Explains hyperkähler manifolds and their properties.
problem None explicitly stated; focuses on introduction and properties.
method Elementary introduction and survey of properties.
result Properties of hyperkähler manifolds discussed.
Study pseudo-hyperkähler geometry of curves in hyperkähler twistor spaces.
problem Understanding the geometry of rational curves in twistor spaces.
method Investigate pseudo-hyperkähler geometry of higher degree rational curves.
result Characterize the pseudo-hyperkähler structure of rational curves.
Study collapsing geometry of hyperkähler 4-manifolds and prove conjectures.
problem Understanding collapsing geometry of hyperkähler 4-manifolds.
method Investigation of collapsing geometry and proving conjectures.
result Proved two conjectures about collapsed limits and asymptotic behavior of hyperkähler 4-manifolds.
The study proves curvature bounds for hyperkähler manifolds.
problem Proving curvature invariants of hyperkähler manifolds.
method Analytical proof in complex dimension four, experimental proof in higher dimensions, verification for known manifolds.
result The conjectured curvature invariants are proven to be positive/negative for all known hyperkähler manifolds up to dimension eight.
Extends parabolic study to flat hyperkähler manifolds.
problem Study problems in hyperhermitian geometry.
method Extends elliptic approach to parabolic setting.
result Solves problems in hyperhermitian geometry.
New HyperKahler structure found for 3-contact distributions on Sasakian manifolds.
problem Finding a HyperKahler structure for 3-contact distributions on Sasakian manifolds.
method Defined a special metric connection and proved curvature properties.
result 3-Sasakian manifolds with constant φα-sectional curvatures have constant holomorphic sectional curvatures in their HyperKahler contact distribution. Parabolic automorphisms on hyperkahler manifolds act ergodically on fibers.
problem Understanding the dynamics of parabolic automorphisms on hyperkahler manifolds.
method Analyzing the action of parabolic automorphisms on the second cohomology group and fibers of Lagrangian fibrations.
result Parabolic automorphisms preserving Lagrangian fibrations act ergodically on the fibers.
Study on stability of hyperkähler flow in 4-manifolds.
problem Stability of hyperkähler flow in 4-manifolds.
method Extending results from mean curvature flow for minimal surfaces to hyperkähler flow.
result Obtained a dynamic stability theorem for hyperkähler flow.
We discuss various aspects of moment map geometry in symplectic and hyperKähler geometry. In particular, we classify complete hyperKähler manifolds of dimension 4n with a tri-Hamiltonian action of a torus of dimension n, without any assumption on the finiteness of the Betti numbers. As a result we find that the hyp…
New construction method for special geometric structures.
problem Creating strong HKT and generalized hyperkähler structures.
method Mapping tori construction of skew-symmetric torsion.
result Non-product structures with parallel skew-symmetric torsion.
Constructing Einstein metrics on bundles over hyperKähler manifolds.
problem Finding Einstein metrics on bundles over hyperKähler manifolds.
method Constructing quaternion-Kähler and hypercomplex structures, studying infinitesimal symmetries, and performing QK reduction.
result Producing several explicit quaternion-Kähler metrics and new proofs of correspondences.
Holomorphic automorphisms on hyperkähler manifolds with high entropy are Kummer examples.
problem Characterizing holomorphic automorphisms with high entropy on hyperkähler manifolds.
method Using Jensen's inequality and properties of stable and unstable distributions, the authors show uniform contraction and expansion, leading to the conclusion that the manifold is birational to a torus quotient.
result Holomorphic automorphisms with high entropy on hyperkähler manifolds are Kummer examples.
This paper explores hyperkähler structures on specific orbits using different methods.
problem Investigate hyperkähler structures on adjoint orbits.
method Infinite dimensional hyperkähler reduction and symplectic geometry.
result Differences and connections between two methods are thoroughly investigated.
Proves C^2,alpha estimates for elliptic equations on hyperkähler manifolds.
problem Elliptic equations on hypercomplex manifolds.
method Proves C^2,alpha estimates under suitable assumptions.
result Solutions to specific elliptic equations on hyperkähler manifolds satisfy C^2,alpha estimates.
Study proves long-term solutions to a specific equation on hyperKähler manifolds.
problem Proving long-term existence and uniqueness of solutions to a parabolic quaternionic Monge-Ampère equation.
method Proved long-term existence and uniqueness using parabolic quaternionic Monge-Ampère type equation.
result Solution converges smoothly to the unique solution of the Monge-Ampère equation.
Study elliptic equations on hyperhermitian manifolds with flat hyperkähler metric.
problem Solving elliptic equations on compact hyperhermitian manifolds.
method Adapting Székelyhidi's approach to the hypercomplex setting.
result Prove a priori estimates for solutions to elliptic equations.
Solves a conjecture about hyperKähler manifolds using quaternionic Monge-Ampère equation.
problem Proving the solvability of a conjecture about hyperKähler manifolds with torsion.
method Solving quaternionic Monge-Ampère equation on hyperKähler manifolds without assuming flatness.
result Proves the conjecture for hyperKähler manifolds with trivial canonical bundle.
Solves long-time solutions for a specific equation on hyperkähler manifolds.
problem Finding solutions to a specific equation on hyperkähler manifolds.
method Introduced a parabolic quaternionic Monge-Ampère equation and proved its long-time solvability.
result Smooth convergence to a solution of the quaternionic Monge-Ampère equation.
We give an equivalent definition of compact locally conformally hyperkähler manifolds in terms of the existence of a nondegenerate complex two-form with natural properties. This is a conformal analogue of Beauville's theorem stating that a compact Kähler manifold admitting a holomorphic symplectic form is hyperkähler.
New minimal 2-spheres found in hyperkähler 4-manifolds, unstable and not holomorphic.
problem Characterizing stable minimal surfaces in hyperkähler 4-manifolds.
method Gluing construction using Scherk and Taub-NUT surfaces, harmonic map parametrization.
result Existence of unstable minimal 2-spheres with degree-1 Gauss lift, not holomorphic.
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…
The paper proves a smooth Birman-Hilden theorem for hyperkähler manifolds.
problem Understanding the smooth mapping class groups of certain 4-manifolds.
method Geometric and Teichmüller-theoretic methods.
result Proves a global Torelli theorem for generalized Enriques manifolds.
We survey briefly the definition of the Rozansky-Witten invariants, and review their relevance to the study of compact hyperkahler manifolds. We consider how various generalisations of the invariants might prove useful for the study of non-compact hyperkahler manifolds, of quaternionic-Kahler manifolds, and of relation…
Study surjectivity of Kirwan map for generalized hyperkähler reduction.
problem Establishing surjectivity of Kirwan map for a specific class of Hamiltonian manifolds.
method Defined a close analogue of hyperkähler reduction for manifolds with equivariant functions under semi-linear G-actions. result Surjectivity of Kirwan map proved for the defined class of manifolds.
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…
Study rigid classes on hyperkahler manifolds, showing general ones are rigid.
problem Characterize rigid classes on compact hyperkahler manifolds.
method Analyze eigenvectors of hyperbolic automorphisms and use BBF form.
result General parabolic classes on hyperkahler manifolds are rigid.
Let S be an infinite-dimensional manifold of all symplectic, or hyperkahler, structures on a compact manifold M, and Diff0 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 generalizes hyperkahler metrics near Lagrangian submanifolds.
problem Constructing hyperkahler structures near complex Lagrangian submanifolds.
method Generalization of Feix-Kaledin theorem and deformations of holomorphic symplectic structures.
result Hyperkahler structures can be constructed on symplectic realizations of holomorphic Poisson manifolds.
Rozansky and Witten proposed a 3-dimensional sigma-model whose target space is a hyperkähler manifold. They conjectured that this theory has an associated TQFT, with Hilbert spaces given by certain cohomology groups of the hyperkähler manifold. On the other hand, there is a certain modified TQFT constructed by Murakami…
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…
A weight system on graph homology was constructed by Rozansky and Witten using a compact hyperkähler manifold. A variation of this construction utilizing holomorphic vector bundles over the manifold gives a weight system on chord diagrams. We investigate these weights from the hyperkähler geometry point of view.
The Kobayashi pseudometric on a complex manifold is the maximal pseudometric such that any holomorphic map from the Poincaré disk to the manifold is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this conjecture fo…
The paper embeds CR manifolds into twistor spaces and constructs neutral hyperkähler metrics.
problem Embedding CR manifolds into twistor spaces and constructing neutral hyperkähler metrics.
method Embedding a real analytic twistor CR manifold into the twistor space of a Poincaré-Einstein metric, constructing the associated Fefferman ambient metric as a neutral hyperkähler metric.
result The construction of neutral hyperkähler metrics associated with twistor CR manifolds.
Given a projective hyperkahler manifold with a holomorphic Lagrangian fibration, we prove that hyperkahler metrics with volume of the torus fibers shrinking to zero collapse in the Gromov-Hausdorff sense (and smoothly away from the singular fibers) to a compact metric space which is a half-dimensional special Kahler ma…
Let M be a compact irreducible hyperkahler manifold, from Bogomolov inequality [V1] we obtain forbidden values of the second Betti number b2 in arbitrary dimension. UPD: Unfortunately, decomposition of dual to BBF-form is not right in the main theorem. Instead of this work, take a look on recent preprints of Sawon…
Study of D2 ALF manifolds via hyperkahler quotients of affine spaces.
problem Resolving flat orbifold quotients of R4. method Infinite-dimensional generalization of Kronheimer's construction, hyperkahler quotients of affine spaces, singular equivariant instantons, stability of Nahm data.
result Construction of the family of D2 ALF manifolds as a deformation of the flat orbifold (R3×S1)/Z2. Simplified argument for second order estimate in quaternionic Calabi-Yau problem.
problem Second order estimates for quaternionic Calabi-Yau problem on hyperkähler manifolds.
method Simplified argument to derive the estimate.
result Simplified derivation of second order estimate.
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, the HyperKahler contact distribution of a 3-Sasakian manifold is studied. To analyze the curvature properties of this distribution, the special metric connection ∇ˉ is defined. This metric connection is completely determined by HyperKahler contact distribution. We prove that HyperKahler conta…
The paper proves a version of the SYZ conjecture for hyperkahler manifolds.
problem Proving the SYZ conjecture for hyperkahler manifolds with specific conditions.
method Introduced a Teichmuller space to parametrize pairs of hyperkahler manifolds up to isotopy, used a version of the global Torelli theorem to deduce the deformation invariance of semiampleness.
result Proved the SYZ conjecture for hyperkahler manifolds with specific conditions.
New hyperKähler orbifolds of Kummer type discovered.
problem Identifying new compact hyperKähler orbifolds.
method Construction of orbifolds and analysis of singularities.
result Only orbifolds with known holomorphic symplectic resolutions lead to known hyperKähler 8-manifolds.
In this paper, we construct complex metric structures on complex hypersurfaces in hyperkahler manifolds. This construction is that in contact geometry.
Study complex structures of hyperkähler manifolds with infinite type.
problem Understanding complex structures of hyperkähler four-manifolds with infinite topological type.
method Analyzing the Gibbons-Hawking ansatz and proving biholomorphic relationships to hypersurfaces or minimal resolutions of singular surfaces.
result For almost all complex structures, the manifold is biholomorphic to a hypersurface in \(\mathbb{C}^3\). For the remaining, it is biholomorphic to the minimal resolution of a singular surface under certain conditions.
New method constructs degenerate Sasakian manifolds from hyperkähler bundles.
problem Constructing degenerate 3-(α,δ)-Sasakian manifolds. method Using fiber products of Boothby-Wang bundles over hyperkähler manifolds.
result No non-trivial compact examples exist, and one family of nilpotent Lie groups with this geometry is identified.
We propose a generalization of the Hodge ddc-lemma to the case of hyperkähler manifolds. As an application of this result we derive the global construction of the fourth order transgression of the Chern character forms of hyperholomorphic bundles over compact hyperkähler manifolds. At the second part of the paper we…
Novel approach to instantons on quaternionic Kähler manifolds.
problem Understanding Yang-Mills instantons on quaternionic Kähler manifolds.
method Extension of harmonic space method for hyperkähler manifolds.
result Established a bijection between instantons and holomorphic maps.
Hyperkahler manifolds with round Kahler cones have unique bimeromorphic models.
problem Existence of round Kahler cones in hyperkahler manifolds.
method Analyzing the Kahler cone and its relation to the Bogomolov-Beauville-Fujiki form.
result Maximal holonomy hyperkahler manifolds with b2>4 have deformations with round Kahler cones. In this article, we study a generalisation of the Seiberg-Witten equations, replacing the spinor representation with a hyperKahler manifold equipped with certain symmetries. Central to this is the construction of a (non-linear) Dirac operator acting on the sections of the non-linear fibre-bundle. For hyperKahler manifo…