Kahler Geometry Research Articles

Nonlinear Whole Parapsychology and Its Mathematical-Physical Methods

Based on the quantum theory, we first discuss some research of parapsychology, whose experiments indicate possible existence of new other worlds. Panpsychism shows also the existence of mind, psi, etc., except general matter-energy-information. Second, we propose the nonlinear whole parapsychology and its three basic rules. Third, we study some mathematical and physical methods, which include the thought field, quantum parapsychology, etc. Fourth, new spaces of parapsychology and the network model are researched. Fifth, we search a mathematical space with complex manifold—Kahler geometry, which may provide wide perspectives to lucubrate new worlds in parapsychology. Chinese parapsychology can combine rich traditional Chinese culture to display various unique phenomena, but parapsychology must be universal for the world.

Finite entropy vs finite energy

Probability measures with either finite Monge-Ampere energy or finite entropy have played a central role in recent developments in Kahler geometry. In this note we make a systematic study of quasi-plurisubharmonic potentials whose Monge-Ampere measures have finite entropy. We show that these potentials belong to the finite energy class ${\mathcal E}^{\frac{n}{n-1}}$, where $n$ denotes the complex dimension, and provide examples showing that this critical exponent is sharp. Our proof relies on refined Moser-Trudinger inequalities for quasi-plurisubharmonic functions.

Some results on the complex Monge-Ampère equation

The complex Monge-Ampere equation has been the subject ofintensive studies in the past fifty years, since its significant applications in complex analysis, complex geometry and Kahler geometry. In this paper, we will introduce our recent work on the Liouville type theorem, $C^{1,~\\alpha}$ and $C^{2,~\\alpha~}$ estimates of the complex Monge-Ampere equation and its applications in Sasakian geometry. These works are joint with S. Dinew, P. F. Guan, C. Li, J. Y. Li and X. W. Zhang.

Scalar curvature as moment map in generalized Kähler geometry

It is known that the scalar curvature arises as the moment map in Kahler geometry. In pursuit of this analogy, we introduce the notion of a moment map in generalized Kahler geometry which gives the definition of a generalized scalar curvature on a generalized Kahler manifold. From the viewpoint of the moment map, we obtain the generalized Ricci form which is a representative of the first Chern class of the anticanonical line bundle. It turns out that infinitesimal deformations of generalized Kahler structures with constant generalized scalar curvature are finite dimensional on a compact manifold. Explicit descriptions of the generalized Ricci form and the generalized scalar curvature are given on a generalized Kahler manifold of type $(0,0)$. Poisson structures constructed from a Kahler action of $T^m$ on a Kahler-Einstein manifold give intriguing deformations of generalized Kahler-Einstein structures. In particular, the anticanical divisor consists of three lines on $C P^2$ in general position yields nontrivial examples of generalized Kahler-Einsein structures

Gauge Linear Sigma Model for Berglund—Hübsch-Type Calabi—Yau Manifolds

We briefly present the results of our computation of special Kahler geometry for polynomial deformations of Berglund–Hubsch type Calabi–Yau manifolds. We also build mirror symmetric Gauge Linear Sigma Model and check that its partition function computed by supersymmetric localization coincides with exponent of the Kahler potential of the special metric.

Blow-ups in generalized complex geometry

Generalized complex geometry is a theory that unifies complex geometry and symplectic geometry into one single framework. It was introduced by Hitchin and Gualtieri around 2002. In this thesis we address the following question: given a generalized complex manifold together with a submanifold, does the blow-up of that submanifold admit again a generalized complex structure? We give the following answers to this question: If the directions that are normal to the submanifold are purely complex, then the blow-up admits a generalized complex structure for which the blow-down map is generalized holomorphic if and only if the induced Lie algebra on the conormal bundle of the submanifold is degenerate. This is a rather restrictive condition, but it is always satisfied if the submanifold is of complex codimension two. On the other extreme, if all the normal directions are symplectic, then we show that the blow-up always admits a generalized complex structure, provided the submanifold is compact. We then also give an example of a submanifold that is not of the above two types, and show that it can not be blown up. Finally, we also address the analogous question in generalized Kahler geometry. Given a generalized Kahler manifold together with a submanifold, we show that under certain geometric constraints on the submanifold the blow-up is again generalized Kahler.

On the CR Poincaré–Lelong Equation, Yamabe Steady Solitons and Structures of Complete Noncompact Sasakian Manifolds

In this paper, we solve the so-called CR Poincare–Lelong equation by solving the CR Poisson equation on a complete noncompact CR (2n + 1)-manifold with nonegative pseudohermitian bisectional curvature tensors and vanishing torsion which is an odd dimensional counterpart of Kahler geometry. With applications of this solution plus the CR Liouvelle property, we study the structures of complete noncompact Sasakian manifolds and CR Yamabe steady solitons.

Propagation of chaos, Wasserstein gradient flows and toric Kähler–Einstein metrics

Motivated by a probabilistic approach to Kahler-Einstein metrics we consider a general nonequilibrium statistical mechanics model in Euclidean space consisting of the stochastic gradient flow of a given (possibly singular) quasiconvex N-particle interaction energy. We show that a deterministic macroscopic evolution equation emerges in the large N-limit of many particles. This is a strengthening of previous results which required a uniform two-sided bound on the Hessian of the interaction energy. The proof uses the theory of weak gradient flows on the Wasserstein space. Applied to the setting of permanental point processes at negative temperature, the corresponding limiting evolution equation yields a driftdiffusion equation, coupled to the Monge-Ampere operator, whose static solutions correspond to toric Kahler-Einstein metrics. This drift-diffusion equation is the gradient flow on the Wasserstein space of probability measures of the K-energy functional in Kahler geometry and it can be seen as a fully nonlinear version of various extensively studied dissipative evolution equations and conservation laws, including the Keller-Segel equation and Burger's equation. In a companion paper, applications to singular pair interactions in one dimension are given.

Interaction of Codazzi Couplings with (Para-)Kähler Geometry

We study Codazzi couplings of an affine connection $$\nabla $$ with a pseudo-Riemannian metric g, a nondegenerate 2-form $$\omega $$ , and a tangent bundle isomorphism L on smooth manifolds, as an extension of their parallelism under $$\nabla $$ . In the case that L is an almost complex or an almost para-complex structure and $$(g, \omega , L)$$ form a compatible triple, we show that Codazzi coupling of a torsion-free $$\nabla $$ with any two of the three leads to its coupling with the remainder, which further gives rise to a (para-)Kahler structure on the manifold. This is what we call a Codazzi-(para-)Kahler structure; it is a natural generalization of special (para-)Kahler geometry, without requiring $$\nabla $$ to be flat. In addition, we also prove a general result that g-conjugate, $$\omega $$ -conjugate, and L-gauge transformations of $$\nabla $$ , along with identity, form an involutive Abelian group. Hence a Codazzi-(para-)Kahler manifold admits a pair of torsion-free connections compatible with the $$(g, \omega , L)$$ . Our results imply that any statistical manifold may admit a (para-)Kahler structure as long as one can find an L that is compatible to g and Codazzi coupled with $$\nabla $$ .

Uniform K-stability, Duistermaat–Heckman measures and singularities of pairs

The purpose of the present paper is to set up a formalism inspired from non-Archimedean geometry to study K-stability. We rst provide a detailed analysis of Duistermaat-Heckman measures in the context of test congurations, characterizing in par- ticular the trivial case. For any normal polarized variety (or, more generally, polarized pair in the sense of the Minimal Model Program), we introduce and study the non-Archimedean analogues of certain classical functionals in Kahler geometry. These functionals are dened on the space of test congurations, and the Donaldson-Futaki invariant is in particular interpreted as the non-Archimedean version of the Mabuchi functional, up to an explicit error term. Finally, we study in detail the relation between uniform K-stability and sin- gularities of pairs, reproving and strengthening Y. Odaka's results in our formalism. This provides various examples of uniformly K-stable varieties.

Nearly Kahler geometry and (2,3,5)-distributions via projective holonomy

We show that any dimension $6$ nearly K\"ahler (or nearly para-K\"ahler) geometry arises as a projective manifold equipped with a $\textrm{G}_2^{(*)}$ holonomy reduction. In the converse direction we show that if a projective manifold is equipped with a parallel $7$-dimensional cross product on its standard tractor bundle then the manifold is: a Riemannian nearly K\"ahler manifold, if the cross product is definite, otherwise, if the cross product has the other algebraic type, the manifold is in general stratified with nearly K\"ahler and nearly para-K\"ahler parts separated by a hypersurface which canonically carries a Cartan $(2,3,5)$-distribution. This hypersurface is a projective infinity for the pseudo-Riemannian geometry elsewhere on the manifold, and we establish how the Cartan distribution can be understood explicitly, and also in terms of conformal geometry, as a limit of the ambient nearly (para\nobreakdash-)K\"ahler structures. Any real-analytic $(2,3,5)$-distribution is seen to arise as such a limit, because we can solve the geometric Dirichlet problem of building a collar structure equipped with the required holonomy-reduced projective structure. A model geometry for these structures is provided by the projectivization of the imaginary (split) octonions. Our approach is to use Cartan/tractor theory to provide a curved version of this geometry, this encodes a curved version of the algebra of imaginary (split) octonions as a flat structure over its projectivization. The perspective is used to establish detailed results concerning the projective compactification of nearly (para\nobreakdash-)K\"ahler manifolds, including how the almost (para\nobreakdash-)complex structure and metric smoothly degenerate along the singular hypersurface to give the distribution there.

Kähler moduli inflation in type IIB compactifications: A random tumble through the Calabi-Yau landscape

In this paper we present an initial exploration of the Calabi-Yau landscape in the context of Kahler moduli inflation. We review how the slow-roll requirement on the scalar potential translates to a geometric constraint on the Kahler geometry of the vacuum. This constraint leads to a hard bound on the moduli space geometry and we consider the effects of this constraint on the string landscape that arises in type IIB string compactifications on an O3/O7 orientifold. Most notably we find that the inflationary constraint is independent of the moduli space dimension and only 6.57% of geometries inspected support high-scale Kahler moduli inflation.

Choquet-Monge-Ampère Classes

We introduce and study Choquet-Monge-Ampere classes on compact Kahler manifolds. They consist of quasi-plurisubharmonic functions whose sublevel sets have small enough asymptotic Monge-Ampere capacity. We compare them with finite energy classes, which have recently played an important role in Kahler Geometry.

Hamiltonian stationary Lagrangian fibrations

Hamiltonian stationary Lagrangian submanifolds (HSLAG) are a natural generalization of special Lagrangian manifolds (SLAG). The latter only make sense on Calabi-Yau manifolds whereas the former are defined for any almost Kahler manifold. Special Lagrangians, and, more specificaly, fibrations by special Lagrangians play an important role in the context of the geometric mirror symmetry conjecture. However, these objects are rather scarce in nature. On the contrary, we show that HSLAG submanifolds, or fibrations, arise quite often. Many examples of HSLAG fibrations are provided by toric Kah-ler geometry. In this paper, we obtain a large class of examples by deforming the toric metrics into non toric almost Kahler metrics, together with HSLAG submanifolds.

Lower order tensors in non-Kähler geometry and non-Kähler geometric flow

In recent years, Streets and Tian introduced a series of curvature flows to study non-Kahler geometry. In this paper, we study how to construct the second-order curvature flows in a uniform way, under some natural assumptions which hold in Streets and Tian’s works. As a result, by classifying the lower order tensors, we classify the second-order curvature flows in almost Hermitian, almost Kahler, and Hermitian geometries in certain sense. In particular, the Symplectic Curvature Flow is the unique way to generalize Ricci Flow on almost Kahlermanifolds.

Hessian geometry and the holomorphic anomaly

We present a geometrical framework which incorporates higher derivative corrections to the action of N = 2 vector multiplets in terms of an enlarged scalar manifold which includes a complex deformation parameter. This enlarged space carries a deformed version of special Kahler geometry which we characterise. The holomorphic anomaly equation arises in this framework from the integrability condition for the existence of a Hesse potential.

Euclidean Embeddings and Riemannian Bergman Metrics

Consider the sum of the first N eigenspaces for the Laplacian on a Riemannian manifold. A basis for this space determines a map to Euclidean space and for N sufficiently large the map is an embedding. In analogy with a fruitful idea of Kahler geometry, we define (Riemannian) Bergman metrics of degree N to be those metrics induced by such embeddings. Our main result is to identify a natural sequence of Bergman metrics approximating any given Riemannian metric. In particular we have constructed finite dimensional symmetric space approximations to the space of all Riemannian metrics. Moreover the construction induces a Riemannian metric on that infinite dimensional manifold which we compute explicitly.

Early universe cosmology, effective supergravity, and invariants of algebraic forms

The presence of light scalars can have profound effects on early universe cosmology, influencing its thermal history as well as paradigms like inflation and baryogenesis. Effective supergravity provides a framework to make quantifiable, model-independent studies of these effects. The Riemanian curvature of the Kahler manifold spanned by scalars belonging to chiral superfields, evaluated along supersymmetry breaking directions, provides an order parameter (in the sense that it must necessarily take certain values) for phenomena as diverse as slow roll modular inflation, non-thermal cosmological histories, and the viability of Affleck-Dine baryogenesis. Within certain classes of UV completions, the order parameter for theories with $n$ scalar moduli is conjectured to be related to invariants of $n$-ary cubic forms (for example, for models with three moduli, the order parameter is given by the ring of invariants spanned by the Aronhold invariants). Within these completions, and under the caveats spelled out, this may provide an avenue to obtain necessary conditions for the above phenomena that are in principle calculable given nothing but the intersection numbers of a Calabi-Yau compactification geometry. As an additional result, abstract relations between holomorphic sectional and bisectional curvatures are utilized to constrain Affleck-Dine baryogenesis on a wide class of Kahler geometries.

Cosmological attractors fromα-scale supergravity

The Planck value of the spectral index can be interpreted as $n_s = 1 - 2/N$ in terms of the number of e-foldings $N$. An appealing explanation for this phenomenological observation is provided by $\alpha$-attractors: the inflationary predictions of these supergravity models are fully determined by the curvature of the Kahler manifold. We provide a novel formulation of $\alpha$-attractors which only involves a single chiral superfield. Our construction involves a natural deformation of no-scale models, and employs these to construct a De Sitter plateau with an exponential fall-off. Finally, we show how analogous structures with a flat Kahler geometry arise as a singular limit of such $\alpha$-scale models.

Edges, orbifolds, and Seiberg–Witten theory

Seiberg-Witten theory is used to obtain new obstructions to the existence of Einstein metrics on 4-manifolds with conical singularities along an embedded surface. In the present article, the cone angle is required to be of the form 2…=p, p a positive integer, but we conjecture that similar results will also hold in greater generality. Recent work on Kahler-Einstein metrics by Chen, Donaldson, Sun, and others (5), (12), (13), (17), (26) has elicited wide interest in the existence and uniqueness problems for Einstein metrics with conical singularities along a submanifold of real codimension 2. This article will show that Seiberg-Witten theory gives rise to interesting obstructions to the existence of 4-dimensional Einstein metrics with conical singularities along a sur- face. These results are intimately tied to known phenomena in Kahler geometry, and reinforce the overarching principle that Kahler metrics play a uniquely privileged role in 4-dimensional Riemannian geometry, to a degree that is simply unparalleled in other dimensions. We now recall the definition (2) of an edge-cone metric on a 4-manifold. Let M be a smooth compact 4-manifold, let § ‰ M be a smoothly embedded compact surface. Near any point p 2 §, we can thus find local coordinates (x 1 ;x 2 ;y 1 ;y 2 ) in which § is given by y 1 = y 2 = 0. Given any such adapted coordinate system, we then introduce an associated transversal polar coordinate system (‰;µ;x 1 ;x 2 ) by setting y 1 = ‰cosµ and y 2 = ‰sinµ. Now fix some positive constant fl > 0. An edge-cone metric g of cone angle