Construct a Hermitian metric on non-Hermitian Yang--Mills moduli spaces near the Hermitian locus.
problem Moduli space of non-Hermitian Yang--Mills connections over a compact Kähler manifold
method Using normalized harmonic metrics
result Near the Hermitian locus, the unobstructed locus carries an almost hypercomplex structure compatible with the associated Riemannian metric.
Complete classification of quaternionic skew-Hermitian symmetric spaces found.
problem Classifying quaternionic skew-Hermitian symmetric spaces.
method Proving the existence of a torsion-free mSO∗(2n)mSp(1)-structure and showing that any homogeneous space is symmetric. result A complete classification of quaternionic skew-Hermitian symmetric spaces for arbitrary n>1. Study of Hermitian metrics on Lie algebroids over complex spaces.
problem Exploring Hermitian metrics on Lie algebroids over analytic spaces.
method Introduced Hermitian metrics on holomorphic Lie algebroids and examined associated characteristics foliations.
result Extended equivariant de Rham cohomology to Hermitian Lie algebroids.
Study various submanifolds in quaternionic skew-Hermitian spaces.
problem Characterize submanifolds in almost quaternionic skew-Hermitian manifolds.
method Construct explicit examples of submanifolds in semisimple quaternionic skew-Hermitian symmetric spaces.
result Explicit examples of submanifolds for each type considered.
It was pointed out that the space of hermitian triples is an anology of the hermitian connection space. Generalizing the Ashtekar - Isham procedure one can quantize the space of hermitian triples as well as the original one. Here we add an example how this similarity can be exploited in a quantum theory of riemannian g…
Study on a metric on Hermitian metrics space, proving diffeomorphisms and completeness.
problem Metric on space of Hermitian metrics on complex vector bundles.
method Compute metric spray, geodesics, curvature, and use Nash-Moser theorem.
result Metric completion of Hermitian metrics space is L2 integrable singular Hermitian metrics.
Surveying locally homogeneous almost-Hermitian spaces with formulas for curvature.
problem Understanding the geometry of locally homogeneous almost-Hermitian spaces.
method Using the framework of varying Lie brackets to compute curvature of Gauduchon connections.
result Explicit formulas and examples for curvature of Gauduchon connections on locally homogeneous almost-Hermitian spaces.
Proves infinitesimal rigidity of Hermitian gravitational instantons.
problem Understanding the moduli space of Hermitian gravitational instantons.
method Proof of infinitesimal rigidity and integrability using boundary conditions and conformal Kähler properties.
result Completes the understanding of Hermitian gravitational instantons, both compact and non-compact.
Study on HYM connections on Kähler manifolds, calculating moduli space virtual dimension.
problem Calculating the moduli space of Hermitian-Yang-Mills connections.
method Analytic proof of stability of Higgs bundles on compact Kähler manifolds.
result Virtual dimension of the moduli space of HYM connections calculated.
Proves gap rigidity theorem for Hermitian symmetric spaces.
problem Gap rigidity problems in compact Hermitian symmetric spaces.
method Dual analogy to Mok's noncompact case theorem, theorem on higher dimensional submanifolds.
result Proves gap rigidity theorem for diagonal curves in tube type spaces.
We determine the space of algebraic pseudo-Hermitian Kähler-Weyl curvature tensors and the space of para-Hermitian Kähler-Weyl curvature tensors in dimension 4 and show that every algebraic possibility is geometrically realizable. We establish the Gray identity for pseudo-Hermitian Weyl manifolds and for para-Hermitian…
Verify conjecture for special Hermitian manifolds.
problem Conjecture about space forms for canonical metric connections.
method Verify conjecture for complex nilmanifolds and Bismut torsion-parallel manifolds.
result Verify conjecture for two special types of Hermitian manifolds.
Classifies Higgs and co-Higgs bundles on symmetric spaces.
problem Classifying Higgs and co-Higgs bundles over Hermitian symmetric spaces.
method Defined homogeneous principal Higgs and co-Higgs bundles, provided a classification up to isomorphism.
result Defined and classified moduli spaces for each type of bundle.
The paper classifies Hermitian manifolds with specific connection properties.
problem Classifying Hermitian manifolds with specific connection properties.
method Algebraic consideration of holonomy systems, structure theorems, and classification theorems.
result The universal cover of such Hermitian manifolds is the product of a complex Lie group and Hermitian symmetric spaces.
Gauduchon's theorem extended to singular spaces with smoothing.
problem Extending Gauduchon's theorem to singular spaces.
method Using smoothing techniques for singular spaces.
result Existence of conformally equivalent metrics on singular spaces.
We call a quaternionic Kaehler manifold with non-zero scalar curvature, whose quaternionic structure is trivialized by a hypercomplex structure, a hyper-Hermitian quaternionic Kaehler manifold. We prove that every locally symmetric hyper-Hermitian quaternionic Kaehler manifold is locally isometric to the quaternionic p…
Compact pseudo-Hermitian spaces have rigid holomorphic isometries.
problem Characterizing the rigidity of pseudo-Hermitian homogeneous spaces.
method Analysis of Tits fibration and automorphism groups of compact spaces.
result Holomorphic isometries of compact pseudo-Hermitian spaces are compact.
We analyze polar actions on Hermitian and quaternion-Kähler symmetric spaces of compact type. For complex integrable polar actions on Hermitian symmetric spaces of compact type we prove a reduction theorem and several corollaries concerning the geometry of these actions. The results are independent of the classificatio…
In this paper, following the constructions of N. R. O'Brian, J. H. Rawnsley and I. Vaisman, we define four almost Hermitian structures (up to conjugation) on the twistor space of a Hermitian surface by using canonical connections, including the Lichnerowicz connection and the Chern connection. We also study the relatio…
Proves regularity of geodesic equation on Hermitian manifolds.
problem Regularity of geodesic equation in mixed volume forms space.
method Ellipticity conditions, uniform Laplacian estimates, explicit subsolutions.
result Existence of unique C1,1 solution to Donaldson equation. From a view point of the moment map, we shall introduce the notion of Einstein-Hermitian generalized connections over a generalized Kähler manifold of symplectic type. We show that moduli spaces of Einstein-Hermitian generalized connections arise as the Kähler quotients. The deformation complex of Einstein-Hermitian ge…
In this paper we address the problem of studying those Kähler manifolds whose first two coefficients of the associated TYZ expansion vanish and we prove that for a locally Hermitian symmetric space this happens only in the flat case. We also prove that there exist nonflat locally Hermitian symmetric spaces where all th…
We give a complete characterization of invariant integrable complex structures on principal bundles defined over hermitian symmetric spaces, using the Jordan algebraic approach for the curvature computations. In view of possible generalizations, the general setup of invariant holomorphic principal fibre bundles is desc…
We obtain the full classification of coisotropic and polar actions of compact Lie group on irreducible Hermitian symmetric spaces.
Study perturbations of submodules in Drury-Arveson space, finding smooth vector bundles with Hermitian connections.
problem Geometry of perturbations in Drury-Arveson space.
method Analysis of smooth vector bundles with Hermitian connections and computation of parallel transport operators.
result Found natural Hermitian connections on perturbed submodules.
Study pseudo-harmonic maps on Weyl manifolds.
problem Characterize conditions for a Hermitian-Weyl manifold's complex structure to be a pseudo-harmonic map.
method Investigate geometric conditions for pseudo-harmonic maps in the context of Hermitian-Weyl manifolds.
result Find conditions for the complex structure to be a pseudo-harmonic map under specific dimensions or conformal structures.
Harmonic Hermitian structures found on specific Riemannian manifolds.
problem Finding conditions for harmonic Hermitian structures on Riemannian manifolds with skew-torsion.
method Geometric conditions on a four-dimensional Hermitian manifold with a metric connection of totally skew-symmetric torsion.
result The complex structure is a harmonic map into the twistor space under certain conditions.
Study deformed Hermitian-Yang-Mills equation on complex projective space blowup.
problem Solving the deformed Hermitian-Yang-Mills equation on complex projective space blowup.
method Expressed the equation as an ODE and solved it using combinatorial methods under an algebraic stability condition.
result Evidence supporting a conjecture on general compact Kahler manifolds.
Inspired by the work of G. Lu on pseudo symplectic capacities we obtain several results on the Gromov width and the Hofer--Zehnder capacity of Hermitian symmetric spaces of compact type. Our results and proofs extend those obtained by Lu for complex Grassmannians to Hermitian symmetric spaces of compact type. We also c…
Develops a universal Hermitian projective calculus for complex hyperbolic two-space
problem Complex hyperbolic geometry
method Algebraic invariant calculus
result Denominator-cleared identities for various geometric quantities
Any pseudo-Hermitian or para-Hermitian manifold of dimension 4 admits a unique Kaehler-Weyl structure; this structure is locally conformally Kaehler if and only if the alternating Ricci tensor vanishes. The alternating Ricci tensor takes values in a certain representation space. In this paper, we show that any algebrai…
We calculate the volume entropy of local Hermitian symmetric spaces of noncompact type in terms of its invariant r, a, b.
Study of complexified Hermitian symmetric spaces and their structures.
problem Understanding the hyperkähler structures and symplectic properties of complexified Hermitian symmetric spaces.
method Explicit diffeomorphisms and moment-critical subsets analysis.
result Almost all complex and symplectic structures are equivalent to the ones on G/K and $T^*\left(G_u/K_0
ight)$ respectively. In \cite{BR1}, \cite{BR2}, a parabolic determinant line bundle on a moduli space of stable parabolic bundles was constructed, along with a Hermitian structure on it. The construction of the Hermitian structure was indirect: The parabolic determinant line bundle was identified with the pullback of the determinant line b…
The grassmannian of hermitian lagrangian spaces in Cn⊕Cn is a natural compactification of the space of hermitian n×n matrices. We describe a Schubert-like, Whitney regular stratification on this space which has a Morse theoretic origin. We prove that these strata define closed subana…
We introduce the notion of tight homomorphism into a locally compact group with nonvanishing bounded cohomology and study these homomorphisms in detail when the target is a Lie group of Hermitian type. Tight homomorphisms between Lie groups of Hermitian type give rise to tight totally geodesic maps of Hermitian symmetr…
New equation approximates Kähler potentials using Hermitian-Yang-Mills metrics.
problem Approximating Kähler potentials in complex domains.
method New Wess-Zumino-Witten type equation and Berndtsson's theorem on direct image bundles.
result Approximation of Kähler potentials by Hermitian-Yang-Mills metrics.
I show that any complex manifold that resembles a rank two compact Hermitian symmetric space (other than a quadric hypersurface) to order two at a general point must be an open subset of such a space.
The paper examines curvature properties of twistor spaces.
problem None explicitly stated in the abstract.
method Review of Riemannian and almost Hermitian geometry of twistor spaces.
result Curvature properties of twistor spaces are explored.
Study on a deformed Hermitian-Yang-Mills equation on compact Kähler manifolds.
problem Existence of solutions to the hypercritical deformed Hermitian-Yang-Mills equation.
method Introduce coerciveness and properness of the J-functional on almost calibrated (1,1)-forms.
result Equivalence of coerciveness and properness to the existence of solutions.
In this paper, we derive the second variation formula of pseudoharmonic maps into any pseudo-Hermitian manifolds. When the target manifold is an isometric embedded CR manifold in complex Euclidean space or a pseudo-Hermitian immersed submanifold in Heisenberg group, we give some conditions on Weingarten maps to obtain …
Study of almost Yamabe solitons on Kaehler submersions.
problem Characterizing almost Yamabe solitons on Kaehler submersions.
method Analyzing conditions for solitons to be shrinking, steady, or expanding.
result Characterizations for almost Yamabe solitons in terms of extrinsic horizontal scalar curvature.
We describe the shape of the symplectic Dirac operators on Hermitian symmetric spaces. For this, we consider these operators as families of operators that can be handled more easily than the original ones.
Introduces Kähler duality between domains in complex space.
problem None explicitly stated; focuses on concept introduction.
method None explicitly stated; focuses on concept introduction.
result Introduces Kähler duality between domains in complex space.
Study parametrized Kähler class for cocycles on Hermitian symmetric spaces.
problem Understanding the cohomology of measurable cocycles on Hermitian symmetric spaces.
method Define and analyze parametrized Kähler class to determine cocycles up to cohomology.
result Parametrized Kähler class completely determines the cocycle up to cohomology.
Study on Hermitian Calabi functional in complexified orbits of symplectic manifolds.
problem Analyzing the Hermitian Calabi functional on complexified orbits of symplectic manifolds.
method Explicit formula for Hessian of Hermitian Calabi functional, semi-positive definiteness proof, and weak parabolicity of Hermitian Calabi flow.
result Hessian of Hermitian Calabi functional is semi-positive definite on complexified orbits.
Study of Hermitian and Gauduchon connections on Lie groups with almost Hermitian structures.
problem Characterizing connections on Lie groups with almost Hermitian structures.
method Analyzing left-invariant Hermitian and Gauduchon connections on Lie groups equipped with almost Hermitian structures.
result Explicit formulas for torsion components and curvature of Gauduchon connections on Lie groups.
The study examines deformations of Ricci-flat ALF spaces, showing they must be Hermitian.
problem Classifying and understanding deformations of Ricci-flat ALF spaces.
method Assuming suitable fall-off conditions, the authors show that deformations must be Hermitian and carry a non-trivial Killing vector field.
result The new Ricci-flat metric must belong to the family of previously classified metrics under mild additional hypotheses.