1}$-bundles over complex flag manifolds. The main feature of our approach is to employ elements of representation theory of complex simple Lie algebras in order to describe and classify these structures. We use these normal almost contact structures to explicitly describe a huge class of compact Hermitian non-Kähler manifolds obtained from products of principal $S^{1}$-bundles over complex flag manifolds. Moreover, we obtain from our de…

Elementary deformations and the hyperKähler-quaternionic Kähler correspondence

The hyperKähler-quaternionic Kähler correspondence constructs quaternionic Kähler metrics from hyperKähler metrics with a rotating circle symmetry. We discuss how this may be interpreted as a combination of the twist construction with the concept of elementary deformation, surveying results of our forthcoming paper. We outline how this leads to a uniqueness statement for the above correspondence and indicate how basic examples of c-map constructions may be realised in this context.

SO and USp Kahler and Hyper-Kahler Quotients and Lumps

We study non-linear sigma models whose target spaces are the Higgs phases of supersymmetric SO and USp gauge theories by using the Kahler and hyper-Kahler quotient constructions. We obtain the explicit Kahler potentials and develop an expansion formula to make use of the obtained potentials from which we also calculate the curvatures of the manifolds. The 1/2 BPS lumps in the U(1) x SO and U(1) x USp Kahler quotients and their effective descriptions are also studied. In this connection, a general relation between moduli spaces of vortices and lumps is discussed. We find a new singular limit of…

The Generalised Complex Geometry of $(p,q)$ Hermitian Geometries

We define $(p,q)$ hermitian geometry as the target space geometry of the two dimensional $(p,q)$ supersymmetric sigma model. This includes generalised Kähler geometry for $(2,2)$, generalised hyperkähler geometry for $(4,2)$, strong Kähler with torsion geometry for $(2,1)$ and strong hyperkähler with torsion geometry for $(4,1)$. We provide a generalised complex geometry formulation of hermitian geometry, generalising Gualtieri's formulation of the $(2,2)$ case. Our formulation involves a chiral version of generalised complex structure and we provide explicit formulae for the map to generalise…

Bianchi type A hyper-symplectic and hyper-Kähler metrics in 4D

We present a simple explicit construction of hyper-Kaehler and hyper-symplectic (also known as neutral hyper-Kaehler or hyper-parakaehler) metrics in 4D using the Bianchi type groups of class A. The construction underlies a correspondence between hyper-Kaehler and hyper-symplectic structures in dimension four.

A twistor sphere of generalized Kahler potentials on hyperkahler manifolds

We consider the generalized Kahler structures (g,J_+,J_-) that arise on a hyperkahler manifold (M,g,I,J,K) when we choose J_+ and J_- from the twistor space of M. We find a relation between semichiral and arctic superfields which can be used to determine the generalized Kahler potential for hyperkahler manifolds whose description in projective superspace is fully understood. We use this relation to determine an S^2-family of generalized Kahler potentials for Euclidean space and for the Eguchi-Hanson geometry. Cotangent bundles of Hermitian symmetric spaces constitute a class of hyperkahler man…

A note about invariant SKT structures and generalized Kahler structures on flag manifolds

We prove that any invariant strong Kahler structure with torsion (SKT structure) on a flag manifold M=G/K of a semisimple compact Lie group G is Kahler. As an application we describe invariant generalized Kahler structures on M.

Non-Kähler Calabi-Yau manifolds

We study the class of compact complex manifolds whose first Chern class vanishes in the Bott-Chern cohomology. This class includes all manifolds with torsion canonical bundle, but it is strictly larger. After making some elementary remarks, we show that a manifold in Fujiki's class C with vanishing first Bott-Chern class has torsion canonical bundle. We also give some examples of non-Kahler Calabi-Yau manifolds, and discuss the problem of defining and constructing canonical metrics on them.

具有平行挠率和常全纯截面曲率的Hermitian联络

Hermitian connections with parallel torsion and constant holomorphic sectional curvature

本文研究具有平行挠率和常全纯截面曲率的Hermitian联络，证明非零常数迫使挠率为零，从而度量是Kähler的且局部为复空间形式，并应用于Chern和Bismut联络。

Extremal almost-Kähler metrics

We generalize the notions of the Futaki invariant and extremal vector field of a compact Kähler manifold to the general almost-Kahler case and show the periodicity of the extremal vector field when the symplectic form represents an integral cohomology class modulo torsion. We also give an explicit formula of the hermitian scalar curvature in Darboux coordinates which allows us to obtain examples of non-integrable extremal almost-Kähler metrics saturating LeBrun's estimates.

On para-Kähler and hyper-para-Kähler Lie algebras

We study Lie algebras admitting para-Kähler and hyper-para-Kähler structures. We give new characterizations of these Lie algebras and we develop many methods to build large classes of examples. Bai considered para-Kähler Lie algebras as left symmetric bialgebras. We reconsider this point of view and improve it in order to obtain some new results. The study of para-Kähler and hyper-para-Kähler is intimately linked to the study of left symmetric algebras and, in particular, those admitting invariant symplectic forms. In this paper, we give many new classes of left symmetric algebras and a comple…

Homogeneous hyper-Hermitian metrics which are conformally hyper-Kähler

Let $g$ be a hyper-Hermitian metric on a simply connected hypercomplex four-manifold $M$. We show that when the isometry group $I(M,g)$ contains a subgroup acting simply transitively on $M$ by hypercomplex isometries then the metric $g$ is conformal to a hyper-Kähler metric. We describe explicitely the corresponding hyper-Kähler metrics and it follows that, in four dimensions, these are the only hyper-Kähler metrics containing a homogeneous metric in its conformal class.

平衡Bismut挠率平行流形上的Fino-Vezzoni猜想

The Fino-Vezzoni conjecture on balanced Bismut torsion-parallel manifolds

该研究针对带平衡Bismut挠率平行Hermite度量的紧复流形，证明若其还容许pluriclosed Hermite度量则必容许Kähler度量，推进了Fino-Vezzoni猜想的相关研究。

Non-umbilical quaternionic contact hypersurfaces in hyper-Kähler manifolds

We show that any compact quaternionic contact (qc) hypersurfaces in a hyper-Kähler manifold which is not totally umbilical has an induced qc structure, locally qc homothetic to the standard 3-Sasakian sphere. We also show that any nowhere umbilical qc hypersurface in a hyper-Kähler manifold is endowed with an involutive 7-dimensional distribution whose integral leafs are locally qc-conformal to the standard 3-Sasakian sphere.

Quaternionic contact hypersurfaces in hyper-Kähler manifolds

We describe explicitly all quaternionic contact hypersurfaces (qc-hypersurfaces) in the flat quaternion space $\Hnn$ and the quaternion projective space. We show that up to a quaternionic affine transformation a qc-hypersurface in $\Hnn$ is contained in one of the three qc-hyperquadrics in $\Hnn$. Moreover, we show that an embedded qc-hypersurface in a hyper-Kähler manifold is qc-conformal to a qc-Einstein space and the Riemannian curvature tensor of the ambient hyper-Kähler metric is degenerate along the hypersurface.

Twistor Spaces for QKT Manifolds

We find that the target space of two-dimensional (4,0) supersymmetric sigma models with torsion coupled to (4,0) supergravity is a QKT manifold, that is, a quaternionic Kähler manifold with torsion. We give four examples of geodesically complete QKT manifolds one of which is a generalisation of the LeBrun geometry. We then construct the twistor space associated with a QKT manifold and show that under certain conditions it is a Kähler manifold with a complex contact structure. We also show that, for every 4k-dimensional QKT manifold, there is an associated 4(k+1)-dimensional hyper-Kähler one.

具有常全纯截面曲率的平衡Bismut挠率平行流形

Balanced Bismut torsion-parallel manifold with constant holomorphic sectional curvature

本文证明了所有维数下平衡Bismut挠率平行流形的常全纯截面曲率猜想，并构造反例否证满秩B猜想。

The quaternionic/hypercomplex-correspondence

Given a quaternionic manifold $M$ with a certain $\mathrm{U}(1)$-symmetry, we construct a hypercomplex manifold $M'$ of the same dimension. This construction generalizes the quaternionic Kähler/hyper-Kähler-correspondence. As an example of this construction, we obtain a compact homogeneous hypercomplex manifold which does not admit any hyper-Kähler structure. Therefore our construction is a proper generalization of the quaternionic Kähler/hyper-Kähler-correspondence.

Metric connections with parallel skew-symmetric torsion

A geometry with parallel skew-symmetric torsion is a Riemannian manifold carrying a metric connection with parallel skew-symmetric torsion. Besides the trivial case of the Levi-Civita connection, geometries with non-vanishing parallel skew-symmetric torsion arise naturally in several geometric contexts, e.g. on naturally reductive homogeneous spaces, nearly Kähler or nearly parallel $\mathrm{G}_2$-manifolds, Sasakian and $3$-Sasakian manifolds, or twistor spaces over quaternion-Kähler manifolds with positive scalar curvature. In this paper we study the local structure of Riemannian manifolds c…

四元数-Kähler 结构的调和流

Harmonic flow of quaternion-Kähler structures

本文建立 8 维流形上 $Sp(2)Sp(1)$-结构的梯度 Dirichlet 流理论，证明近单调性、小能量长时间存在性，并构造调和 QK 孤立子及显式收敛解。

具有常数全纯截面曲率的 Bismut 挠率平行度量

Bismut torsion parallel metrics with constant holomorphic sectional curvature

本文针对紧 Hermitian 流形常数全纯截面曲率猜想，证明其对所有非平衡 BTP 流形成立，并推出对所有 Vaisman 流形成立。

广义凯勒与超凯勒商

Generalized Kahler and hyper-Kahler quotients

本文提出一种基于广义黎曼度量的广义凯勒与超凯勒结构约化理论，证明其可特殊化为通常的凯勒与超凯勒约化，并用广义凯勒约化重新解释通常的超凯勒商。

超凯勒瞬子、对称性与平坦空间

Hyper-Kähler instantons, symmetries, and flat spaces

本文分类了允许共形凯勒结构的超凯勒4流形，证明其由扭量初等态组合构成，且必须为吉本斯-霍金型，并唯一确定了具有非常数基林-矢量的半平坦Taub-NUT瞬子。

A Geometric Interpretation of the χ-{y} Genus on Hyper-Kahler Manifolds

The group SL(2) acts on the space of cohomology groups of any hyper-Kahler manifold X. The χ_{y} genus of a hyper-Kahler X is shown to have a geometric interpretation as the super trace of an element of SL(2). As a by product one learns that the generalized Casson invariant for a mapping torus is essentially the χ_{y} genus.

On the Hermitian structures of the sequence of tangent bundles of an affine manifold endowed with a Riemannian metric

Let $(M,\nabla,\langle\;,\;\rangle)$ be a manifold endowed with a flat torsionless connection $\nabla$ and a Riemannian metric $\langle\;,\;\rangle$ and $(T^kM)_{k\geq1}$ the sequence of tangent bundles given by $T^kM=T(T^{k-1}M)$ and $T^1M=TM$. We show that, for any $k\geq1$, $T^kM$ carries a Hermitian structure $(J_k,g_k)$ and a flat torsionless connection $\nabla^k$ and when $M$ is a Lie group and $(\nabla,\langle\;,\;\rangle)$ are left invariant there is a Lie group structure on each $T^kM$ such that $(J_k,g_k,\nabla^k)$ are left invariant. It is well-known that $(TM,J_1,g_1)$ is Kähler if…

Generalized orthotoric Kahler surfaces

In this paper we describe QCH Kähler surfaces $(M,g,J)$ of generalized orthotoric type. We introduce a distinguished orthonormal frame on $(M,g)$ and give the structure equations for $(M,g,J)$. In the case when $I$ is conformally Kähler and $(M,g,J)$ is not hyperkähler we integrate these structure equations and rediscover the orthotoric Kähler surfaces. We also investigate the hyperkäler case. We prove in a simple way that if $(M,g,J)$ is a hyperkähler surface with a degenerate Weyl tensor $W^-$ (i.e. QCH hyperkähler surface) then among all hyperkäler structures on $(M,g)$ there exists a Kähle…

The moduli space of flat G-bundles on a compact hyper-Kahler manifold

We prove that if G is a compact connected Lie group and X is a compact connected hyper-Kahler manifold, then the L^2 metric on (the smooth locus of) the moduli space of flat G-bundles on X is a hyper-Kahler metric.

P=W for Lagrangian fibrations and degenerations of hyper-Kähler manifolds

We identify the perverse filtration of a Lagrangian fibration with the monodromy weight filtration of a maximally unipotent degeneration of compact hyper-Kähler manifolds.

Harmonic morphisms and moment maps on hyper-Kähler manifolds

We characterise the actions, by holomorphic isometries on a Kähler manifold with zero first Betti number, of an abelian Lie group of dim\geq 2, for which the moment map is horizontally weakly conformal (with respect to some Euclidean structure on the Lie algebra of the group). Furthermore, we study the hyper-Kähler moment map $ϕ$ induced by an abelian Lie group T acting by triholomorphic isometries on a hyper-Kähler manifold M, with zero first Betti number, thus obtaining the following: If dim T=1 then $ϕ$ is a harmonic morphism. Moreover, we illustrate this on the tangent bundle of the comple…

六维近Kähler流形的超曲面

Hypersurfaces of six-dimensional nearly Kähler manifolds

该研究在六维齐次近Kähler流形框架下，证明$\u211d^6$是唯一存在常截面曲率超曲面的外围空间，并刻画了该类流形中Sasakian等四类特殊超曲面，推进了近Kähler几何的子流形研究。

Hyper-Kähler quotients of solvable Lie groups

In this paper we apply the hyper-Kähler quotient construction to Lie groups with a left invariant hyper-Kähler structure under the action of a closed abelian subgroup by left multiplication. This is motivated by the fact that some known hyper-Kähler metrics can be recovered in this way by considering different Lie group structures on $\H^p \times \H^q$ ($\H$: the quaternions). We obtain new complete hyper-Kähler metrics on Euclidean spaces and give their local expressions.

On the Kähler structures over Quot schemes

Let $S^n(X)$ be the $n$-fold symmetric product of a compact connected Riemann surface $X$ of genus $g$ and gonality $d$. We prove that $S^n(X)$ admits a Kähler structure such that all the holomorphic bisectional curvatures are nonpositive if and only if $n < d$. Let ${\mathcal Q}_X(r,n)$ be the Quot scheme parametrizing the torsion quotients of ${\mathcal O}^{\oplus r}_X$ of degree $n$. If $g \geq 2$ and $n \leq 2g-2$, we prove that ${\mathcal Q}_X(r,n)$ does not admit a Kähler structure such that all the holomorphic bisectional curvatures are nonnegative.

The adapted hyper-Kähler structure on the crown domain

Let $\,Ξ\,$ be the crown domain associated with a non-compact irreducible hermitian symmetric space $\,G/K$. We give an explicit description of the unique $\,G$-invariant adapted hyper-Kähler structure on $\,Ξ$,$\ $i.$\,$e.$\ $compatible with the adapted complex structure $\,J_{ad}\,$ and with the $\,G$-invariant Kähler structure of $\,G/K$. We also compute invariant potentials of the involved Kähler metrics and the associated moment maps.

On a variational theorem of Gauduchon and torsion-critical manifolds

In 1984, Gauduchon considered the functional of $L^2$-norm of his torsion $1$-form on a compact Hermitian manifold. He obtained the Euler-Lagrange equation for this functional, and showed that in dimension $2$ the critical metrics must be balanced (namely with vanishing torsion $1$-form). In this note we extend his result to higher dimensions, and show that critical metrics are balanced in all dimensions. We also consider the $L^2$-norm of the full Chern torsion, and show by examples that there are critical points of this functional that are not Kähler.

The Casimir operator of a metric connection with skew-symmetric torsion

For any triple $(M^n, g, \nabla)$ consisting of a Riemannian manifold and a metric connection with skew-symmetric torsion we introduce an elliptic, second order operator $Ω$ acting on spinor fields. In case of a reductive space and its canonical connection our construction yields the Casimir operator of the isometry group. Several non-homogeneous geometries (Sasakian, nearly Kähler, cocalibrated $\mathrm{G}_2$-structures) admit unique connections with skew-symmetric torsion. We study the corresponding Casimir operator and compare its kernel with the space of $\nabla$-parallel spinors.

Existence of Weak Conical Kähler-Einstein Metrics Along Smooth Hypersurfaces

The existence of \emph{weak conical Kähler-Einstein} metrics along smooth hypersurfaces with angle between $0$ and $2π$ is obtained by studying a smooth continuity method and a \emph{local Moser's iteration} technique. In the case of negative and zero Ricci curvature, the $C^0$ estimate is unobstructed; while in the case of positive Ricci curvature, the $C^0$ estimate obstructed by the properness of the \emph{twisted K-Energy}. As soon as the $C^0$ estimate is achieved, the local Moser iteration could improve the \emph{rough bound} on the approximations to a \emph{uniform $C^2$ bound}, thus pr…

非Kähler簇上的Gauduchon度量与Hermite-Einstein度量

Gauduchon metrics and Hermite-Einstein metrics on non-Kähler varieties

本文证明了任意紧Hermiteian簇上Gauduchon度量的存在性，借此在非Kähler紧正规簇上定义无挠凝聚层的斜率稳定性，并证明了斜率稳定反射层奇异Hermite-Einstein度量的存在性与唯一性。

Special Kähler structures, cubic differentials and hyperbolic metrics

We obtain necessary conditions for the existence of special Kähler structures with isolated singularities on compact Riemann surfaces. We prove that these conditions are also sufficient in the case of the Riemann sphere and, moreover, we determine the whole moduli space of special Kähler structures with fixed singularities. The tool we develop for this aim is a correspondence between special Kähler structures and pairs consisting of a cubic differential and a hyperbolic metric.

HyperKahler and quaternionic Kahler manifolds with S^1-symmetries

We study relations between quaternionic Riemannian manifolds admitting different types of symmetries. We show that any hyperKahler manifold admitting hyperKahler potential and triholomorphic action of S^1 can be constructed from another hyperKahler manifold (of lower dimention) with an action of S^1 which fixes one complex structure and rotates the other two and vice versa. We also study corresponding quaternionic Kahler manifolds equipped with a quaternionic Kahler action of the circle. In particular we show that any positive quaternionic Kahler manifold with S^1 symmetry admits a Kahler metr…

A note on the Kähler and Mori cones of hyperkähler manifolds

In the present paper we prove that, on a hyperkähler manifold, walls of the kähler cone and extremal rays of the Mori cone are determined by all divisors satisfying certain numerical conditions.

Some Remarks on Nijenhuis Bracket, Formality, and Kähler Manifolds

One (actually, almost the only effective) way to prove formality of a differentiable manifold is to be able to produce a suitable derivation $δ$ such that $dδ$-lemma holds. We first show that such derivation $δ$ generates a (1,1)-tensor field (we denote it by $R$). Then, we show that the supercommutation of $d$ and $δ$ (which is a natural, essentially necessary condition to get a $dδ$-lemma) is equivalent to vanishing of the Nijenhujis torsion of $R$. Then, we are looking for sufficient conditions that ensure the $dδ$-lemma holds: we consider the cases when $R$ is self adjoint with respect to …

Extension de la torsion analytique holomorphe aux fibrés en droites intégrables

Let $X$ be a compact Kähler manifold. We extend the notion of Quillen metric to the set of integrable line bundles on $X$. In particular, we prove that the notion of holomorphic analytic torsion extends to integrable line bundles $\bar{L}$, such that $H^q(X,L)=0$ for $q\geq 1$.

有界对称域的Kähler双曲性研究

On the Kähler-hyperbolicity of bounded symmetric domains

该研究刻画了有界对称域的Kähler双曲性长度为特殊Bergman位势常梯度长度确定的唯一常数，并建立了任意Bergman位势梯度长度的$L^\infty$范数下界的刻画。

Real projective structures on Riemann surfaces and new hyper-Kähler manifolds

The twistor space of the moduli space of solutions of Hitchin's self-duality equations can be identified with the Deligne-Hitchin moduli space of $λ$-connections. We use real projective structures on Riemann surfaces to prove the existence of new components of real holomorphic sections of the Deligne-Hitchin moduli space. Applying the twistorial construction we show the existence of new hyper-Kähler manifolds associated to any compact Riemann surface of genus $g\geq2$. These hyper-Kähler manifolds can be considered as moduli spaces of (certain) singular solutions of the self-duality equations.