arXiv research

This paper explores hyperkähler structures on specific orbits using different methods.

A hyperKähler potential is a function rho that is a Kähler potential for each complex structure compatible with the hyperKähler structure. Nilpotent orbits in a complex simple Lie algebra are known to carry hyperKähler metrics admitting such potentials. In this paper, we explicitly calculate the hyperKähler potential w…

This paper connects complex hyperkähler structures to Donaldson-Thomas invariants.

New HyperKahler structure found for 3-contact distributions on Sasakian manifolds.

The paper generalizes hyperkahler metrics near Lagrangian submanifolds.

Promotes Poisson deformations to hyperkähler structures.

New construction method for special geometric structures.

We introduce the notion of hypersymplectic structure on a Courant algebroid and we prove the existence of a one-to-one correspondence between hypersymplectic and hyperkähler structures. This correspondence provides a simpler way to define a hyperkähler structure on a Courant algebroid. We show that hypersymplectic stru…

This dissertation explores T-duality between hyperkähler structures and branes on algebraic integrable systems.

Constructing Einstein metrics on bundles over hyperKähler manifolds.

Let M be a hyperkähler manifold. The S^2-family of complex structures compatible with the hyperkähler metric can be assembled into a single complex structure on Z=MxS^2; the resulting complex manifold is known as the twistor space of M. We describe the analogous construction for generalized complex structures in the se…

In this survey article we describe the geometry of toric hyperkähler varieties, which are hyperkähler quotients of the quaternionic vector spaces by tori. In particular, we discuss the Betti numbers, the cohomology ring, and variation of hyperkähler structures of these spaces with many improved results and proofs.

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 manifol…

The paper studies nilpotent structures in oriented neutral vector bundles and neutral hyperKähler structures.

Flow proves hypersymplectic structure on $T^4$ converges to hyperkähler.

Study complex structures of hyperkähler manifolds with infinite type.

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 struc…

In this paper, we construct complex metric structures on complex hypersurfaces in hyperkahler manifolds. This construction is that in contact geometry.

A non-linear generalization of the Dirac operator in 4-dimensions, obtained by replacing the spinor representation with a hyperKahler manifold admitting certain symmetries, is considered. We show that the existence of a covariantly constant, generalized spinor defines a Kahler structure on the base 4-dimensional manifo…

It is known that nilpotent orbits in a complex simple Lie algebra admit hyperKähler metrics with a single function that is a global potential for each of the Kähler structures (a hyperKähler potential). In an earlier paper the authors showed that nilpotent orbits in classical Lie algebras can be constructed as finite-d…

A flow from hypersymplectic to hyperkähler structures is described.

We discuss hypercomplex and hyperkähler structures obtained from higher degree curves in complex spaces fibring over ${\mathbb{P}}^1$.

The twist construction is a geometric T-duality that produces new manifolds from old, works well with for example hypercomplex structures and is easily inverted. It tends to destroy properties such as the hyperKähler condition. On the other hand modifications preserve the hyperKähler property, but do not have an obviou…

We study the problem of existence of geometric structures on compact complex surfaces that are related to split quaternions. These structures, called para-hypercomplex, para-hyperhermitian and para-hyperkähler are analogs of the hypercomplex, hyperhermitian and hyperkähler structures in the definite case. We show that …

Study counts rational curves on hyperKähler ALE 4-manifolds.

Study of a metric on cotangent bundle spaces of Kähler quotients.

Let O be a nilpotent orbit in g^C where G is a compact, simple group and g=Lie(G). It is known that O carries a unique G-invariant hyperKähler metric admitting a hyperKähler potential compatible with the Kirillov-Kostant-Souriau symplectic form. In this work, the hyperKähler potential is explicitly calculated when O is…

The paper describes a new type of Kähler surfaces and their properties.

Holomorphic automorphisms on hyperkähler manifolds with high entropy are Kummer examples.

New method recovers hyperkähler metrics from twistor models.

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 t…

Defines special Joyce structures for ASK manifolds encoding real HK structures.

Exotic hypercomplex structures on a torus are proven to not exist.

We shall obtain unobstructed deformations of four geometric structures: Calabi-Yau, HyperKähler, $\G$ and Spin(7) structures in terms of closed differential forms (calibrations). We develop a direct and unified construction of smooth moduli spaces of these four geometric structures and show that the local Torelli type …

Inspired by the work of Leung-Wan, we study the mean curvature flow in hyperkähler manifolds starting from hyper-Lagrangian submanifolds, a class of middle dimensional submanifolds, which contains the class of complex Lagrangian submanifolds. For each hyper-Lagrangian submanifold, we define a new energy concept called …

Solves a conjecture about hyperKähler manifolds using quaternionic Monge-Ampère equation.

Study calibrated geometry in hyperkähler cones and their related spaces.

Study para-hyperKähler geometry of anti-de Sitter structures.

The paper studies hyperkähler structures and adapted complex structures using the Monge-Ampère equation.

Quantizes the standard hyperkähler space R^(4n) without a point.

In this paper we show that the dimensionally reduced Seiberg-Witten equations lead to a Higgs field and study the resulting moduli spaces. The moduli space arising out of a subset of the equations, shown to be non-empty for a compact Riemann surface of genus g >= 1, gives rise to a family of moduli spaces carrying a hy…

We shall develop a new deformation theory of geometric structures in terms of closed differential forms. This theory is a generalization of Kodaira -Spencer theory and further we obtain a criterion of unobstructed deformations. We apply this theory to certain geometric structures: Calabi-Yau, HyperKähler, $\G$ and $\Sp…

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 com…

New insights connect strong coupling SYM amplitudes to hyperkähler geometry.

Compactifies metrics on K3 surfaces with algebraic description.

The paper proves a smooth Birman-Hilden theorem for hyperkähler manifolds.

In this paper we propose and investigate in full generality new notions of (continuous, non-isometric) symmetry on hyperkähler spaces. These can be grouped into two categories, corresponding to the two basic types of continuous hyperkähler isometries which they deform: tri-Hamiltonian isometries, on one hand, and rotat…

Let $M$ be a compact complex manifold. The corresponding Teichmuller space $\Teich$ is a space of all complex structures on $M$ up to the action of the group of isotopies. The group $Γ$ of connected components of the diffeomorphism group (known as the mapping class group) acts on $\Teich$ in a natural way. An ergodic c…