Fixed point indices and periodic points of holomorphic mappings

Let Δ ⁿ be the ball |x| <  1 in the complex vector space , let be a holomorphic mapping and let M be a positive integer. Assume that the origin is an isolated fixed point of both f and the Mth iteration f ^M of f. Then for each factor m of M, the origin is again an isolated fixed point of f ^m and the fixed point index of f ^m at the origin is well defined, and so is the (local) Dold’s index [Invent. Math. 74(3), 419–435 (1983)] at the origin:

Rigidity of proper holomorphic mappings between nonequidimensional bounded symmetric domains

We prove that any proper holomorphic mapping from to is necessarily a totally geodesic isometric embedding with respect to their Bergman metrics and therefore is the standard linear embedding up to their automorphisms. The proof of the main result in this paper will be achieved by a differential-geometric study of a special class of complex geodesic curves on the bounded symmetric domains with respect to their Bergman metrics. Received: 31 October 2000; in final form: 2 July 2001/ Published online: 4 April 2002     

Oka's principle for holomorphic submersions with sprays

We classify, up to a local isometry, all non-K?hler almost K?hler 4-manifolds for which the fundamental 2-form is an eigenform of the Weyl tensor, and whose Ricci tensor is invariant with respect to the almost complex structure. Equivalently, such almost K?hler 4-manifolds satisfy the third curvature condition of A. Gray. We use our local classification to show that, in the compact case, the third curvature condition of Gray is equivalent to the integrability of the corresponding almost complex structure. Received: 1 October 2001 / Published online: 17 June 2002     

An algebraic characterization of holomorphic nondegeneracy for real algebraic hypersurfaces and its application to CR mappings

We give a new algebraic characterization of holomorphic nondegeneracy for embedded real algebraic hypersurfaces in , . We then use this criterion to prove the following result about real analyticity of smooth CR mappings: any smooth CR mapping H between a real analytic hypersurface and a rigid polynomial holomorphically nondegenerate hypersurface is real analytic, provided the map H is not totally degenerate in the sense of Baouendi and Rothschild. Received September 19, 1997     

On singular fibres of Lagrangian fibrations over holomorphic symplectic manifolds

We classify singular fibres over general points of the discriminant locus of projective Lagrangian fibrations over 4-dimensional holomorphic symplectic manifolds. The singular fibre F is the following either one: F is isomorphic to the product of an elliptic curve and a Kodaira singular fibre up to finite unramified covering or F is a normal crossing variety consisting of several copies of a minimal elliptic ruled surface of which the dual graph is Dynkin diagram of type or . Moreover, we show all types of the above singular fibres actually occur. Received: 10 March 2000 / Revised version: 29 September 2000 / Published online: 24 September 2001     

A topological proof of cancellation theorem for ${\vec C}^2$

We give a new proof of the Cancellation Theorem for which says that if V is a smooth complex affine surface such that is isomorphic to then V is isomorphic to The proof is based on the topological method of Mumford-Ramanujam. Received: 2 November 2000 / in final form: 10 September 2001 / Published online: 1 February 2002     

On symplectic cobordisms

In this note we make several observations concerning symplectic cobordisms. Among other things we show that every contact 3-manifold has infinitely many concave symplectic fillings and that all overtwisted contact 3-manifolds are “symplectic cobordism equivalent”. Received: 26 March 2001 / Revised version: 1 May 2001 / Published online: 28 February 2002     

Mountain pass critical points for Paneitz-Branson operators

Given a smooth compact Riemannian manifold of dimension , we study fourth order equations involving Paneitz-Branson type operators and the critical Sobolev exponent. Received: 19 June 2001 / Accepted: 27 September 2001 / Published online: 28 February 2002 Esposito's research is supported by M.U.R.S.T. under the national project Variational methods and nonlinear differential equations.     

Multiplicity of holomorphic functions

The main result of this paper concerns an analytic version of Birkhoff's ergodic theorem. It allows us to define asymptotic multiplicities given any algebraically stable rational map . This is the key for our study of the singularities of the Green current T associated to f. We characterize the points where the Lelong number of T is strictly positive. Received: 30 July 1999     

Projective resolutions associated to projections

In this paper we will describe projective resolutions of d dimensional Cohen–Macaulay spaces X by means of a projection of X to a hypersurface in d+1-dimensional space. We will show that for a certain class of projections, the resulting resolution is minimal. Received: 22 February 1999     

Relative Discrete Spectrum and Joinings

 A joining characterization of ergodic isometric extensions is given. We also give a simple joining proof of a relative version of the Halmos-von Neumann theorem. Research partly supported by KBN grant 2 P03A 002 14 (1998). Received June 5, 2001; in revised form March 4, 2002     

On holomorphic harmonic morphisms

We study holomorphic harmonic morphisms from K?hler manifolds to almost Hermitian manifolds. When the codomain is also K?hler we get restrictions on such maps in the case of constant holomorphic curvature. We also prove a Bochner-type formula for holomorphic harmonic morphisms which, under certain curvature conditions of the domain, gives insight to the structure of the vertical distribution. We thus prove that when the domain is compact and non-negatively curved, the vertical distribution is totally geodesic. Received: 28 May 2001     

The central limit theorem for Markov chains with normal transition operators, started at a point

The central limit theorem and the invariance principle, proved by Kipnis and Varadhan for reversible stationary ergodic Markov chains with respect to the stationary law, are established with respect to the law of the chain started at a fixed point, almost surely, under a slight reinforcing of their spectral assumption. The result is valid also for stationary ergodic chains whose transition operator is normal. Received: 28 March 2000 / Revised version: 25 July 2000 /?Published online: 15 February 2001     

Stability and error estimates for the $\theta$-method for strongly monotone and infinitely stiff evolution equations

Summary.   For evolution equations with a strongly monotone operator we derive unconditional stability and discretization error estimates valid for all . For the -method, with , we prove an error estimate , if , where is the maximal integration step for an arbitrary choice of sequence of steps and with no assumptions about the existence of the Jacobian as well as other derivatives of the operator , and an optimal estimate under some additional relation between neighboring steps. The first result is an improvement over the implicit midpoint method , for which an order reduction to sometimes may occur for infinitely stiff problems. Numerical tests illustrate the results. Received March 10, 1999 / Revised version received April 3, 2000 / Published online February 5, 2001     

Univalent holomorphic mappings on a complex manifold with a C1 exhaustion function

Let M be a complex manifold with a exhaustion function ρ. Assume that ρ attains its minimum at a single point o and dρ≠ 0 on M\{o}. In this case, we will give a necessary condition and a sufficient condition for locally biholomorphic mappings on M to be biholomorphic and to have Φ-like images. Received: 15 May 1998 / Revised version: 8 March 1999     

Harder–Narasimhan reduction for principal bundles over a compact Kähler manifold

Generalizing the Harder–Narasimhan filtration of a vector bundle it is shown that a principal G-bundle over a compact K?hler manifold admits a canonical reduction of its structure group to a parabolic subgroup of G. Here G is a complex connected reductive algebraic group; in the special case where , this reduction is the Harder–Narasimhan filtration of the vector bundle associated to by the standard representation of . The reduction of in question is determined by two conditions. If P denotes the parabolic subgroup, L its Levi factor and the canonical reduction, then the first condition says that the principal L-bundle obtained by extending the structure group of the P-bundle using the natural projection of P to L is semistable. Denoting by the Lie algebra of the unipotent radical of P, the second condition says that for any irreducible P-module V occurring in , the associated vector bundle is of positive degree; here is considered as a P-module using the adjoint action. The second condition has an equivalent reformulation which says that for any nontrivial character of P which can be expressed as a nonnegative integral combination of simple roots (with respect to any Borel subgroup contained in P), the line bundle associated to for is of positive degree. The equivalence of these two conditions is a consequence of a representation theoretic result proved here. Received: 10 November 1999 / Revised version: 31 October 2001 / Published online: 26 April 2002     

Plurisubharmonic Defining Functions,Good Vector Fields,and Exactness of a Certain One Form

 We show that the approaches to global regularity of the -Neumann problem via the methods listed in the title are equivalent when the conditions involved are suitably modified. These modified conditions are also equivalent to one that is relevant in the context of Stein neighborhood bases and Mergelyan type approximation. Received 14 May 2001; in revised form 8 November 2001