Gravitational instantons and the topology of 4-dimensional manifolds

设M是紧致Kaehier流形，omega是Kaehler形式，R(Omega)是omega的Ricci形式。根据这个定义，M是Einstein-Kaehler流形当且仅当R(Omega)=k omega，对于某个常数k，M称为引力瞬子。(当M是开的时，假定在无穷远处有一些边界条件。)Futaki教授发现了一个新的不变量f，它将Einstein-Kaehler度规的存在与M.f的拓扑联系起来，定义如下。设H(M)是由M的所有全纯自同构组成的李群，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&lt；不是&gt；h(M)的成员，f(X)定义为X的散度乘以第m次外积(其中m是M的复数维)的积分。以及从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omega的选择，并且f是爱因斯坦-Kaehler度规存在的障碍。在我们的论文《A.Futaki和K.Tsuboi，关于一些积分不变量，Lefschetz数和诱导映射》中，Tokyo J.Math.Vol.11 No.2(1988)，pp289-302“，我们将f与一类椭圆复形联系起来，阐明了f成为李群同态的机制，它不依赖于Kaeliler型omega的选择。在我们的论文《A.Futaki和K.Tsuboi，紧致复流形的Eta不变量和自同构》中，Adv.Stud.在纯数学中。卷19()，第1-20页“，我们把F(其中F是f到H(M)的升力)联系起来。利用一定的ETA不变量，得到了F的一个计算公式.利用这个公式，我们尝试构造了Calabl猜想的一个反例：“Kaehler流形M当C_1(M)&gt；0且h(M)={0}时，M允许爱因斯坦-Kaehler度量.”但到目前为止，我们还没有成功地构造出这个例子.

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.