Special Lagrangian Geometry in irreducible symplectic 4-folds

可解辛李代数的完整描述

A complete description of solvable symplectic Lie algebras

通过辛双扩张过程，完整刻画了可解辛李代数，证明了任何此类代数要么是辛不可约的，要么可通过有限次线或平面的辛双扩张从辛不可约李代数构造得到，并分类了维数不超过6的辛李代数。

Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds I

We define relative Gromov-Witten invariants and establish a general gluing theory of pseudo-holomorphic curves for symplectic cutting and contact surgery. Then, we use our general gluing theory to study the change of GW-invariants of Calabi-Yau 3-folds tranform under flops and extremal transitions. We prove a complete formula for the change of GW-invariants of any genus transform under flop and a general type I extremal transition. Other extremal transition will be handled in a subsequent paper.

Irreducible symplectic 4-folds and Eisenbud-Popescu-Walter sextics

Eisenbud Popescu and Walter have constructed certain special 4-dimensional sextic hypersurfaces as Lagrangian degeneracy loci. We prove that the natural double cover of a generic EPW-sextic is a deformation of the Hilbert square of a K3-surface and that the family of such varieties is locally complete for deformations that keep the hyperplane class of type (1,1) - thus we get an example similar to that (discovered by Beauville and Donagi) of the Fano variety of lines on a cubic 4-fold. Conversely suppose that X is an irreducible symplectic 4-fold numerically equivalent to the Hilbert square of…

Stability of the Type IIA flow and its applications in symplectic geometry

In this paper the dynamical stability of the Type IIA flow with no source near its stationary points is established. These stationary points had been shown previously by the authors to be Ricci-flat Kähler metrics on Calabi-Yau 3-folds. The dynamical stability of the Type IIA flow is then applied to prove the stability under symplectic deformations of the Kähler property for Calabi-Yau 3-folds.

K3 surfaces with non-symplectic involution and compact irreducible G_2-manifolds

We consider the connected-sum method of constructing compact Riemannian 7-manifolds with holonomy G_2 developed in math.DG/0012189. The method requires pairs of projective complex threefolds endowed with anticanonical K3 divisors, the latter `matching' via a certain non-holomorphic map. Suitable examples of threefolds were previously obtained in math.DG/0012189 by blowing up curves in Fano threefolds. In this paper, we give further suitable algebraic threefolds using theory of K3 surfaces with non-symplectic involution due to Nikulin. These threefolds are not obtainable from Fano threefolds, a…

Symplectic spinors and Hodge theory

Results on symplectic spinors and their higher spin versions, concerning representation theory and cohomology properties are presented. Exterior forms with values in the symplectic spinors are decomposed into irreducible modules including finding the hidden symmetry (Schur--Weyl--Howe type duality) given by a representation of the Lie superalgebra $\mathfrak{osp}(1|2)$ in this case. We also determine ranges of the induced exterior symplectic spinor derivatives when they are restricted to bundles induced by the irreducible submodules mentioned. This duality is used to decompose curvature tensor…

Totally non-symplectic Anosov actions on tori and nilmanifolds

We show that sufficiently irreducible totally non-symplectic Anosov actions of higher rank abelian groups on tori and nilmanifolds are smoothly conjugate to affine actions.

On the geography and botany of irreducible nonspin 4-manifolds with abelian fundamental group

The geography and botany of smooth/symplectic nonspin 4-manifolds with abelian fundamental group are addressed.

辛结构在Sasakian三维流形的特征流形上

Symplectic structure on the character varieties of Sasakian threefolds

本文研究Sasakian三维流形的特征流形上的辛结构，构造了自然的代数2形式并证明其闭合性，进而证明其在SU(r)同调空间上的限制为辛结构。

On invariants of almost symplectic connections

We study the irreducible decomposition under Sp(2n, R) of the space of torsion tensors of almost symplectic connections. Then a description of all symplectic quadratic invariants of torsion-like tensors is given. When applied to a manifold M with an almost symplectic structure, these instruments give preliminary insight for finding a preferred linear almost symplectic connection on M . We rediscover Ph. Tondeur's Theorem on almost symplectic connections. Properties of torsion of the vectorial kind are deduced.

Symplectic Lie Groups I-III

We develop the structure theory of symplectic Lie groups based on the study of their isotropic normal subgroups. The article consists of three main parts. In the first part we show that every symplectic Lie group admits a sequence of subsequent symplectic reductions to a unique irreducible symplectic Lie group. The second part concerns the symplectic geometry of cotangent symplectic Lie groups and the theory of Lagrangian extensions of flat Lie groups. In the third part of the article we analyze the existence problem for Lagrangian normal subgroups in nilpotent symplectic Lie groups.

辛和的结合性

Associativity properties of the symplectic sum

本文应用一个4重和运算，给出了两两辛和的结合性规则。这使我们能够表明，某些复纯辛$4$ -流形的椭圆表面实际上是辛变形等价的。我们还表明，爆破点可以从一个辛和的一边到另一边，而不改变所得到的流形的辛变形类的交易。

Symplectic cutting of Kaehler manifolds

We obtain estimates on the character of the cohomology of an $S^1$-equivariant holomorphic vector bundle over a Kaehler manifold $M$ in terms of the cohomology of the Lerman symplectic cuts and the symplectic reduction of $M$. In particular, we prove and extend inequalities conjectured by Wu and Zhang. The proof is based on constructing a flat family of complex spaces $M_t$ such that $M_t$ is isomorphic to $M$ for $t\not=0$, while $M_0$ is a singular reducible complex space, whose irreducible components are the Lerman symplectic cuts.