Curvature and $$L^p$$ L p Bergman Spaces on Complex Submanifolds in $$\pmb {{\mathbb {C}}^N}$$ C N
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Curvature and Lp Bergman Spaces on Complex Submanifolds in CN Bo-Yong Chen1 · Yuanpu Xiong1 Received: 19 February 2020 / Accepted: 24 September 2020 © Mathematica Josephina, Inc. 2020
Abstract Let M be a closed complex submanifold in C N with the complete Kähler metric induced by the Euclidean metric. Several finiteness theorems on the L p Bergman space of holomorphic sections of a given Hermitian line bundle L over M and the associated L 2 cohomology groups are obtained. Some infiniteness theorems are also given in order to test the accuracy of finiteness theorems. As applications we obtain some rigidity results concerning growth of curvatures. Keywords L p Bergman space · L 2 -cohomology group · Curvature · Complex submanifold Mathematics Subject Classification 32Q15 · 32Q28
1 Introduction The interest of closed complex submanifolds in C N is twofold. The first is that these complex manifolds are exactly all Stein manifolds—a central subject in several complex variables; the second is that they are minimal submanifolds in Euclidean spaces which enjoy a rich geometry. In what follows we shall denote by M a closed complex submanifold of dimension n in C N and g the restriction of the Euclidean metric on M. A basic question is Problem 1 What is the relation between function theory and geometry of M?
Supported by NSF grant 11771089 and Gaofeng grant from School of Mathematical Sciences, Fudan University.
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Yuanpu Xiong [email protected] Bo-Yong Chen [email protected]
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Department of Mathematical Sciences, Fudan University, Shanghai 200433, China
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B.-Y. Chen, Y. Xiong
A beautiful result in this direction is the cerebrated result of Stoll [21] that M is affine-algebraic if and only if |M(x, r )| = O(r 2n ) for some (and any) x, where M(x, r ) is the intersection of M with Euclidean ball B(x, r ) ⊂ C N , and | · | means the volume with respect to g. It is natural to expect to characterize affine-algebraicity through growth of curvature. However, Vitter [22] found examples of smooth affinealgebraic varieties that some of them have Ricci curvature decaying faster than quadradically while others do not. On the other hand, Cornalba–Griffiths [6] characterized the property that a Hermitian vector bundle over a smooth affine-algebraic variety has an algebraic structure through growth of curvature. The goal of this paper is to give some finiteness theorems on the L p Bergman space and L 2 cohomology groups associated to a Hermintian line bundle over a closed complex submanifold in C N through growth of curvature or affine-algebraicity. We remark that a large literature exists for vanishing theorems and finiteness theorems for holomorphic sections of vector bundles and high cohomology groups on noncompact complete Kähler manifolds (see, e.g., [1,7,15,16], or [17], Chapter 2, § 3). Definition 1 For a Hermitian line bundle (L, h) over M we define the L p Bergman 0 space pHL p (M, L) to be the set of holomorphic sections s of L over M satisfying M |s|h dVg < ∞. Here | · |h is the point-wise norm with respect
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