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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Quotients by $ {\bf C}\sp\ast \times{\bf C}\sp\ast$ actions


Authors: Andrzej Białynicki-Birula and Andrew John Sommese
Journal: Trans. Amer. Math. Soc. 289 (1985), 519-543
MSC: Primary 32M05; Secondary 14L30
MathSciNet review: 784002
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Abstract: Let $ T \approx {{\mathbf{C}}^\ast} \times {{\mathbf{C}}^\ast}$ act meromorphically on a compact Kähler manifold $ X$, e.g. algebraically on a projective manifold. The following is a basic question from geometric invariant theory whose answer is unknown even if $ X$ is projective.

PROBLEM. Classify all $ T$-invariant open subsets $ U$ of $ X$ such that the geometric quotient $ U \to U/T$ exists with $ U/T$ a compact complex space (necessarily algebraic if $ X$ is).

In this paper a simple to state and use solution to this problem is given. The classification of $ U$ is reduced to finite combinatorics. Associated to the $ T$ action on $ X$ is a certain finite $ 2$-complex $ \mathcal{C}(X)$. Certain $ \{ 0,1\} $ valued functions, called moment measures, are defined in the set of $ 2$-cells of $ \mathcal{C}(X)$. There is a natural one-to-one correspondence between the $ U$ with compact quotients, $ U/T$, and the moment measures.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1985-0784002-8
PII: S 0002-9947(1985)0784002-8
Article copyright: © Copyright 1985 American Mathematical Society