Tetraplectic structures, tri-momentum maps, and quaternionic flag manifolds

Analytic Torsion for Quaternionic manifolds and related topics

In this paper we show that the Ray-Singer complex analytic torsion is trivial for even dimensional Calabi-Yau manifolds. Then we define the quaternionic analytic torsion for quaternionic manifolds and prove that they are metric independent. In dimension four, the quaternionic analytic torsion equals to the self-dual analytic torsion. For higher dimensional manifolds, the self-dual analytic torsion is a conformal invariant.

四元数复流形与 $S^1$-作用的固定点集

Quaternionic complex manifolds and fixed-point sets of $S^{1}$-actions

本文研究了四元数流形上 $S^1$-作用的固定点集，推导了与兼容复结构相关的陈类方程，并探讨了超复结构的存在性问题。

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…

Quaternionic Dolbeault complex and vanishing theorems on hyperkahler manifolds

Let (M,I,J,K) be a hyperkahler manifold of real dimension 4n, and L a non-trivial holomorphic line bundle on (M,I). Using the quaternionic Dolbeault complex, we prove the following vanishing theorem for holomorphic cohomology of L. If the Chern class c_1(L) lies in the closure $\hat K$ of the dual Kahler cone, then $H^i(L)=0$ for i>n. If c_1(L) lies in the opposite cone $-\hat K$, then $H^i(L)=0$ for i<n. Finally, if $c_1(L)$ is neither in $\hat K$ nor in $-\hat K$, then $H^i(L)=0$ for $i\neq n$.

Completeness of projective special Kähler and quaternionic Kähler manifolds

We prove that every projective special Kähler manifold with \emph{regular boundary behaviour} is complete and defines a family of complete quaternionic Kähler manifolds depending on a parameter $c\ge 0$. We also show that, irrespective of its boundary behaviour, every complete projective special Kähler manifold with \emph{cubic prepotential} gives rise to such a family. Examples include non-trivial deformations of non-compact symmetric quaternionic Kähler manifolds.

Plurisubharmonic functions on hypercomplex manifolds and HKT-geometry

A hypercomplex manifold is a manifold equipped with a triple of complex structures $I, J, K$ satisfying the quaternionic relations. We define a quaternionic analogue of plurisubharmonic functions on hypercomplex manifolds, and interpret these functions geometrically as potentials of HKT (hyperkähler with torsion) metrics, and prove a quaternionic analogue of A.D. Aleksandrov and Chern-Levine-Nirenberg theorems.

局部共形超凯勒流形上的HKT几何

HKT geometry on locally conformally hyperkähler manifolds

研究紧致局部共形超凯勒流形上的HKT几何，证明四元数蒙日-安培方程可解，确认HKT-Ricci流最大存在时间猜想，并在四元数Hopf曲面上构造非平凡HKT孤子及证明归一化流收敛。

具有H-可解李代数的超复幂零流形中的复曲线

Complex curves in hypercomplex nilmanifolds with H-solvable Lie algebras

本文研究幂零李代数上的超复结构，提出四元数可解性概念并猜想所有超复结构均满足该性质，进而证明在相应的紧超复幂零流形中一般复结构下不存在复曲线。

Complete hyperkaehler 4n-manifolds with a local tri-Hamiltonian R^n-action

We classify those manifolds mentioned in the title which have finite topological type. Namely we show any such connected M is isomorphic to a hyperkaehler quotient of a flat quaternionic vector space by an abelian group. We also show that a compact connected and simply connected 3-Sasakian manifold of dimension 4n-1 whose isometry group has rank n+1 is isometric to a 3-Sasakian quotient of a sphere by a torus. As a corollary, a compact connected quaternion-Kaehler 4n-manifold with positive scalar curvature and isometry group of rank n+1 is isometric to the quaternionic projective space or the …

射影特殊复流形的内蕴刻画

Intrinsic characterization of projective special complex manifolds

本文通过构造锥形特殊复流形及其 $\mathbb{C}^{\ast}$-商，给出射影特殊复流形的内蕴刻画，推广了 Mantegazza 的结果，并应用于构造 Obata-Ricci-平坦超复结构，进而揭示四元数结构平坦性与 c-射影 Weyl 张量消失之间的等价关系。

Holomorphic Lagrangian fibrations on hypercomplex manifolds

A hypercomplex manifold is a manifold equipped with a triple of complex structures satisfying the quaternionic relations. A holomorphic Lagrangian variety on a hypercomplex manifold with trivial canonical bundle is a holomorphic subvariety which is calibrated by a form associated with the holomorphic volume form; this notion is a generalization of the usual holomorphic Lagrangian subvarieties known in hyperkaehler geometry. An HKT (hyperkaehler with torsion) metric on a hypercomplex manifold is a metric determined by a local potential, in a similar way to the Kaehler metric. We prove that a ba…

Subvarieties of hypercomplex manifolds with holonomy in SL(n,H)

A hypercomplex manifold M is a manifold with a triple I,J,K of complex structure operators satisfying quaternionic relations. For each quaternion L=aI +bJ+cK, L^2=-1, L is also a complex structure operator on M, called an induced complex structure. We are studying compact complex subvarieties of (M,L), when L is a generic induced complex structure. Under additional assumptions (Obata holonomy contained in SL(n,H), existence of an HKT metric), we prove that (M,L) contains no divisors, and all complex subvarieties of codimension 2 are trianalytic (that is, also hypercomplex).

Group manifolds and homogeneous spaces with HKT geometry: the role of automorphisms

We present a new simple proof of the fact that certain group manifolds as well as certain homogeneous spaces G/H of dimension 4n admit a quaternionic triple of integrable complex structures that are covariantly constant with respect to the same torsionful Bismut connection, i.e. exhibit the HKT geometry. The key observation is that different complex structures are interrelated by automorphisms of the Lie algebra. To construct the quaternion triples, one only needs to construct the proper automorphisms, which is a more simple problem.

关于超Kähler流形上的Alesker-Verbitsky猜想

On the Alesker-Verbitsky conjecture on hyperKähler manifolds

本研究求解超Kähler流形上的四元数蒙日-安培方程，证明了Alesker-Verbitsky猜想的假设形式，创新在于无需假设超复结构平坦性，还得到了HKT度量的卡拉比-丘型定理。