Researchers found a canonical form for pairs of Hermitian and antilinear operators.
problem Simultaneous normalization of pairs of Hermitian and antilinear operators in differential geometry.
method Finding a canonical form for pairs of Hermitian and antilinear operators.
result Generalized previous results on simultaneous normalization of such pairs.
Pluriclosed flow preserves Hermitian-symplectic structures and forms, with topological constraints.
problem Preserving Hermitian-symplectic structures under pluriclosed flow.
method Consideration of an extra evolution equation determined by the Bismut-Ricci form.
result Obtained topological obstruction to long-time existence in arbitrary dimensions.
The paper extends holomorphic forms on generalized Hermitian manifolds.
problem Understanding holomorphic forms on generalized Hermitian manifolds.
method Developed a criterion for holomorphic forms and used it to extend ∂-closed forms. result Invariance of generalized Hodge numbers in deformations of compact generalized Hermitian manifolds.
Study generalizes map properties between Hermitian manifolds preserving specific forms.
problem Understanding maps between Hermitian manifolds that preserve certain forms.
method Generalizing results from Chan-Yuan [2025] to new maps.
result Obtained further rigidity and non-existence theorems.
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. Study of Lee form exterior derivative in almost Hermitian manifolds.
problem Understanding the exterior derivative of Lee form in almost Hermitian manifolds.
method Analyzes the Lee form θ of almost Hermitian manifolds and its exterior derivative dθ in terms of intrinsic torsion and Riemannian curvature tensor components. result Proves that the Rω-component of dθ is always zero and provides expressions for other components. Torsion objects of von Neumann categories describe the phenomen "spectrum near zero" discovered by S. Novikov and M. Shubin. In this paper we classify Hermitian forms on torsion objects of a finite von Neumann category. We prove that any such form can be represented as a discriminant form of a degenerate Hermitian form…
Study calculates curvatures in holomorphic fibrations using degenerate Hermitian forms.
problem Calculating curvatures in holomorphic fibrations with degenerate Hermitian forms.
method Theory of Chern connections and curvature forms for degenerate Hermitian forms on holomorphic vector bundles.
result Positive holomorphic sectional curvature in Grassmannian bundles if the base does.
Paper develops formulas and theorems in Hermitian geometry.
problem None explicitly stated in the abstract.
method Develops second variational formulas and index forms in Hermitian geometry.
result Establishes results analogous to classical theorems in Riemannian geometry.
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.
Study shows compact Vaisman manifolds cannot have certain special Hermitian metrics.
problem Compact Vaisman manifolds and their compatibility with special Hermitian structures.
method Proof of non-existence of specific Hermitian metrics on compact Vaisman manifolds.
result Compact Vaisman manifolds cannot admit special Hermitian metrics like special k-Gauduchon metrics or pluriclosed metrics. The study examines harmonic forms on almost Hermitian manifolds and complex surfaces.
problem Analyzing harmonic forms on almost Hermitian manifolds and complex surfaces.
method Using techniques from Bott-Chern and Aeppli numbers, the study generalizes harmonic forms from complex and symplectic manifolds to almost Hermitian manifolds.
result Bott-Chern and Aeppli numbers of compact complex surfaces depend only on the topology of the underlying manifold.
Study differential operators on specific manifolds and their harmonic forms.
problem Understanding harmonic forms on almost-Hermitian manifolds.
method Analysis of differential operators, Hodge Theory, and cohomologies.
result Comparison of harmonic forms and cohomologies with classical ones.
Maps associate metrics to forms on Kähler manifolds.
problem Mapping metrics to forms on Kähler manifolds.
method Proving any positive hermitian form can be an L2-inner product of a metric, applying to Fubini-Study map.
result Fubini-Study map is injective.
Study on harmonic forms on almost Hermitian 4-manifolds, calculating dimensions and invariants.
problem Understanding harmonic forms on almost Hermitian 4-manifolds.
method Analyzing Bott-Chern and ∂ˉ harmonic forms, calculating dimensions and invariants. result Dimensions of harmonic forms on almost Hermitian 4-manifolds are determined.
In the theory of so called "Covariant Quantum Mechanics" a basic role is played by Hermitian vector fields on a complex line bundle in the frameworks of Galilei and Einstein spacetimes. In fact, it has been proved that the Lie algebra of Hermitian vector fields is naturally isomorphic to a Lie algebra of "special funct…
The paper investigates differential geometry of CR manifolds with new metrics.
problem Investigate differential geometry of CR manifolds with new metrics.
method Introduce a new Riemannian metric and canonical connection, derive curvature and torsion properties.
result Generalize known results in Riemannian geometry to the pseudo-Hermitian case.
Study on special Hermitian metrics and their stability.
problem Existence and stability of Hermitian metrics with specific properties.
method Analysis of Hermitian metrics with $∂ar{∂}ω^k=0$ for k=1 to n−1. result Stability of metrics at blow-up and deformations.
Study on stability and continuity of solutions to complex Monge-Ampère equations on compact Hermitian manifolds.
problem Stability and continuity of solutions to degenerate complex Monge-Ampère equations.
method Analysis of Hölder continuity and global continuity of solutions.
result Established uniform diameter bound for the twisted Chern-Ricci flow.
Study shows that dimension of Dolbeault harmonic forms is not always equal to B- on certain 4-manifolds.
problem Determining the dimension of Dolbeault harmonic (1,1)-forms on almost Hermitian 4-manifolds.
method Provided examples and proved non-equality of h∂1,1 and b− for certain structures. result Dimension of Dolbeault harmonic (1,1)-forms is not always equal to B- on almost Hermitian 4-manifolds.
A Hermitian symplectic manifold is a complex manifold endowed with a symplectic form ω, for which the bilinear form ω(I⋅,⋅) is positive definite. In this work we prove ddc-lemma for 1- and (1,1)-forms for compact Hermitian symplectic manifolds of dimension 3. This shows that Albanese map for such manifol…
Study scalar curvatures in almost Hermitian geometry and derive inequalities and characterization results.
problem Characterize and study scalar curvatures in almost Hermitian manifolds.
method Explicit formulas for Hermitian scalar curvatures, inequalities of total scalar curvatures, and characterization results.
result Derive inequalities and characterization results for specific types of metrics.
Proves unique ALE instanton with toric Hermitian structure.
problem Classify Ricci flat ALE instantons with toric Hermitian non-Kähler structure.
method Direct global analysis of Tod form in Weyl-Papapetrou coordinates, avoiding toric Kähler geometry.
result Eguchi-Hanson instanton is the only smooth, Ricci flat, ALE instanton with toric Hermitian non-Kähler structure.
Surveying hermitian integral geometry, the paper describes new kinematic formulas for complex space forms.
problem Understanding curvature measures and valuations on complex spaces.
method Analyzing valuations and curvature measures on complex space forms, deriving kinematic formulas.
result New local kinematic formulas for hermitian geometry, containing more information than global formulas.
We prove that if the fundamental 4-form of an almost-quaternionic Hermitian manifold (M, Q, g) of dimension at least eight satisfies the conformal-Killing equation, then (M, Q, g) is quaternionic-Kahler.
The covariant derivative of the Kähler form of an almost pseudo-Hermitian or of an almost para-Hermitian manifold satisfies certain algebraic relations. We show, conversely, that any 3-tensor which satisfies these algebraic relations can be realized geometrically.
Study on Hermitian manifolds with curvature, finding geometric properties.
problem Understanding the structure of Hermitian manifolds with semipositive Griffiths curvature.
method Combining HCF, torsion-twisted connection properties, and geometric observations.
result Null spaces of the Chern-Ricci form generate a holomorphic, integrable distribution.
On a Hermitian manifold we construct a symmetric (1,1)- tensor H using the torsion and the curvature of the Chern connection. On a compact balanced Hermitian manifold we find necessary and sufficient conditions in terms of the tensor H for a harmonic 1-form to be analytic and for an analytic 1-form to be harm…
This paper classifies LCSKT almost abelian Lie algebras in 6 dimensions.
problem Characterizing LCSKT structures on almost abelian Lie algebras.
method Analyzing the LCSKT condition and its compatibility with other Hermitian structures.
result Classification of LCSKT almost abelian Lie algebras in dimension 6.
New operators generalize Michelsohn's on almost Hermitian manifolds.
problem Generalizing differential operators to almost Hermitian manifolds.
method Introducing two differential operators on sections of the complex Clifford bundle over compact almost Hermitian manifolds.
result Surprising Kähler-like symmetries in the kernel of the Laplacians of these operators.
Survey connects singularity invariants to link pairings.
problem Understanding connections between singularities and link pairings.
method Use of Hermitian Variation Structures.
result Unified understanding of Picard--Lefschetz invariants and Blanchfield forms.
Study Dolbeault harmonic forms on Lie group quotients with specific structures.
problem Characterize the space of Dolbeault harmonic (1,1)-forms on compact Lie group quotients.
method Analyze left invariant almost Hermitian structures on 4D Lie groups and their quotients.
result Dimension of Dolbeault harmonic (1,1)-forms depends on existence of a specific anti-self-dual form.
On non-Kähler manifolds the notion of harmonic maps is modified to that of Hermitian harmonic maps in order to be compatible with the complex structure. The resulting semilinear elliptic system is {\it not} in divergence form. The case of noncompact complete preimage and target manifolds is considered. We give conditio…
A Hermitian metric on a complex manifold is called strong Kähler with torsion (SKT) if its fundamental 2-form ω is ∂∂ˉ-closed. We review some properties of strong KT metrics also in relation with symplectic forms taming complex structures. Starting from a 2n-dimensional SKT Lie algebra $\mathfr…
We obtain conditions on the Lee form under which a holomorphic map between almost Hermitian manifolds is a harmonic map or morphism. Then we discuss under what conditions (i) the image of a holomorphic map from a cosymplectic manifold is also cosymplectic, (ii) a holomophic map with Hermitian image defines a Hermitian …
We use Chern-Weil theory for Hermitian holomorphic vector bundles with canonical connections for explicit computation of the Chern forms of trivial bundles with special non-diagonal Hermitian metrics. We prove that every del-dellbar exact real form of the type (k,k) on an n-dimensional complex manifold X arises as a di…
KT-geometry is the geometry of a Hermitian connection whose torsion is a 3-form. HKT-geometry is the geometry of a hyper-Hermitian connection whose torsion is a 3-form. We identify non-trivial conditions for a reduction theory for these types of geometry.
With the aid of the theory of Jordan triple systems, we construct an explicit bi-symplectomorphism between a Hermitian symmetric space of non-compact type and $\C^n$ equipped with both the flat Kaehler-form and the Fubini-Study form. Our symplectomorphism is an explicit version of the symplectomorphism between Kaehler …
A Theorem of Kirichenko states that the torsion 3-form of the characteristic connection of a nearly Kähler manifold is parallel. On the other side, any almost hermitian manifold of type G1 admits a unique connection with totally skew symmetric torsion. In dimension six, we generalize Kirichenko's Theorem an…
Regularities and stability shown for a specific type of complex parallelizable manifolds.
problem Stability and regularity of Chern-flat metrics on complex parallelizable manifolds.
method Study of Hermitian metrics governed by the second Chern-Ricci form on compact complex manifolds.
result Chern-flat metrics are dynamically stable on compact complex parallelizable manifolds.
This study calculates the average number of common zeros of holomorphic functions on complex manifolds.
problem Calculating the average number of common zeros of holomorphic functions.
method Defined a Hermitian mixed volume for a mix of non-negative Hermitian forms and proved the average number of common zeros equals this mixed volume.
result The average number of common zeros of holomorphic functions equals the mixed volume of the manifold.
Reflection principles for harmonic and holomorphic maps from Hermitian symmetric spaces.
problem Establishing reflection principles for harmonic and holomorphic maps between Riemannian manifolds.
method Introducing recursive real forms and proving reflection principles for harmonic and holomorphic maps from Hermitian symmetric spaces.
result Reflection principles for harmonic and holomorphic maps from a class of Hermitian symmetric spaces.
New positivity condition for Hermitian manifold curvature.
problem Generalizing curvature positivity to non-Kähler manifolds.
method Introducing a new positivity condition and deriving a Bochner formula.
result Positivity condition on certain generalized Hopf and Vaisman manifolds.
We study almost Hermitian 4-manifolds with holonomy algebra, for the canonical Hermitian connection, of dimension at most one. We show how Riemannian 4-manifolds admitting five orthonormal symplectic forms fit therein and classify them. In this set-up we also fully describe almost Kaehler 4-manifolds.
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.
We generalize Conway's approach to integral binary quadratic forms on Q to study integral binary hermitian forms on quadratic imaginary extensions of Q. In Conway's case, an indefinite form that doesn't represent 0 determines a line ("river") in the spine T associated with SL(2,Z) in the hyperbolic plane. In our genera…
The paper classifies invariant structures on complex almost Abelian groups.
problem Investigating invariant geometric structures on almost Abelian Lie groups.
method Explicit formulas for Haar measures, modular function, and generator fields were derived.
result All invariant tensor fields have constant coefficients in the invariant frame.
We show that the non Hermitian Black-Scholes Hamiltonian and its various generalizations are eta-pseudo Hermitian. The metric operator eta is explicitly constructed for this class of Hamitonians. It is also shown that the effective Black-Scholes Hamiltonian and its partner form a pseudo supersymmetric system.