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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
Citations:
Zbl 0682.53064
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