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.
We show that Calabi-Yau manifolds are emergent from the commutative limit of six-dimensional noncommutative Hermitian U(1) instantons. Therefore we argue that the noncommutative Hermitian U(1) instantons correspond to quantized Calabi-Yau manifolds.
Survey on complex Monge-Ampère equations in Hermitian contexts.
problem Complex Monge-Ampère equations in Hermitian settings.
method Survey of results from many authors over 15 years.
result Overview of known results on complex Monge-Ampère equations.
New Kähler metrics generalize Calabi's and relate to Fano manifolds.
problem Existence of extremal Kähler metrics on Fano manifolds.
method Introducing σ-extremal Kähler metrics and relating them to multiplier Hermitian-Einstein metrics. result Existence of σ-extremal Kähler metrics implies existence of multiplier Hermitian-Einstein metrics on Fano manifolds. New bounds for geometric flows of Hermitian metrics established.
problem Regularity of geometric flows of Hermitian metrics.
method Establishing a C1 a priori bound for smooth curves of Hermitian metrics. result New regularity result for Hermitian curvature flows, including the second Chern-Ricci flow.
Flow smooths Chern-Ricci-flat metrics on Hermitian manifolds.
problem Smooth Chern-Ricci-flat metrics on Hermitian manifolds.
method An analogue of the Calabi flow for compact Hermitian manifolds with vanishing first Bott-Chern class.
result The flow converges to the unique Chern-Ricci-flat metric under certain conditions.
New metrics found on non-Kähler Calabi-Yau manifolds.
problem Finding Levi-Civita Ricci-flat metrics on non-Kähler Calabi-Yau manifolds.
method Constructing new metrics on specific types of non-Kähler Calabi-Yau manifolds.
result Examples of Levi-Civita Ricci-flat metrics on various non-Kähler Calabi-Yau manifolds.
New metrics found on non-Kähler Calabi-Yau manifolds.
problem Constructing Ricci-flat metrics on non-Kähler Calabi-Yau manifolds.
method Using t-Gauduchon metrics on principal torus bundles over rational homogeneous varieties. result Examples of new metrics on non-Kähler Calabi-Yau manifolds.
Study Bismut connection curvatures and solve Yamabe and Calabi-Yau problems.
problem Yamabe problem and Calabi-Yau with torsion metrics for Bismut connection.
method Analysis of Bismut scalar and Ricci curvatures, construction of examples.
result Existence of metrics with constant Bismut scalar curvature.
We address the issue why Calabi-Yau manifolds exist with a mirror pair. We observe that the irreducible spinor representation of the Lorentz group Spin(6) requires us to consider the vector spaces of two-forms and four-forms on an equal footing. The doubling of the two-form vector space due to the Hodge duality doubles…
We study and construct non-abelian hermitian Yang-Mills (HYM) instantons on Calabi-Yau cones. By means of a particular isometry preserving ansatz, the HYM equations are reduced to a novel Higgs-Yang-Mills flow on the Einstein-Kahler base. For any 2d-dimensional Calabi-Yau cone, we find explicit solutions of the flow eq…
Study finds isolated SL submanifolds on non-Kähler Calabi-Yau threefolds.
problem Existence of special Lagrangian submanifolds in non-Kähler Calabi-Yau spaces.
method Introduced perturbed special Lagrangian submanifolds and used Sard-Smale technique to prove existence.
result Existence of isolated moduli spaces of perturbed special Lagrangian submanifolds.
Paper proves solvability condition for complex equation on special submanifolds.
problem Solvability condition for supercritical deformed Hermitian-Yang-Mills equation.
method Used integrals on subvarieties to provide necessary and sufficient condition.
result Confirms mirror version of Thomas-Yau conjecture about special Lagrangian submanifolds.
The paper proves estimates for Hermitian metrics and shows curvature blow-up on complex manifolds.
problem Estimating curvature blow-up in Hermitian metrics.
method Local Calabi and higher order estimates for continuity equations.
result Chern scalar curvature blows up at a finite-time singularity on compact complex manifolds.
We show existence of unique smooth solutions to the Monge-Ampere equation for (n-1)-plurisubharmonic functions on Hermitian manifolds, generalizing previous work of the authors. As a consequence we obtain Calabi-Yau theorems for Gauduchon and strongly Gauduchon metrics on a class of non-Kahler manifolds: those satisfyi…
The study examines geometric properties of complex Hermitian manifolds and their holonomy groups.
problem Understanding the geometric properties and restrictions of Hermitian manifolds and their holonomy groups.
method Analyzing the representation of restricted holonomy groups and their geometric consequences.
result Established criteria for when a Hermitian manifold is Kähler or projective based on its holonomy group.
New metrics found on non-Kähler complex manifolds.
problem Finding metrics on non-Kähler complex manifolds.
method Reinterpretation of Bismut Hermitian-Einstein condition and associated holomorphic Courant algebroid.
result Infinitely many non-Kähler manifolds without Bismut Hermitian-Einstein metrics.
The paper studies deformations of Calabi-Yau manifolds using Gauduchon metrics.
problem Deformations of Calabi-Yau manifolds under co-polarised conditions.
method Analyzes local deformations of Calabi-Yau ∂∂ˉ-manifolds using Gauduchon metrics and constructs a new hp-HS form. result Proves the p-SKT h-∂∂ˉ-property is deformation open. The paper studies conditions for graphs connecting level sets of harmonic polynomials.
problem Conditions for graphs connecting level sets of harmonic polynomials.
method Algebraic properties and Kempf-Ness functional construction.
result Stability condition equivalent to the existence of a solution to the deformed Hermitian-Yang-Mills equation.
The paper studies harmonic complex structures and special metrics on Sasakian manifolds.
problem Analyzing harmonic complex structures and special metrics on products of Sasakian manifolds.
method Examining 2-parameter families of Hermitian structures (Ja,b,ga,b) and their properties. result The complex structure Ja,b is harmonic with respect to ga,b, and conditions for various types of special metrics are determined. We prove a priori estimates for a class of transverse fully nonlinear equations on Sasakian manifolds and give some geometric applications such as the transversion Calabi-Yau theorem for transverse balanced and (strongly) Gauduchon metrics. We also explain that similar results hold on compact oriented, taut, transverse…
Study stability of non-Kähler Calabi-Yau metrics using critical points of generalized Einstein Hilbert action.
problem Stability of critical points of the generalized Einstein Hilbert action in non-Kähler Calabi-Yau theory.
method Analysis of Bismut Hermitian Einstein manifolds and Bismut flat pluriclosed steady solitons, proving stability conditions.
result All Bismut Hermitian Einstein manifolds are linearly stable, and all Bismut flat pluriclosed steady solitons with positive Ricci curvature are linearly strictly stable.
In this paper, we obtain the Bedford-Taylor interior C2 estimate and local Calabi C3 estimate for the solutions to complex Monge-Ampère equations on Hermitian manifolds.
Extends three circle theorem to almost Hermitian manifolds.
problem Three circle theorem for Kähler manifolds to almost Hermitian manifolds.
method General maximum principle established to prove three circle theorem.
result Sharp dimension estimates and Liouville theorems for holomorphic functions.
Solutions to Strominger system found for square of Kähler class.
problem Finding solutions to Strominger system with specific balanced classes.
method Deforming Calabi-Yau and Hermitian-Yang-Mills metrics.
result Classes that are squares of Kähler metrics admit solutions.
Motivated from mathematical aspects of the superstring theory, we introduce a new equation on a balanced, hermitian manifold, with zero first Chern class. Solving the equation, one will obtain, in each Bott--Chern cohomology class, a balanced metric which is hermitian Ricci--flat. This can be viewed as a differential f…
The paper explores curvature constraints on Kodaira dimension for specific almost Hermitian manifolds.
problem Investigating Riemannian curvature constraints on the Kodaira dimension of compact almost Hermitian manifolds.
method Analyzing compact almost Hermitian manifolds in the Gray-Hervella class and Hermitian manifolds with nonnegative scalar curvature.
result For compact almost Hermitian manifolds with nonnegative scalar curvature, the Kodaira dimension is either -∞ or 0, with specific conditions.
Study of Hermitian structures on toric suspensions of balanced manifolds.
problem Exploring Hermitian structures on specific types of manifolds.
method Analysis of toric suspensions of Calabi-Yau and hyperkähler manifolds under holomorphic automorphisms.
result Suspensions of hyperkähler manifolds do not admit certain Hermitian metrics.
Study shows continuity of non-Kähler Calabi-Yau conifold transitions.
problem Understanding the geometry of Calabi-Yau conifold transitions.
method Use of balanced and Hermitian-Yang-Mills metrics to analyze conifold transitions.
result The conifold transition is continuous in the Gromov-Hausdorff topology.
Sheng and Zuo's characteristic forms are invariants of a variation of Hodge structure. We show that they characterize Gross's canonical variations of Hodge structure of Calabi-Yau type over (Hermitian symmetric) tube domains.
We introduce a class of hermitian metrics with {\em Lee potential}, that generalize the notion of l.c.K. metrics with potential introduced in \cite{ov} and show that in the classical examples of Calabi and Eckmann of complex structures on $S^{2p+1}\x S^{2q+1}$, the corresponding hermitian metrics are of this type. Thes…
The flow proves a theorem for Fano manifolds.
problem Proving a theorem for Fano manifolds using the prescribed Hermitian-Yang-Mills flow.
method Using the prescribed Hermitian-Yang-Mills flow to prove the Donaldson-Uhlenbeck-Yau theorem.
result The flow converges to a Hermitian metric satisfying the prescribed tensor condition.
The paper finds explicit instantons on a specific 6-manifold.
problem Finding explicit solutions to the Hermitian Yang-Mills equations.
method Dimensional reduction and reformulation of the equations on quotient spaces.
result Explicit abelian instantons and deformed Hermitian Yang-Mills connections are found.
In this note, using Calabi's method, we construct rotationally symmetric Kahler-Ricci solitons on the total space of direct sum of fixed hermitian line bundle and its projective compactification, where the curvature of hermitian line bundle is Kahler-Einstein. These examples generalize the construction of Koiso, Cao an…
New obstruction found for Hull-Strominger system solutions.
problem Existence of solutions to the Hull-Strominger system.
method Construction of Hermitian-Einstein metric on string algebroid.
result Definition of Futaki invariants obstructing solutions.
This paper shows hypercommutative algebras on Calabi-Yau manifolds are formal.
problem Formality of hypercommutative algebras on Calabi-Yau manifolds.
method Using purity of mixed Hodge structures.
result The canonical hypercommutative algebra on compact Calabi-Yau manifolds is formal.
Calabi-Yau theorem extended to Vaisman manifolds.
problem Uniqueness of Vaisman metrics and their characterization.
method Analyzing the Lee form and Lee class properties.
result Vaisman metrics uniquely determined by volume and Lee class.
Researchers find unique metrics solving complex PDEs for constant scalar curvature.
problem Finding metrics with constant scalar curvature in complex manifolds.
method Proving existence and uniqueness of smooth functions f that solve a fourth-order nonlinear PDE related to the Calabi functional. result Critical metrics minimize the Calabi functional and have constant Chern scalar curvature.
Introduces new equations linking Kähler-Einstein and Hermitian-Yang-Mills theories.
problem Existence of solutions to coupled Kähler-Einstein and Hermitian-Yang-Mills equations.
method Moment map interpretation, Futaki invariant, Matsushima-Lichnerowicz theorem, deformation results.
result Nontrivial solutions produced under certain conditions.
New insights prevent certain types of metrics on compact spaces.
problem Preventing the existence of specific types of metrics on compact spaces.
method Computing cohomology and analyzing stability of metrics for the pluriclosed flow.
result Prevents the existence of non-flat homogeneous Bismut Hermitian Einstein metrics on C-spaces.
We consider non-Kaehler compact complex manifolds which are homogeneous under the action of a compact Lie group of biholomorphisms and we investigate the existence of special (invariant) Hermitian metrics on these spaces. We focus on a particular class of such manifolds comprising the case of Calabi-Eckmann manifolds a…
The paper examines SKT and CYT metrics on Lie groups.
problem Existence and properties of SKT and CYT metrics on Lie groups.
method Study left-invariant metrics on compact semi-simple Lie groups.
result Left-invariant metrics that are both CYT and SKT must be Bismut flat.
Study shows stability of tangent bundle through conifold transitions.
problem Stability of tangent bundle through conifold transitions.
method Hermitian-Yang-Mills metric and conformally balanced metrics.
result Tangent bundle T1,0Xt admits a Hermitian-Yang-Mills metric Ht near vanishing cycles of Xt. We introduce a new geometric flow of Hermitian metrics which evolves an initial metric along the second derivative of the Chern scalar curvature. The flow depends on the choice of a background metric, it always reduces to a scalar equation and preserves some special classes of Hermitian structures, as balanced and Gaud…
We first study the degeneration of a sequence of Hermitian-Yang-Mills metrics with respect to a sequence of balanced metrics on a Calabi-Yau threefold X^ that degenerates to the balanced metric constructed by Fu, Li, and Yau on the complement of finitely many (-1,-1)-curves in X^. Then under some assumpti…
In this article, we examine the behavior of the Riemannian and Hermitian curvature tensors of a Hermitian metric, when one of the curvature tensors obeys all the symmetry conditions of the curvature tensor of a Kähler metric. We will call such metrics G-Kähler-like or Kähler-like, for lack of better terminologies. Such…
We compute the hessian of the natural Hermitian form successively on the Calabi family of a hyperkähler manifold, on the twistor space of a 4-dimensional anti-self-dual Riemannian manifold and on the twistor space of a quaternionic Kähler manifold. We show a strong convexity property of the cycle space of twistor lines…
We exhibit a transformation taking special Lagrangian submanifolds of a Calabi-Yau together with local systems to vector bundles over the mirror manifold with connections obeying deformed Hermitian-Yang-Mills equations. That is, the transformation relates supersymmetric A- and B-cycles. In this paper, we assume that th…