Gravitational instantons and the topology of 4-dimensional manifolds

引力瞬子和 4 维流形的拓扑

• 批准号: 62540024
• 负责人: TSUBOI Kenji
• 金额: $ 1.47万
• 依托单位: Tokyo Univ. of Fisheries
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for General Scientific Research (C)
• 财政年份: 1987
• 资助国家: 日本
• 起止时间: 1987 至 1989
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-62540024/
• 关键词: Einstein-Kaehler manifold; Gravitational instanton; Futaki invariant; Elliptic complex; Eta invariant; Calabi's conjecture; 4次元多様体; 複多面体群; Self-intersection form; momentmap; eta不変量; Einstein-Kahler計量; 動力インスタントン; self-intersection form; momen

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.

设 M 为紧凯希尔流形，omega 为凯勒形式，R(omega) 为 omega 的 Ricci 形式。根据定义，M 是一个爱因斯坦-凯勒流形当且仅当 R(omega)=k omega 对于某个常数 k。 k=O 的爱因斯坦-凯勒流形 M 称为引力瞬时子。 （当 M 开时，假定无穷远的一些边界条件。）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 的拓扑联系起来。f 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<不是成员> h(M)，f(X) 的定义是 X 的散度乘以第 m 个外积（其中 m 是 M 的复数维数）的积分。 R(omega) 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不依赖于凯勒形式欧米伽的选择，并且f是爱因斯坦-凯勒度量存在的障碍。在我们的论文“A.Futaki 和 K.Tsuboi，关于一些积分不变量 Lefschetz 数和归纳图”中，Tokyo J. Math。 Vol.11 No.2 (1988), pp 289-302」，我们将f与某个椭圆复形联系起来，阐明了f成为不依赖于Kaeliler形式omega选择的李群同态的机制。在我们的论文“A.Futaki 和 K.Tsuboi，紧复流形的 Eta 不变量和自同构，Adv.螺柱。在纯数学中。 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猜想的反例：「如果C_1（M）> 0，则凯勒流形M承认Einsteill-Kaehler度量，并且 h(M)= {0}.” 但到目前为止我们还没有成功构建这个例子。

项目成果

Akito Futaki and Kenji Tsuboi: "On some integral invariants, Lefschetz numbers and induction maps" Tokyo J. Math., Vol. 11 No.2, PP289-302, 1988.

Akito Futaki 和 Kenji Tsuboi：“关于一些积分不变量、Lefschetz 数和归纳图”Tokyo J. Math.，Vol.1。

A.Futaki-K.Tsuboi: Tokyo J.Math.11-2. 289-302 (1988)

A.Futaki-K.Tsuboi：东京 J.Math.11-2。

A.Futaki-K.Tsuboi: Adv.in Pureand Appl.Math.

A.Futaki-K.Tsuboi：纯数学和应用数学高级。

Akito Futaki and Kenji Tauboi: "Eta invariunts and automorphisms of compact complex munifolds" Auv.Stud.in Pure Math.19. 1-20 (1989)

Akito Futaki 和 Kenji Tauboi：“紧复多形的 Eta 不变量和自同构”Auv.Stud.in Pure Math.19。

Akito Futaki and Kenji Tsuboi: "Eta invariants and automorphisms of compact complex manifolds" Adv. Stud. in Pure Math., Vol. 19, PP1-20, 1989.

Akito Futaki 和 Kenji Tsuboi：“紧复流形的 Eta 不变量和自同构”Adv.