PAUL EHRLICH

L O C A L CONVEX D E F O R M A T I O N S ,

H E R M I T I A N M E T R I C S ,

AND H E R M I T I A N C O N N E C T I O N S *

The purpose of this appendix to [2] is to discuss the non-extension of the Riemannian Ricci curvature deformation theorems in [2] to Hermitian connections for Hermitian metrics. However the Riemannian results of [2] do extend to the Levi-Civita connection of a Hermitian metric. To avoid confusion, we first recall some standard definitions and facts.

Let (M, J) be a fixed complex manifold with a fixed complex structure o r. Let g be a Hermitian metric for (M, J), i.e., g(Jx, Jy)=g(x, y) for all x, y~TM. Since g is in particu!ar a Riemannian metric for M, g determines a unique Levi-Civita connection D with Dg=O and Tor(D)=0. (Here Tor(D)(X, Y ) = D x Y - D r X - [ X , Y] for X, Y vector fields on M.) We denote by ric(g) and sc(g) respectively the Ricci curvature and scalar curvature determined by D and g. The fundamental 2-form ~b(g) of the Hermitian manifold (M, J, g) is given by q~(g) (x, y): =g(Jx, y). Ifd~(g) =0, then (M, or, g) is called Kfihler. It is well known that DJ=O iff (M, J, g) is K~ihler. We note also the elementary

FACT 1. (M, or, g) K~hler, gl =Jg . Then (M, J, g~) is Kfihler i f f f is a con- stant function.

Let (M, J, g) be Hermitian. For the purpose of Hermitian (but non- K/ihler) geometry, the Levi-Civita connection D of g and the curvature tensors determined by D and g are not the right objects to utilize because DJ¢ O. However it is possible to define a unique connection V for (M, or, g) called the Hermitian connection of (M, J, g) which satisfies V g = V J = 0 and Tot (V)(Jx, y)= Tot (V)(x, Jy) for all x, y eTM. It is known that V is torsion free iff (3/, or, g) is K/ihler iff D =V. We will use the notation S(g) for the Ricci tensor determined by g and V, and R(g) for the scalar curvature determined by g and V respectively.

If go is a Hermitian metric for (M, J), then any metric conformal to go is a Hermitian metric for (M, J). Thus using the method of local convex deformations from [2], we have for the Ricci curvature of the Levi-Civita connection for (M, J, go).

* This work was done under the program of the Sonderforschungsbereich 'Theoretische Mathematik' at the University of Bonn.

Geometriae Dedicata 5 (1976) 27-29. All Rights Reserved Copyright © 1976 by D. Reidel Publishing Company, Dordrecht-Holland

2 8 P A U L E H R L I C H

THEOREM 1. Let (M", J, go), n/> 2, be a compact Hermitian manifoM with ric(go)~>0 (resp. ric(go)~<0) and all Rieei curvatures at some point positive (resp. negative). Then (M, J) admits a Hermitian metric g with ric(g)>0 (resp. ric(g) < 0) everywhere.

Note. If (M", J, go) is K/ihler and S(go)>~O, S(go)>0 at some point, Theorem 1 produces a Hermitian metric g for (M, J) with tic(g)>0 every- where. (Remember S(go)=ric(go) since go is Kiihler.) However, by Fact 1, g will never be Kiihler so S(g)#ric(g) in general and we will not in general have S(g) > 0. This is important in view of Theorem 4 below. Since (M, J, g) is only Hermitian and not Kahler, rio(g)>0 implies nothing about the first Chern class cl(M) of (M, J). The Ricci form determined by S(g), not rio(g) represents the first Chern class in the de Rham cohomology ring in the Hermitian (non-K~hler) case.

We have also as in [3] using local convex deformations

THEOREM 2. Let (M", J, go), n/> 2, be a compact Hermitian manifold with sc (go) ~> 0 and sc (go) (P) > 0 (resp. sc (go) ~< 0 and sc (go) (P) < 0)for some p ~ M. Then (M, J) admits a Hermitian metric g with sc(g)>0 (resp. sc(g)<0) everywhere.

Note. The same caveat as for Thin. 1 applies here. We note that Melvin

has shown by the calculus of variations that if ~ R(go)<<. O, then Berger, [1 ], J M

(M, J) admits a Hermitian metric obtained from go by a conformal change of metric with R(g)< 0 everywhere.

As in [3], Thin. 4, we obtain from Thm. 2 of this appendix and work of Bourguignon cited in [3].

THEOREM 3. Let (M", J, go) be Kiihler with R(go)= sc(go)/> 0. Suppose the ricei tensor of go does not vanish identically. Then (M, J) admits a Hermitian metric g with sc(g) > 0 everywhere.

Proof In view of [3], we need only remark that (M, go) K/~hler implies tic (go) (Jx, Jy) = tic (go) (x, y) so that g(t): = go - t ric (go) is a deformation of go through Hermitian metrics. Q.E.D.

Now we remark that the analogue of Theorem 1 is in general false for the Ricci curvature S(go) of the Hermitian connection determined by (M, J, go). In [4], p. •22, S.T. Yau has constructed a K~ihler metric go for a Hirzebruch surface 272 (a real 4-dimensional manifold) with S(go)=ric(go)>~O. It is a standard fact that

FACT 2. Since (c1(X2))2~0, there is some point p~Z2 with all go-Ricci curvatures at p positive.

LOCAL CONVEX DEFORMATIONS 29

However, X2 admits no Hermitian metric g with S(g) > 0 everywhere because there is a holomology class, y~H2 (X2:Z) with c1(2~2) [y] = 0, (see [4], p. 222).

Also T. Ochiai has informed us that direct calculations show that ric (go) > 0 offy.

For emphasis we summarize the above remarks as

T H E O R E M 4 (T.Ochiai, S.T.Yau). There exists a compact, simply con- nected Kdhler manifoM (M, J, go) with S(go) = rio(go) 1> 0 and all Ricci curva- tures at some point positive such that no Hermitian metric g for (M, J) can have S(g) > 0 everywhere.

ACKNOWLEDGEMENTS

We would like to thank T. Ochiai (Bonn) and B. Smyth (Bonn) for several conversations on complex geometry and S. T. Yau for calling our attention to his example in [4].

BIBLIOGRAPHY

1. Melvin Berger, 'Constant Scalar Curvature Metrics for Complex Manifolds', to appear in the Proceedings of the Amer. Math. Soe. Summer Institute in Differential Geometry, Stanford, 1973.

2. Ehrlich, P.E., 'Metric Deformations and Curvature I: Local Convex Deformations', this issue of Geometriae Dedicata.

3. Ehrlich, P.E., 'Deformations of Scalar Curvature', submitted to Geometriae Dedicata. 4. Yau, S.T., 'On the Curvature of Compact Hermitian Manifolds', Inventiones Math.

25 (1974), 213-239.

Author's address:

Paul E. Ehrlich, Mathematisches Institut der Universit/it Bonn, 53 Bonn, WegelerstraBe 10, Federal Republic of Germany

(Received February 1, 1975)