Gravitational instantons and the topology of 4-dimensional manifolds

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Let M be a compact Kaehier manifold, omega the Kaehler form and R(omega) the Ricci form of omega. By the definition, M is an Einstein-Kaehler manifold iff R(omega)=k omega for some constant k. An Einstein-Kaehler manifold M with k=O is called a gravitational instantons. (When M is open, some boundary conditions at infinity is assumed.) Professor Akito Futaki discovered a new invariant f which relates the existence of Einstein-Kaehler metrics with the topology of M. f is defined as follows. Let H(M) be the Lie group which consists of all holomorphic automorphisms of M and h(M) the Lie algebra of H(M) which consists of all holomorphic vector fields on M. For X<not a member of> h(M), f(X) is defined by the integration of the divergence of X multiplied by the m-th exterior product (Where m is the complex dimension of M.) of R(omega) and thus a Lie algebra homomorphism from h(M) to the trivial Lie algebra of complex numbers. Prof. Futaki proved that f does not depend on the choice of Kaehler forms omega and that f is an obstruction to the existence of Einstein-Kaehler metrics. In our paper 「A.Futaki and K.Tsuboi, On some integral invariants Lefschetz numbers and induction maps, Tokyo J. Math. Vol.11 No.2 (1988), pp 289-302」 , we related f with a certain elliptic complex and clarified the mechanism of that f becomes a Lie group homomorphism which does not depend on the choice of Kaeliler forms omega. And in our paper 「A.Futaki and K.Tsuboi, Eta invariants and automorphisms of compact complex manifolds, Adv. Stud. in Pure Math. Vol.19(1989), pp 1-20」 , we related F (where F is the lift of f to H(M).) with a certain eta invariants and obtained a calculation formula of F. Using this formula, we tried to construct a counter-example of the following Calabl's conjecture: 「A Kaehler manifold M admits an Einsteill-Kaehler metric if C_1 (M) > 0 and h(M)= {0}.」 But so far we have not yet succeeded in constructing the example.