Symplectic, complex and Kähler structures on four-dimensional generalized symmetric spaces

We obtain the full classification of invariant symplectic, (almost) complex and Kähler structures, together with their paracomplex analogues, on four-dimensional pseudo-Riemannian generalized symmetric spaces. We also apply these results to build some new examples of five-dimensional homogeneous K-contact, Sasakian, K-paracontact and para-Sasakian manifolds.     

Parabolic equations of Von Karman type on Kähler manifolds

We study a parabolic version of a system of Von Karman type on a compact Kähler manifold of arbitrary dimension. We provide local in time regular solutions, which can be extended to global bounded ones if the data of the problem are small.     

Stratified Kähler structures on adjoint quotients

Given a compact Lie group, endowed with a bi-invariant Riemannian metric, its complexification inherits a Kähler structure having twice the kinetic energy of the metric as its potential, and Kähler reduction with reference to the adjoint action yields a stratified Kähler structure on the resulting adjoint quotient. Exploiting classical invariant theory, in particular bisymmetric functions and variants thereof, we explore the singular Poisson-Kähler geometry of this quotient. Among other things we prove that, for various compact groups, the real coordinate ring of the adjoint quotient is generated, as a Poisson algebra, by the real and imaginary parts of the fundamental characters. We also show that singular Kähler quantization of the geodesic flow on the reduced level yields the irreducible algebraic characters of the complexified group.     

Parabolic equations of Von Karman type on Kähler manifolds, II

We complete the study of a parabolic version of a system of Von Karman type on a compact Kähler manifold of complex dimension m. We consider a family of problems k(P). We prove existence of local in time solutions when k=0. When −m?k<0, we define a notion of weak solution, and give some uniqueness and existence results.     

Polar actions on compact rank one symmetric spaces are taut

We prove that the orbits of a polar action of a compact Lie group on a compact rank one symmetric space are tautly embedded with respect to Z ₂-coefficients.The second author was supported in part by FAPESP and CNPq.     

On stability and continuity of bounded solutions of degenerate complex Monge-Ampère equations over compact Kähler manifolds

We obtain a stability estimate for the degenerate complex Monge-Ampère operator which generalizes a result of Ko?odziej (2003) [12]. In particular, we obtain the optimal stability exponent and also treat the case when the right-hand side is a general Borel measure satisfying certain regularity conditions. Moreover, our result holds for functions plurisubharmonic with respect to a big form, thus generalizing the Kähler form setting in Ko?odziej (2003) [12]. Independently, we also provide more detail for the proof in Zhang (2006) [18] on continuity of the solution with respect to a special big form when the right-hand side is Lp-measure with p>1.     

Kähler structure on moduli spaces of principal bundles

Let M be a moduli space of stable principal G-bundles over a compact Kähler manifold (X,ωX), where G is a reductive linear algebraic group defined over C. Using the existence and uniqueness of a Hermite-Einstein connection on any stable G-bundle P over X, we have a Hermitian form on the harmonic representatives of H¹(X,ad(P)), where ad(P) is the adjoint vector bundle. Using this Hermitian form a Hermitian structure on M is constructed; we call this the Petersson-Weil form. The Petersson-Weil form is a Kähler form, a fact which is a consequence of a fiber integral formula that we prove here. The curvature of the Petersson-Weil Kähler form is computed. Some further properties of this Kähler form are investigated.     

Kähler non-collapsing,eigenvalues and the Calabi flow

We first prove a new compactness theorem of Kähler metrics, which confirms a prediction in [17]. Then we establish several eigenvalue estimates along the Calabi flow. Combining the compactness theorem and these eigenvalue estimates, we generalize the method developed for the Kähler–Ricci flow in [22] to obtain several new small energy theorems of the Calabi flow.     

Holomorphic vector bundles on compact complex non-algebraic surfaces

LetX be a smooth complex compact surface without non-constant meromorphic functions. Here we prove the existence of rank holomorphic vector bundles onX containing exactly one rank one saturated subsheaf.

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Sunto SiaX una superficie complessa compatta non singolare senza funzioni meromorfe non costanti. In questo lavoro si domstra cheX possiede molti fibrati olomorfi di rango 2 contenenti un unico fibrato in rette.

     

Almost Kähler 4-dimensional Lie groups with J-invariant Ricci tensor

The J-invariance of the Ricci tensor is a natural weakening of the Einstein condition in almost Hermitian geometry. The aim of this paper is to determine left-invariant strictly almost Kähler structures (g,J,Ω) on real 4-dimensional Lie groups such that the Ricci tensor is J-invariant. We prove that all these Lie groups are isometric (up to homothety) to the (unique) 4-dimensional proper 3-symmetric space.     

Nearly Kähler manifolds with vanishing Tricerri-Vanhecke Bochner curvature tensor

We study the local structures of nearly Kähler manifolds with vanishing Bochner curvature tensor as defined by Tricerri and Vanhecke (TV). We show that there does not exist a TV Bochner flat strict nearly Kähler manifold in 2n(?10) dimension and determine the local structures of the manifolds in 6 and 8 dimensions.     

Ricci solitons on compact Kähler surfaces

We classify the Kähler metrics on compact manifolds of complex dimension two that are solitons for the constant-volume Ricci flow, assuming that the curvature is slightly more positive than that of the single known example of a soliton in this dimension.

     

The g-periodic subvarieties for an automorphism g of positive entropy on a compact Kähler manifold

For a compact Kähler manifold X and a strongly primitive automorphism g of positive entropy, it is shown that X has at most ρ(X) of g-periodic prime divisors. When X is a projective threefold, every prime divisor containing infinitely many g-periodic curves, is shown to be g-periodic (a result in the spirit of the Dynamic Manin-Mumford conjecture as in Zhang (2006) [17]).     

Metrization of free groups on ultrametric spaces

We consider ultrametrizations of free topological groups of ultrametric spaces. A construction is defined that determines a functor in the category UMET₁ of ultrametric spaces of diameter ?1 and nonexpanding maps. This functor is the functorial part of a monad in UMET₁ and we provide a characterization of the category of its algebras.     

Algebraic deformations of compact Kähler surfaces

A short proof is given of Kodaira's result that every compact kähler surface is a deformation of an algebraic surface under the extra assumption that all infinitesimal deformations are unobstructed.     

Volume renormalization for complete Einstein-Kähler metrics

For a strictly pseudoconvex domain in a complex manifold we define a renormalized volume with respect to the approximately Einstein complete Kähler metric of Fefferman. We compute the conformal anomaly in complex dimension two and apply the result to derive a renormalized Chern-Gauss-Bonnet formula. Relations between renormalized volume and CR Q-curvature are also investigated.     

Holomorphic approximation on compact pseudoconvex complex manifolds

Let be a smoothly bounded compact pseudoconvex complex manifold of finite type in the sense of D’Angelo such that the complex structure of M extends smoothly up to bM. Let m be an arbitrary nonnegative integer. Let f be a function in H(M)∩ H_m(M), where H_m(M) is the Sobolev space of order m. Then f can be approximated by holomorphic functions on in the Sobolev space H_m(M). Also, we get a holomorphic approximation theorem near a boundary point of finite type.