Ergodicity of laminations with singularities in Kähler surfaces

Let $\mathcal F $ be a holomorphic foliation on $\mathcal M $ , a homogeneous compact Kähler surface, with only hyperbolic singularities. Let $\mathcal L $ be a closed set saturated by leaves of the foliation, containing singularities and with every leaf dense on it. If there are no positive closed currents directed by $\mathcal L $ , then there is a unique positive harmonic current directed by $\mathcal L $ of mass one. This result was previously obtained for $\mathbb CP ^2$ by Fornæss and Sibony and we obtain the result for the rest of homogeneous compact Kähler surfaces.     

ALE Ricci-flat K?hler metrics and deformations of quotient surface singularities

Let N₀=\mathbb C²/H{N_0={\mathbb C}^2/H} be an isolated quotient singularity with H ì U(2){H\subset U(2)} a finite subgroup. We show that for any \mathbb Q{\mathbb Q} -Gorenstein smoothings of N ₀ a nearby fiber admits ALE Ricci-flat K?hler metrics in any K?hler class. Moreover, we generalize Kronheimer’s results on hyperk?hler 4-manifolds (J Differ Geom 29(3):685–697, 1989), by giving an explicit classification of the ALE Ricci-flat K?hler surfaces. We construct ALF Ricci-flat K?hler metrics on the above non-simply connected manifolds. These provide new examples of ALF Ricci-flat K?hler 4-manifolds, with cubic volume growth and cyclic fundamental group at infinity.     

A note on L. Zhou's result on Finsler surfaces with K = 0 and J = 0

In this note, we show that the examples of non-Berwaldian Landsberg surfaces with vanishing flag curvature, obtained in [5], are in fact Berwaldian. Consequently, Bryant's claim is still unverified.     

Kac–Moody $${\widetilde{E}_k}$$-bundles over elliptic curves and del Pezzo surfaces with singularities of type A

Given an elliptic curve Σ, flat E _k -bundles over Σ are in one-to-one correspondence with smooth del Pezzo surfaces of degree 9 − k containing Σ as an anti-canonical curve. This correspondence was generalized to Lie groups of any type. In this article, we show that there is a similar correspondence between del Pezzo surfaces of degree 0 with an A _d -singularity containing Σ as an anti-canonical curve and Kac–Moody [(E)\tilde]_k{\widetilde{E}_{k}}-bundles over Σ with k = 8 − d. In the degenerate case where surfaces are rational elliptic surfaces, the corresponding [(E)\tilde]_k{\widetilde{E}_k}-bundles over Σ can be reduced to E _k -bundles.     

Surfaces with $${K^2=2\mathcal{X}-2}$$ and pg ≥ 5

This note describes minimal surfaces S of general type satisfying p _g  ≥ 5 and K ² = 2p _g . For p _g  ≥ 8 the canonical map of such surfaces is generically finite of degree 2 and the bulk of the paper is a complete characterization of such surfaces with non birational canonical map. It turns out that if p _g  ≥ 13, S has always an (unique) genus 2 fibration, whose non 2-connected fibres can be characterized, whilst for p _g  ≤ 12 there are two other classes of such surfaces with non birational canonical map.     

Graphs with ∏ 1 0 (K)Y-sections

Summary We prove that a Borel subset of the product of two internal setsX andY all of whoseY-sections are ₁ ⁰ (K)( ₁ ⁰ (K)) sets is the intersection (union) of a countable sequence of Borel graphs with internalY-sections. As a consequence we prove some standard results about the domains of graphs in the product of two topological spaces all of whose horizontal section are compact (open) sets. A version of classical Vitali-Lusin theorem for those types of graphs is given as well as a new proof (and an extension) of a classical result of Kunugui.1980 Mathematics Subject Classification (1985) Revision). Primary 03H05. Secondary 04A15, 54H05Dedicated to the meamory of Goran KrkiResearch supported by the Serbian Science Foundation through a grant from the Mathematical Institute in Belgrade, Yugoslavia     

K 0 of hypersurfaces defined by x 1 2 + ... + x n 2 = ±1

Let k be a field of characteristic ≠ 2 and let Q _n,m (x ₁, ..., x _n , y ₁, ..., y _m ) = x ₁ ² +...+x _n ² ? (y ₁ ² +...+y _m ² ) be a quadratic form over k. Let R(Q _n,m ) = R _n,m = k[x ₁, ..., x _n , y ₁, ..., y _m ]/(Q _n,m ? 1). In this note we will calculate $\tilde K_0 \left( {R_{n,m} } \right)$ for every n,m ≥ 0. We will also calculate CH ₀(R _n,m ) and the Euler class group of R _n,m when k = ?.     

关于方程Dx~2±1=y~p,xy≠0

对于Diophantus方程 Dx~2 1=y~p,xy≠0,p>ε是素数,(1) 当D=2时。它仅有整数解x=±11,y=3(p=5)(参阅[1])。而当D>2无平方因子时,Nagell证明了:设ph(-D),这里h(-D)表示虚二次域Q((1/2)D)的类数,则方程(1)给出2|y。     

On the elliptic curve y 2 = x 3 −2rDx and factoring integers

Let D=pq be the product of two distinct odd primes.Assuming the parity conjecture,we construct infinitely many r≥1 such that E2rD:y2=x3-2rDx has conjectural rank one and vp(x([k]Q))≠vq(x([k]Q))for any odd integer k,where Q is the generator of the free part of E(Q).Furthermore,under the generalized Riemann hypothesis,the minimal value of r is less than c log4 D for some absolute constant c.As a corollary,one can factor D by computing the generator Q.     

Γ-Transformation of Parabolic Kählerian Spaces Related by an Almost Geodesic Mapping π2 (e = 0)

For parabolic Kählerian spaces, we obtain a new form of the main equations and construct a -transformation that enables one to convert a certain pair of related parabolic Kählerian spaces into an infinite sequence of different related parabolic Kählerian spaces.     

Algebraic surfaces of general type withK 2=2p g−1,p g≥5

This paper mainly deals with minimal algebraic surfaces of general type withK ²=2p _g–1. We prove that forp _g7 all these surfaces are birational to a double cover of some rational surfaces, and all but a finite classes of them have a unique fibration of genus 2; then we study their structures by determining their branch loci and singular fibres. We study similarly for surfaces withp _g=5, 6. Lastly we show that whenp _g13 all these surfaces are simply-connected.     

A semilinear elliptic equation with a mild singularity at u = 0: Existence and homogenization

In this paper we consider singular semilinear elliptic equations whose prototype is the following

$\{\begin{array}{cc}?divA(x)Du=f(x)g(u)+l(x)\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=0\hfill & \text{on}?\mathrm{\Omega },\hfill \end{array}$

where Ω is an open bounded set of ${\mathbb{R}}^N,N\ge 1$, $A\in L^{\infty }{(\mathrm{\Omega })}^{N\times N}$ is a coercive matrix, $g:[0,+\infty [\to [0,+\infty ]$ is continuous, and $0\le g(s)\le \frac{1}{s^{\gamma }}+1$ for every $s>0$, with $0<\gamma \le 1$ and $f,l\in L^r(\mathrm{\Omega })$, $r=\frac{2N}{N+2}$ if $N\ge 3$, $r>1$ if $N=2$, $r=1$ if $N=1$, $f(x),l(x)\ge 0$ a.e. $x\in \mathrm{\Omega }$.We prove the existence of at least one nonnegative solution as well as a stability result; we also prove uniqueness if $g(s)$ is nonincreasing or “almost nonincreasing”.Finally, we study the homogenization of these equations posed in a sequence of domains ${\mathrm{\Omega }}^{\epsilon }$ obtained by removing many small holes from a fixed domain Ω.     

Compactness theorems and the Calabi flow on Kähler surfaces with stable tangent bundle

Abstract. In this paper, we prove some compactness theorems and collapse phenomenon on compact K?hler surfaces with stable tangent bundle. We then apply the results to the Calabi flow. More precisely, we prove, under suitable curvature conditions, the longtime existence and asymptotic convergence for solutions of the Calabi flow on compact K?hler surfaces admitting no nonzero holomorphic tangent vector fields and with stable tangent bundle. We also give some examples where the Calabi flow blows up. Received January 7, 1999 / Revised February 2, 2000 / Published online July 20, 2000     

Convex hulls of surfaces with boundaries and corners and singularities of transitivity zone in ℝ<Superscript>3</Superscript>

Generic singularities of the boundary of the local transitivity set of a control system on two- and three-dimensional manifolds are classified. The indicatrices of the system are assumed to be given by generic equations and inequalities.     

Higher differentiability for the solutions of nonlinear elliptic systems with lower-order terms and L 1,θ -data

In this paper we study the higher differentiability of the solutions to a class of nonlinear systems of elliptic partial differential equations with a lower-order term having natural growth with respect to the gradient and with data belonging to a suitable Morrey space.     

Invariant Geometric Objects of the Canonical Almost-Geodesic Mapping π2 (e = 0)

For the canonical almost-geodesic mapping ₂ (e = 0), we prove an analog of the Beltrami theorem in the theory of geodesic mappings. We introduce canonical ₂-flat spaces and obtain metrics for them in a special coordinate system.