Compact Hermitian Research Articles

Weak solutions of complex Hessian equations on compact Hermitian manifolds

We prove the existence of weak solutions of complex $m$-Hessian equations on compact Hermitian manifolds for the non-negative right-hand side belonging to $L^{p}$, $p>n/m$ ($n$ is the dimension of the manifold). For smooth, positive data the equation has recently been solved by Székelyhidi and Zhang. We also give a stability result for such solutions.

$E_1$-degeneration and $d'd''$-lemma

For a double complex $(A, d', d'')$, we show that if it satisfies the $d'd''$-lemma and the spectral sequence $\{E^{p, q}_r\}$ induced by $A$ does not degenerate at $E_0$, then it degenerates at $E_1$. We apply this result to prove the degeneration at $E_1$ of a Hodge-de Rham spectral sequence on compact bi-generalized Hermitian manifolds that satisfy a version of $d'd''$-lemma.

Normal family theory and Gauss curvature estimate of minimal surfaces in $\mathbb{R}^m$

In this paper, we first extend Zalcman’s principle of normality to the families of holomorphic mappings from Riemann surfaces to a compact Hermitian manifold. We then use this principle to derive an estimate for Gauss curvatures of the minimal surfaces in $\mathbb{R}^m$ whose Gauss maps satisfy some property $\mathcal{P}$, in the spirit of Bloch’s heuristic principle in complex analysis. Consequently, we recover and simplify the known results about value distribution properties of the Gauss map of minimal surfaces in $\mathbb{R}^m$.

Pluripotential theory on compact Hermitian manifolds

Pluripotential theory on compact Hermitian manifolds

An Equivalence of Scalar Curvatures on Hermitian Manifolds

For a Kahler metric, the Riemannian scalar curvature is equal to twice the Chern scalar curvature. The question we address here is whether this equivalence can hold for a non-Kahler Hermitian metric. For such metrics, if they exist, the Chern scalar curvature would have the same geometric meaning as the Riemannian scalar curvature. Recently, Liu–Yang showed that if this equivalence of scalar curvatures holds even in average over a compact Hermitian manifold, then the metric must in fact be Kahler. However, we prove that a certain class of non-compact complex manifolds do admit Hermitian metrics for which this equivalence holds. Subsequently, the question of to what extent the behavior of said metrics can be dictated is addressed and a classification theorem is proved.

Cotangent bundles over all the Hermitian symmetric spaces

We construct the N = 2 supersymmetric nonlinear sigma models on the cotangent bundles over all the compact and non-compact Hermitian symmetric spaces. In order to construct them we use the projective superspace formalism which is an N = 2 off-shell superfield formulation in four-dimensional space-time. This formalism allows us to obtain the explicit expression of N = 2 supersymmetric nonlinear sigma models on the cotangent bundles over any Hermitian symmetric spaces in terms of the N =1 superfields, once the Kähler potentials of the base manifolds are obtained. Starting with N = 1 supersymmetric Kähler nonlinear sigma models on the Hermitian symmetric spaces, we extend them into the N = 2 supersymmetric models by using the projective superspace formalism and derive the general formula for the cotangent bundles over all the compact and non-compact Hermitian symmetric spaces. We apply to the formula for the non-compact Hermitian symmetric space E7/E6 × U(1)1.

On the volume of a pseudo-effective class and semi-positive properties of the Harder–Narasimhan filtration on a compact Hermitian manifold

This paper divides into two parts. Let $(X,\omega)$ be a compact Hermitian manifold. Firstly, if the Hermitian metric $\omega$ satisfies the assumption that $\partial\overline{\partial}\omega^k=0$ for all $k$, we generalize the volume of the cohomology class in the K\{a}hler setting to the Hermitian setting, and prove that the volume is always finite and the Grauert-Riemenschneider type criterion holds true, which is a partial answer to a conjecture posed by Boucksom. Secondly, we observe that if the anticanonical bundle $K^{-1}_X$ is nef, then for any $\varepsilon>0$, there is a smooth function $\phi_\varepsilon$ on $X$ such that $\omega_\varepsilon:=\omega+i\partial\overline{\partial}\phi_\varepsilon>0$ and Ricci$(\omega_\varepsilon)\geq-\varepsilon\omega_\varepsilon$. Furthermore, if $\omega$ satisfies the assumption as above, we prove that for a Harder-Narasimhan filtration of $T_X$ with respect to $\omega$, the slopes $\mu_\omega(\mathcal{F}_i/\mathcal{F}_{i-1})\geq 0$ for all $i$, which generalizes a result of Cao which plays a very important role in his studying of the structures of K\{a}hler manifolds.

Eigenvalues of vector fields, Bott's residue formula and integral invariants

Given a compatible vector field on a compact connected almost-complex manifold, we show in this article that the multiplicities of eigenvalues among the zero point set of this vector field have intimate relations. We highlight a special case of our result and reinterpret it as a vanishing-type result in the framework of the celebrated Atiyah–Bott–Singer localization formula. This new point of view, via the Chern–Weil theory and a strengthened version of Bott's residue formula, can lead to an obstruction to Killing real holomorphic vector fields on compact Hermitian manifolds in terms of a curvature integral.

The complex Monge–Ampère type equation on compact Hermitian manifolds and applications

We prove the existence and uniqueness of continuous solutions to the complex Monge–Ampère type equation with the right hand side in Lp, p>1, on compact Hermitian manifolds. Next, we generalise results of Eyssidieux, Guedj and Zeriahi [17,18] to compact Hermitian manifolds which a priori are not in the Fujiki class. These generalisations lead to a number of applications: we obtain partial results on a conjecture of Tosatti and Weinkove [40] and on a weak form of a conjecture of Demailly and Paun [11].

CLASSIFICATION OF SMOOTH SCHUBERT VARIETIES IN THE SYMPLECTIC GRASSMANNIANS

Abstract. A Schubert variety in a rational homogeneous variety G/P isdefined by the closure of an orbit of a Borel subgroup Bof G. In general,Schubert varieties are singular, and it is an old problem to determinewhich Schubert varieties are smooth. In this paper, we classify all smoothSchubert varieties in the symplectic Grassmannians. 1. IntroductionA rational homogeneous manifold S= G/Pis a projective manifold, wherea connected complex semisimple group Gacts transitively. Under the actionof a Borel subgroup Bof G, Shas finitely many orbits. The closure of a B-orbit in Sis called a Schubert variety of S. In general, Schubert varieties aresingular, and it is an old problem to determine which Schubert varieties aresmooth. Lakshmibai-Weyman and Brion-Polo have studied the singular lociof Schubert varieties of S, when Sis a compact Hermitian symmetric space([9] and [2]). In particular, they showed that in this case any smooth Schubertvariety in Sis a homogeneous submanifold of Sassociated to a subdiagramof the marked Dynkin diagram of S. For example, a Schubert variety of theGrassmannian Gr(k,V) of k-subspaces in a vector space V is smooth if andonly if it is a linearly embedded sub-Grassmannian.More generally, when Sis associated to a long simple root, we have:Theorem 1.1 (Proposition 3.7 of Hong-Mok [4]). Let S= G/P be a rationalhomogeneous manifold associated to a long simple root. Then any smooth Schu-bert variety in Sis a homogeneous submanifold of Sassociated to a subdiagramof the marked Dynkin diagram of S.On the other hand, when Sis associated to a short simple root, there is asmooth Schubert variety that is not homogeneous. Let V be a vector spacewith a symplectic form ω, i.e., a nondegenerate skew-symmetric bilinear form.

The complex Monge–Ampère equation on some compact Hermitian manifolds

The complex Monge–Ampère equation on some compact Hermitian manifolds

Higher-Order Compact-Flow Field-Dependent Variation (HOC-FDV) Method for Solving Two-Dimensional Navier-Stokes Equations

In this article, a new, higher-order accurate method, namely higher-order compact-flow field-dependent variation (HOC-FDV) method, has been developed to solve two-dimensional Navier-Stokes equations. The HOC-FDV scheme is of third-order accuracy in time and fourth-order in space. The spatial derivatives in the flow field-dependent variation (FDV) equations proposed by Chung are approximated using higher-order compact (HOC) Hermitian (Pade) scheme. The solution procedure at each time step consists of a system of block tri-diagonal matrices which can be solved efficiently in a standard manner. Several numerical examples are tested to examine the accuracy and capability of the new scheme to capture the shock and to simulate accurately separation and discontinuity.

Supersymmetry and cotangent bundle over non-compact exceptional Hermitian symmetric space

Abstract We construct $$ \mathcal{N}=2 $$ N = 2 supersymmetric nonlinear sigma models on the cotangent bundles over the non-compact exceptional Hermitian symmetric spaces $$ \mathrm{\mathcal{M}}={E}_{6\left(-14\right)}/\mathrm{SO}(10)\times \mathrm{U}(1) $$ ℳ = E 6 − 14 / SO 10 × U 1 and E 7(−25) /E 6 × U(1). In order to construct them we use the projective superspace formalism which is an $$ \mathcal{N}=2 $$ N = 2 off-shell superfield formulation in four-dimensional space-time. This formalism allows us to obtain the explicit expression of $$ \mathcal{N}=2 $$ N = 2 supersymmetric nonlinear sigma models on the cotangent bundles over any Hermitian symmetric spaces in terms of the $$ \mathcal{N}=1 $$ N = 1 superfields, once the Kähler potentials of the base manifolds are obtained. We derive the $$ \mathcal{N}=1 $$ N = 1 supersymmetric nonlinear sigma models on the Kähler manifolds $$ \mathrm{\mathcal{M}} $$ ℳ . Then we extend them into the $$ \mathcal{N}=2 $$ N = 2 supersymmetric models with the use of the result in arXiv:1211.1537 developed in the projective superspace formalism. The resultant models are the $$ \mathcal{N}=2 $$ N = 2 supersymmetric nonlinear sigma models on the cotangent bundles over the Hermitian symmetric spaces $$ \mathrm{\mathcal{M}} $$ ℳ . In this work we complete constructing the cotangent bundles over all the compact and non-compact Hermitian symmetric spaces.

ON DIFFERENTIAL INVARIANTS OF HYPERPLANE SYSTEMS ON NONDEGENERATE EQUIVARIANT EMBEDDINGS OF HOMOGENEOUS SPACES

Given a complex submanifold M of the projective space P(T), the hyperplane system R on M characterizes the projective embedding of M into P(T) in the following sense: for any two nondegenerate com- plex submanifolds MP(T) and M ' � P(T ' ), there is a projective linear transformation that sends an open subset of M onto an open subset of M ' if and only if (M,R) is locally equivalent to (M ' ,R ' ). Se-ashi developed a theory for the differential invariants of these types of systems of linear differential equations. In particular, the theory applies to systems of lin- ear differential equations that have symbols equivalent to the hyperplane systems on nondegenerate equivariant embeddings of compact Hermitian symmetric spaces. In this paper, we extend this result to hyperplane sys- tems on nondegenerate equivariant embeddings of homogeneous spaces of the first kind.

Aeppli and Bott–Chern cohomology for bi-generalized Hermitian manifolds and d′d″-lemma

We define Aeppli and Bott–Chern cohomology for bi-generalized complex manifolds and show that they are finite dimensional for compact bi-generalized Hermitian manifolds. For totally bounded double complexes ( A , d ′ d ″ ) , we show that the validity of d ′ d ″ -lemma is equivalent to having the same dimension of several cohomology groups. Some calculations of Bott–Chern cohomology groups of some bi-generalized Hermitian manifolds are given.

Vaisman structures on compact solvmanifolds

Let \((M, g, J)\) be a compact Hermitian manifold and \(\Omega \) the fundamental 2-form of \((g, J)\). A Hermitian manifold \((M, g, J)\) is said to be locally conformal Kahler if there exists a closed 1-form \(\omega \) such that \(d\Omega =\omega \wedge \Omega \). In this paper, we prove that a Vaisman structure on compact locally conformal Kahler solvmanifolds is depend on only the form of the fundamental 2-form \(\Omega \).

MOMENT MAPS AND ISOPARAMETRIC HYPERSURFACES IN SPHERES — HERMITIAN CASES

We are studying a relationship between isoparametric hypersurfaces in spheres with four distinct principal curvatures and the moment maps of certain Hamiltonian actions. In this paper, we consider the isoparametric hypersurfaces obtained from the isotropy representations of compact irreducible Hermitian symmetric spaces of rank two. We prove that the Cartan-M\"unzner polynomials of these hypersurfaces can be written as squared-norms of the moment maps for some Hamiltonian actions. The proof is based on the structure theory of symmetric spaces.

On the Dolbeault-Dirac operator of quantized symmetric spaces

The Dolbeault complex of a quantized compact Hermitian symmetric space is expressed in terms of the Koszul complex of a braided symmetric algebra of Berenstein and Zwicknagl. This defines a spectral triple quantizing the Dolbeault‐Dirac operator associated to the canonical spin c structure.

Cotangent bundle over all the compact Hermitian symmetric spaces and projective superspace

We construct the N = 2 supersymmetric nonlinear sigma model on the cotangent bundle over the compact Hermitian symmetric space E7/E6 × U(1) by using the projective superspace formalism which is an off-shell superfield formulation in four-dimensional space-time. We also give a simple formula giving the hyper-Kahler potential of the cotangent bundle over all the compact Hermitian symmetric spaces.

Hermitian harmonic maps and non-degenerate curvatures

In this paper, we study the existence of various harmonic maps from Hermitian manifolds to Kaehler, Hermitian and Riemannian manifolds respectively. By using refined Bochner formulas on Hermitian (possibly non-Kaehler) manifolds, we derive new rigidity results on Hermitian harmonic maps from compact Hermitian manifolds to Riemannian manifolds, and we also obtain the complex analyticity of pluri-harmonic maps from compact complex manifolds to compact Kaehler manifolds (and Riemannian manifolds) with non-degenerate curvatures, which are analogous to several fundamental results in [28,Siu], [14,Jost-Yau] and [26, Sampson].