COBORDISM, A-GENUS AND EINSTEIN-K•HLER METRICS Bang-Yen Chen*

1. Topological obstruction. Let M n be an n-dimensional complex manifold (all complex manifolds are assumed to be compact, connected and of complex dimensions). We denote by c i and Pi the i-th Chern class and the i-th Pontryagin class, respectively. The Euler characteristic x(Mn) of M n is given by x(M n) = Cn[Mn]. The Bernoulli polynomial B•n)(zl,...,Zn) is defined by zit oo tJ II•=1 exp(zit) _ 1 = 2;j = 0 •B•n)(zl,...,Zn). If the c i are regarded as the elementary symmetric functions in z 1,...,zn,... the Todd genus T(M n) of M n is given by [6] T(M n) = [•-.)nB•)(z 1.....Zn)[Mnl ß The Atiyah-Singer Index Theorem [8] says that the Todd genus is a birational invariant and it is equal to the arithmetic genus of M n. If n is even, the A-genus of M n is defined by [6] A(M n) = FIj = 1 sinh 2V, X-J- [Mn], where the Pi are regarded as the i-th elementary symmetric functions in x 1,x2,'",xj,'". In [1,2,3,4,7], some topological conditions are known for a 2 or 3-dimensional complex manifolds to admit an Einstein-Kahler metric. Topological conditions for a 4-dimensional complex manifold to admit an Einstein-Kahler metric seem to be unknown. In õ2, we shall prove the following theorem which gives a simple topological obstruction to Einstein-Kiihler metric