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 on hyperKähler moment maps and hypertoric manifolds.
problem Classify hyperKähler manifolds with specific properties.
method Tri-Hamiltonian actions of tori, symplectic and hyperKähler geometry.
result HyperKähler moment maps have connected fibers and are surjective.
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 paper constructs complex structures on hypersurfaces in hyperkahler manifolds.
problem No specific problem stated; focuses on construction of structures.
method Construction of complex metric structures on hypersurfaces in hyperkahler manifolds.
result The construction of complex structures on hypersurfaces in hyperkahler manifolds.
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.
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.
The paper explores structures on 4-manifolds using hyperKahler geometry.
problem Exploring structures on 4-manifolds using hyperKahler geometry.
method Non-linear generalization of the Dirac operator, covariantly constant generalized spinor, harmonic spinor, generalized Seiberg-Witten equations.
result Existence of a covariantly constant, generalized spinor defines a Kahler structure on the base 4-dimensional manifold.
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.
The paper describes hyperkähler geometry of cotangent bundles using rank-1 projections.
problem Understanding hyperkähler geometry of cotangent bundles via algebraic methods.
method Algebraic description via the scheme of rank-1 projections, isometric embeddings, and generalizations.
result Explicit isometric embeddings and generalizations of hyperkähler geometry.
Surveying recent developments in Hitchin moduli space geometry.
problem Understanding the asymptotic geometry of Hitchin moduli space.
method Introduction to Hitchin moduli space and hyperkähler geometry.
result Recent developments in asymptotic geometry of Hitchin moduli space.
Study calibrated geometry in hyperkähler cones and their related spaces.
problem Characterize submanifolds in hyperkähler cones and related spaces.
method Systematic study of calibrated geometry in hyperkähler cones, 3-Sasakian manifolds, and twistor spaces.
result Obtain new characterizations of complex Lagrangian and complex isotropic cones in hyperkähler cones.
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.
Defines (p,q) hermitian geometry and formulates it in generalised complex geometry.
problem Defines (p,q) hermitian geometry and formulates it in generalised complex geometry. method Formulates (p,q) hermitian geometry in generalised complex geometry. result Provides explicit formulae for the map to generalised geometry.
New flow connects symplectic maps to hyperKähler geometry.
problem Understanding symplectic maps and their geometry.
method Established a correspondence between symplectic diffeomorphisms and hyperKähler moment maps.
result Introduced a new flow, the modified moment map flow.
Survey and generalization of implosion and contraction in symplectic and hyperkähler geometry.
problem Exploring implosion and contraction in symplectic and hyperkähler geometry.
method Survey and extension of implosion construction to general reductive groups, interpretation in Moore-Tachikawa category, generalization of contraction construction.
result Generalization of implosion and contraction concepts to hyperkähler and complex symplectic situations.
Characterizes Hilbert schemes and their geometric properties.
problem Understanding transverse Hilbert schemes and their geometric properties.
method Characterization through bi-Poisson structures and hyperkähler geometry.
result Characterization of transverse Hilbert schemes and description of their hyperkähler geometry.
The resolved conifold geometry is linked to a special Kähler manifold and an instanton-corrected hyperkähler manifold.
problem Understanding the geometry of the resolved conifold and its associated structures.
method Explicit description of ASK and instanton-corrected HK manifolds, relating them to twistor coordinates and solving Riemann-Hilbert problems.
result The instanton-corrected hyperkähler manifold realizes a smoothing of the semi-flat HK metric associated with the ASK geometry.
New representation of curves helps prove complex geometry result.
problem Understanding cohomologically stable curves in projective space.
method Using commuting matrix polynomials to represent curves and show isomorphism to hyperkähler quotient.
result Hilbert scheme isomorphic to a hyperkähler quotient.
The Ooguri-Vafa space is linked to framed wild harmonic bundles via hyperkähler geometry.
problem Relating the Ooguri-Vafa space to Hitchin moduli spaces.
method Interpreting the Ooguri-Vafa space as framed wild harmonic bundles over CP1.
result The Ooguri-Vafa space is a moduli space of framed wild harmonic bundles.
We introduce an analogue in hyperkahler geometry of the symplectic implosion, in the case of SU(n) actions. Our space is a stratified hyperkahler space which can be defined in terms of quiver diagrams. It also has a description as a non-reductive geometric invariant theory quotient.
Study topology and geometry of hyperkähler quotients using Morse theory.
problem Topology and geometry of hyperkähler quotients and related non-compact Kähler quotients.
method Employ Morse theory with moment maps, Lojasiewicz inequality, Kirwan surjectivity, and Jeffrey-Kirwan residue formula.
result Obtained new proofs and explicit inductive procedures for Betti numbers of fixed-point sets.
New definition for a special type of complex manifolds.
problem Characterizing compact locally conformally hyperkähler manifolds.
method Equivalent definition using a nondegenerate complex two-form.
result Established a conformal analogue of Beauville's theorem.
Under the action of the c-map, special Kahler manifolds are mapped into a class of quaternion-Kahler spaces. We explicitly construct the corresponding Swann bundle or hyperkahler cone, and determine the hyperkahler potential in terms of the prepotential of the special Kahler geometry.
Study para-hyperKähler geometry of anti-de Sitter structures.
problem Understand the geometry of anti-de Sitter structures.
method Investigate para-hyperKähler structures and their relations.
result Found neutral pseudo-Riemannian metric and symplectic structures.
This dissertation explores T-duality between hyperkähler structures and branes on algebraic integrable systems.
problem Investigates T-duality between hyperkähler structures and branes on algebraic integrable systems.
method Uses techniques of generalized geometry and Fourier-Mukai transform to show T-duality between semi-flat hyperkähler structures and generalized branes.
result Shows T-duality between semi-flat hyperkähler structures and generalized branes on algebraic integrable systems.
New insights connect strong coupling SYM amplitudes to hyperkähler geometry.
problem Understanding strong coupling SYM amplitudes.
method Integrable systems, pseudo-hyperkähler geometry, twistor theory.
result Remainder function is a pseudo-Kähler scalar in hyperkähler geometry.
Hyperkähler reduction explained via frame bundles.
problem Describing hyperkähler reduction using frame bundles.
method Using frame bundles and connections, tracing reductions and analyzing fibers.
result Fibers of hyperkähler reduction are totally geodesic.
New collapsing metrics on K3 surface modeled on ALF gravitational instantons.
problem Constructing new collapsing hyperkähler metrics on K3 surface.
method Deforming approximate hyperkähler metrics by gluing ALF gravitational instantons to a background metric.
result The collapsing metrics are modeled by ALF gravitational instantons of dihedral and cyclic types.
Abstract: Linking field theory to Floer theory via regularization.
problem Finding periodic solutions of Hamilton's equation.
method Regularization scheme for polysymplectic formalism linking Euclidean field theory to hyperkähler Floer theory.
result Proved a cuplength estimate.
Study improves cone conjecture on hyperkahler manifolds using homogeneous space dynamics.
problem Improving the Morrison-Kawamata cone conjecture on hyperkahler manifolds.
method Ergodic theory on homogeneous spaces.
result Beauville-Bogomolov square of MBM classes is bounded by a number depending only on the manifold's deformation class.
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…
Reviews uses of nonlinear sigma models in various systems.
problem Understanding nonlinear sigma models in different physical contexts.
method General discussion and focus on geometrical interpretations.
result Connection between sigma models and various geometries.
Study asymptotics of hyperkähler geometry on singular fibers of Hitchin moduli space.
problem Asymptotic hyperkähler geometry of SL2(C)-Hitchin moduli space over singular fibers. method Extension of exponential convergence results to locally fiducial Higgs bundles and subintegrable systems.
result Hyperkähler metric converges exponentially to semi-flat metric on subintegrable systems.
We show that on Hilbert scheme of n points on $\C^2$, the hyperkähler metric construsted by H. Nakajima via hyperkähler reduction is the Quasi-Asymptotically Locally Euclidean (QALE in short) metric constructed by D. Joyce.
Let G be a compact simple Lie group and the O the minimal nilpotent orbit in g^C. We determine all G-invariant Kähler potentials for hyperKähler metrics compatible with the KKS complex symplectic form on O.
Hyperbolic geometry on ample cone of hyperkahler manifolds.
problem Understanding the ample cone structure of hyperkahler manifolds via hyperbolic geometry.
method Analyzing the Bogomolov-Beauville-Fujiki quadratic form and its restriction to the rational Hodge lattice, leading to a hyperbolic orbifold.
result Existence of finitely many geodesic hypersurfaces cutting the orbifold into polyhedral pieces, each isometric to a quotient of the ample cone of a birational model.
Extending work of Bielawski-Dancer and Konno, we develop a theory of toric hyperkahler varieties, which involves toric geometry, matroid theory and convex polyhedra. The framework is a detailed study of semi-projective toric varieties, meaning GIT quotients of affine spaces by torus actions, and specifically, of Lawren…
We discuss a general framework for cutting constructions and reinterpret in this setting the work on non-Abelian symplectic cuts by Weitsman. We then introduce two analogous non-Abelian modification constructions for hyperkähler manifolds: one modifies the topology significantly, the other gives metric deformations. We…
Simplified proof of K3 surface period map surjectivity.
problem Surjectivity of period map on K3 surfaces.
method Utilizes hyperkähler geometry and collapsing techniques.
result Simple proof of Todorov's result on K3 surfaces.
Explains complex adjoint orbits in Lie theory and geometry.
problem Understanding adjoint orbits in Lie theory and geometry.
method Expository introduction to adjoint orbits of complex semisimple groups.
result Provides insights into properties of semisimple and nilpotent orbits.
The paper proves finiteness for stable Lagrangian fibrations with a given divisor.
problem Finiteness of deformation classes of hyperkähler Lagrangian fibrations.
method Survey and proof of finiteness for stable Lagrangian fibrations with a given discriminant divisor.
result Finiteness for stable Lagrangian fibrations with a given discriminant divisor.
Study generalizes Seiberg-Witten equations on hyperKahler manifolds.
problem Generalizing Seiberg-Witten equations on hyperKahler manifolds.
method Construction of a non-linear Dirac operator and derivation of transformation formula for the Dirac operator under conformal changes.
result Generalized Seiberg-Witten equations can be expressed as a second order PDE in terms of almost-complex structure and a conformal factor.
We develop the foundation of the complex symplectic geometry of Lagrangian subvarieties in a hyperkahler manifold. We establish a characterization, a Chern number inequality, topological and geometrical properties of Lagrangian submanifolds. We discuss a category of Lagrangian subvarieties and its relationship with the…
Exponential decay observed between two metrics on a moduli space.
problem Analyzing the difference between two metrics on a moduli space.
method Proving exponential decay along generic rays in the Hitchin moduli space.
result Exponential decay of the difference between gL2 and gsf. In this paper we study the birational geometry of HyperKaehler manifolds by combining the method of minimal model program and the traditional approach of symplectic geometry.
Alternative proof of a theorem using parabolic Monge-Ampère equation in HKT geometry.
problem Proving a theorem about solutions to the quaternionic Monge-Ampère equation.
method Generalizing the parabolic Monge-Ampère equation to HKT geometry and proving existence and convergence of solutions.
result Existence and convergence of solutions to the equation 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.
New sigma models use (4,0) supersymmetry for hyperkähler target spaces.
problem Constructing sigma models with (4,0) off-shell supersymmetry. method Formulated (4,0) supermultiplets, constructed sigma models with hyperkähler target spaces. result Explicit construction of target space geometries using (4,0) supersymmetry.