Let S be an infinite-dimensional manifold of all symplectic, or hyperkahler, structures on a compact manifold M, and $Diff_0$ the connected component of its diffeomorphism group. The quotient $S/\Diff_0$ is called the Teichmuller space of symplectic (or hyperkahler) structures on M. MBM classes on a hyperkahler manifold M are cohomology classes which can be represented by a minimal rational curve on a deformation of M. We determine the Teichmuller space of hyperkahler structures on a hyperkahler manifold, identifying any of its connected components with an open subset of the Grassmannian $SO(b_2-3,3)/SO(3)\times SO(b_2-3)$ consisting of all Beauville-Bogomolov positive 3-planes in $H^2(M, R)$ which are not orthogonal to any of the MBM classes. This is used to determine the Teichmuller space of symplectic structures of Kahler type on a hyperkahler manifold of maximal holonomy. We show that any connected component of this space is naturally identified with the space of cohomology classes $v\in H^2(M,\R)$ with $q(v,v)>0$, where $q$ is the Bogomolov-Beauville-Fujiki form on $H^2(M,\R)$.
Let M be a closed symplectic manifold of volume V. We say that M admits an unobstructed symplectic packing by balls if any collection of symplectic balls (of possibly different radii) of total volume less than V admits a symplectic embedding to M. In 1994 McDuff and Polterovich proved that symplectic packings of Kahler manifolds can be characterized in terms of the Kahler cones of their blow-ups. When M is a Kahler manifold which is not a union of its proper subvarieties (such a manifold is called Campana simple) these Kahler cones can be described explicitly using the Demailly and Paun struct…
Equivariant Integration formulæ in HyperKähler Geometry
Lisa Jeffrey和Frances Kirwan发展了一个辛约化的积分理论。也就是说，给定一个具有辛群作用的辛流形，他们开发了一种方法，将约化形式的积分拉回到原始空间上的群等变形式的积分。我们寻求一个类似的辛积分公式开发的hyperKahler的情况下。这几乎是直截了当的，但我们必须克服这样的障碍，如缺乏一个超达布定理和缺乏紧凑性的情况下，超卡勒约化。
Given a closed surface endowed with a volume form, we equip the space of compatible Riemannian structures with the structure of an infinite-dimensional symplectic manifold. We show that the natural action of the group of volume-preserving diffeomorphisms by push-forward has a group-valued momentum map that assigns to a Riemannian metric the canonical bundle. We then deduce that the Teichmüller space and the moduli space of Riemann surfaces can be realized as symplectic orbit reduced spaces.
A hyperbolic 0-metric on a surface with boundary is a hyperbolic metric on its interior, exhibiting the boundary behavior of the standard metric on the Poincaré disk. Consider the infinite-dimensional Teichmüller spaces of hyperbolic 0-metrics on oriented surfaces with boundary, up to diffeomorphisms fixing the boundary and homotopic to the identity. We show that these spaces have natural symplectic structures, depending only on the choice of an invariant metric on sl(2,R). We prove that these Teichmüller spaces are Hamiltonian Virasoro spaces for the action of the universal cover of the group…
Let $(M,I,J,K)$ be a hyperkahler manifold, and $Z\subset (M,I)$ a complex subvariety in $(M,I)$. We say that $Z$ is trianalytic if it is complex analytic with respect to $J$ and $K$, and absolutely trianalytic if it is trianalytic with respect to any hyperkähler triple of complex structures $(M,I,J',K')$ containing $I$. For a generic complex structure $I$ on $M$, all complex subvarieties of $(M,I)$ are absolutely trianalytic. It is known that a normalization $Z'$ of a trianalytic subvariety is smooth; we prove that $b_2(Z')$ is no smaller than $b_2(M)$ when $M$ has maximal holonomy (that is, $…
Hyperkahler quotients by non-free actions are typically highly singular, but are remarkably still partitioned into smooth hyperkahler manifolds. We show that these partitions are topological stratifications, in a strong sense. We also endow the quotients with global Poisson structures which induce the hyperkahler structures on the strata. Finally, we give a local model which shows that these quotients are locally isomorphic to linear complex-symplectic reductions in the GIT sense. These results can be thought of as the hyperkahler analogues of Sjamaar-Lerman's theorems for symplectic reduction…
Donaldon constructed a hyperkähler moduli space $\mathcal{M}$ associated to a closed oriented surface $Σ$ with $\textrm{genus}(Σ) \geq 2$. This embeds naturally into the cotangent bundle $T^*\mathcal{T}(Σ)$ of Teichmüller space or can be identified with the almost-Fuchsian moduli space associated to $Σ$. The later is the moduli space of quasi-Fuchsian threefolds which contain a unique incompressible minimal surface with principal curvatures in $(-1,1)$. Donaldson outlined various remarkable properties of this moduli space for which we provide complete proofs in this paper: On the cotangent-bun…
In 1995, Dan Guan constructed examples of non-Kahler, simply-connected holomorphically symplectic manifolds. An alternative construction, using the Hilbert scheme of Kodaira-Thurston surface, was given by F. Bogomolov. We investigate topology and deformation theory of Bogomolov-Guan manifolds and show that it is similar to that of hyperkahler manifolds. We prove the local Torelli theorem, showing that holomorphically symplectic deformations of BG-manifolds are unobstructed, and the corresponding period map is locally a diffeomorphism. Using the local Torelli theorem, we prove the Fujiki formul…
We recall the fat-graph description of Riemann surfaces $Σ_{g,s,n}$ and the corresponding Teichmüller spaces $\mathfrak T_{g,s,n}$ with $s>0$ holes and $n>0$ bordered cusps in the hyperbolic geometry setting. If $n>0$, we have a bijection between the set of Thurston shear coordinates and Penner's $λ$-lengths and we can induce, on the one hand, the Poisson bracket on $λ$-lengths from the Poisson bracket on shear coordinates introduced by V.V.Fock in 1997 and, on the other hand, a symplectic structure $Ω_{\text{WP}}$ on the set of extended shear coordinates from Penner's symplectic structure on …
In this paper, we review or introduce several differential structures on manifolds in the general setting of real and complex differential geometry, and apply this study to Teichmüller theory. We focus on bi-Lagrangian i.e. para-Kähler structures, which consist of a symplectic form and a pair of transverse Lagrangian foliations. We prove in particular that the complexification of a real-analytic Kähler manifold has a natural complex bi-Lagrangian structure. We then specialize to moduli spaces of geometric structures on closed surfaces, and show that old and new geometric features of these spac…
In this paper, we construct the moduli space of marked oper structures on a closed, oriented smooth surface of negative Euler characteristic as a holomorphic fiber bundle over Teichmüller space. We prove that the holonomy map from the space of marked oper structures to the moduli space of reductive flat bundles is a holomorphic immersion, generalizing the known results for the moduli space of marked complex projective structures. Finally, we prove that the symplectic structure on the moduli space of marked complex projective structures extends to a pre-symplectic structure on the moduli space …
Extending Donaldson's HyperKähler Structure via Minimal Surfaces into $\mathrm{PSL}(2,\mathbb{C})$
University of Turin（都灵大学）；University of California Riverside（加州大学河滨分校）；Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM)（数学科学研究所（西班牙高等科研理事会-马德里自治大学-马德里理工大学-马德里 Complutense 大学））；University of Pisa（比萨大学）
There is a known hyperkähler structure on any complexified Hermitian symmetric space $G/K$, whose construction relies on identifying $G/K$ with both a (co)adjoint orbit and the cotangent bundle to the compact Hermitian symmetric space $G_u/K_0$. Via a family of explicit diffeomorphisms, we show that almost all of the complex structures are equivalent to the one on $G/K$; via a family of related diffeomorphisms, we show that almost all of the symplectic structures are equivalent to the one on $T^*\left(G_u/K_0\right)$. We highlight the intermediate Kähler structures, which share a holomorphic a…
Let $M$ be the moduli space of complex Lagrangian submanifolds of a hyperKähler manifold $X$, and let $\varpi : \widehat{\mathcal{A}} \rightarrow M$ be the relative Albanese over $M$. We prove that $\widehat{\mathcal{A}}$ has a natural holomorphic symplectic structure. The projection $\varpi$ defines a completely integrable structure on the symplectic manifold $\widehat{\mathcal{A}}$. In particular, the fibers of $\varpi$ are complex Lagrangians with respect to the symplectic form on $\widehat{\mathcal{A}}$. We also prove analogous results for the relative Picard over $M$.
