Gravitational instantons and the topology of 4-dimensional manifolds

设M是紧化的Kaehler流形，是Kaehler形式R是Ricci形式。根据定义，M是一个爱因斯坦-凯勒流形，当R()=k时，对于某个常数k。一个k=O的爱因斯坦-凯勒流形M称为引力瞬子。（当M是开的，在无穷远处假定一些边界条件。）Akito Futaki教授发现了一个新的不变量f，它将爱因斯坦-凯勒度量的存在性与m的拓扑结构联系起来。设H(M)是由M的所有全纯自同构和H(M)组成的李群，H(M)是由M上的所有全纯向量场组成的H(M)的李代数，对于X<不是> H(M)的成员，f(X)由X的散度乘以R（）的第M个外积（其中M是M的复维数）的积分定义，因此是一个从H(M)到复数的平凡李代数的李代数同态。Futaki教授证明了f不依赖于Kaehler形式的选择，f是爱因斯坦-Kaehler度量存在的障碍。在我们的论文中，A。关于积分不变量Lefschetz数和归纳映射的若干问题，数学学报。Vol.11 No.2 (1988), pp 289-302”，我们将f与某椭圆复合体联系起来，阐明了f成为不依赖于Kaeliler形式选择的李群同态的机理。在我们的论文中，A。李建平，紧致复流形的自同构与不变量，第2期。在纯数学。Vol.19(1989), pp 1-20”，我们将F（其中F是F到H(M)的升力）与一定的eta不变量联系起来，得到了F的计算公式。利用这个公式，我们试图构造以下Calabl推测的一个反例：“如果C_1 (M) > 0且H(M) = {0}， Kaehler流形M承认爱因斯坦-Kaehler度规。“但到目前为止，我们还没有成功地构建出这个例子。

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.