Compact Kahler surfaces with harmonic W^-.
by
Wlodzimierz Jelonek
Cracow University of Technology,Poland

Oral Communication

We prove in the paper that every irreducible compact Kähler surface with \de W^-=0 is Kähler-Einstein or is biholomorphic to a ruled extremal Kähler surface. It is well known that self-dual Kähler 4-manifolds (M, g, J) are Bochner-flat and their Ricii tensor \rho satisfies a condition (*) \n_X\rho(X, X)=\frac13X\taug(X, X) where \tau is the scalar curvature of (M, g). This property was studied by A. Gray. A. Gray called Riemannian manifolds satisfying (*) the \AC manifolds. The author has proved that there is a close relation between Einstein-Weyl manifolds and certain class of \AC manifolds. In particular there are many non-trivial compact examples of \AC-manifolds. It is well known that compact self-dual Kähler surfaces are locally symmetric and that Kähler manifold with harmonic Weyl tensor has a parallel Ricci tensor (this last result due to S. Tanno holds locally). In the present paper we shall show that every Kähler surface with harmonic anti-self dual part W^- of the Weyl tensor W (i.e. such that \de W^-=0) is an \AC-manifold. We also prove that every compact Kähler surface (M, g, J) with harmonic anti-selfdual Weyl tensor (\de W^-=0) has constant scalar curvature ( and thus is Einstein or is locally a product of two Riemannian surfaces with constant sectional curvatures) or is a ruled surface with extremal Kähler metric with non-constant scalar curvature and admits an opposite Hermitian structure \bJ such that (M, g, \bJ) satisfies a (G₂) condition of A.Gray and is conformal to an extremal Kähler surface. J.P. Bourguignon has proved that every compact four manifold (M, g) with harmonic curvature tensor \de R=0 and non-vanishing signature is Einstein. We prove in the present paper that for Kähler surfaces (M, g, J) the same result holds under much weaker assumption \de W^-=0. We say that (M, g, J) is a Kähler manifold if its Kähler form \0(X, Y)=g(JX, Y) is closed (d\0 = 0) and (M, J) is a complex manifold. We consider 4-dimensional Kähler manifolds (M, g, J) which we shall also call Kähler surfaces and hermitian surfaces (M, g, J) i.e. almost hermitian four-manifolds with integrable almost complex structure. Such manifolds are always oriented and we choose an orientation in such a way that \0 is self-dual form (i.e. \0 in \w⁺M). The Hodge star operator * (which depends on the orientation of M) defines an endomorphism *\w M --> \w M with *²=id and we denote by \w⁺, \w^- its eigen-subbundles corresponding to 1, -1 respectively. Let us define B=\frac12(\R-*\R*); W=\frac12(\R+*\R*); W⁺=\frac12(W+*W);W^-=\frac12(W-*W). Then

\R = \frac\tau12Id+B+W++W-.

The Lee form \th of (M, g, J) is defined by the equality d\0 = \th\w \0. We have \th = -\delta\0 o J. The conformal scalar curvature \kappa of a Hermitian manifold (M, g, J) is conformally covariant of weight -2 and is related to the Riemannian scalar curvature \tau of (M, g) by

\kappa = \tau-\frac32(|\th|2+2\de\th).

By an \AC- manifold we mean (after A.Gray) a Riemannian manifold (M, g) satisfying the condition

CX Y Z\nX\rho(Y, Z)=\frac 2(dimM+2)CX Y ZX\taug(Y, Z), \tag *

where \rho is the Ricci tensor of (M, g) and C means the cyclic sum. The author has proved that a Riemannian manifold (M, g) is an \AC manifold if and only if the Ricci endomorphism Ric of (M, g) is of the form Ric=S+\frac2n+2\tauId where S is a Killing tensor, \tau is the scalar curvature and n=dimM. Let us recall that a (1, 1) tensor S on a Riemannian manifold (M, g) is called a Killing tensor if g(\n S(X, X), X)= for all X in TM. Finally let us recall that a Riemannian manifold with Killing Ricci tensor is called after A. Gray an \A-manifold ( it is an \AC-manifold with constant scalar curvature).

Proposition 1. Let us assume that (M, g, J) is a Kähler \AC-manifold. Then the following relation holds

\gather \nX\rho(Y, Z)=\frac 1(2dim M+4)(g(X, Y)Z\tau+g(X, Z)Y\tau+2g(Y, Z)X\tau\tag 2.1-g(JX, Y)(JZ)\tau-g(JX, Z)(JY)\tau).\endgather

Corollary. Let (M, g, J) be a Kähler surface with \de W^-=0 and constant scalar curvature. Then (M, g, J) is Kähler Einstein manifold or it is locally the product of two Riemannian surfaces (complex lines) of constant sectional curvatures.

Theorem 1. Let (M, g, J) be a compact Kähler surface with \de W^-=0. Let us assume that (M, g) is real analytic as Riemannian manifold. Then (M, g) has constant scalar curvature or a scalar curvature of (M, g) is nonconstant and (M, J) is a ruled surface, (M, g, J) is an extremal Kähler surface and there exists an opposite Hermitian structure \bJ on M such that (M, g, \bJ) is a G₂ manifold.

We prove that \kappa = 0 on M or \bth = -\frac23dln|\kappa|=2dln|\lb-\mu| and consequently \kappa = C(\lb-\mu)^-3 for some constant C in R-{0}.

Corollary. Let (M, g, J) be a compact Kähler surface with \de W^-=0. If the signature \sigma(M) is different from zero (\sigma(M) =/= 0) and (M, g) is real analytic then (M, g, J) is a Kähler-Einstein manifold.

Institute of Mathematics

Cracow University of Technology

Warszawska 24

31-155 Krak/ow, POLAND.

Date received: May 22, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-51.