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Gromov-Hausdorff limits of Kähler manifolds and the finite generation conjecture. (English) Zbl 1380.32025
Let $$(X,g)$$ be a complete noncompact Kähler manifold with nonnegative bisectional curvature. A holomorphic function $$f:X\to\mathbb{C}$$ is said to have polynomial growth of “degree” $$d\in\mathbb{R}_{\geq 0}$$ if there exist $$x_0\in X$$, $$C>0$$ such that $$|f(x)|\leq C(1+d_g(x_0,x))^d$$, for all $$x\in X$$. The union of all holomorphic functions on $$X$$ with polynomial growth forms a ring, and the main result of this paper, which resolves a conjecture of G. Tian and S. T. Yau [Adv. Ser. Math. Phys. 1, 574–629 (1987; Zbl 0682.53064)], is that this ring is a finitely generated $$\mathbb{C}$$-algebra. As a corollary, the author proves that if $$X$$ furthermore has maximal volume growth, then there exists a nonconstant holomorphic function with polynomial growth on $$X$$ (thus resolving a conjecture of Ni), and furthermore $$X$$ is biholomorphic to an affine algebraic variety. This is consistent with another conjecture of Yau, which states that a complete noncompact Kähler manifold with positive bisectional curvature should be biholomorphic to Euclidean space.

##### MSC:
 32Q15 Kähler manifolds 53C55 Global differential geometry of Hermitian and Kählerian manifolds
Zbl 0682.53064
Full Text:
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