The Michigan Mathematical Journal

Topology of Kähler Manifolds with Weakly Pseudoconvex Boundary

Brian Weber

Abstract

We study Kähler manifolds-with-boundary, not necessarily compact, with weakly pseudoconvex boundary, each component of which is compact. If such a manifold $K$ has $l\ge 2$ boundary components (possibly $l=\infty$), then it has the first Betti number at least $l-1$, and the Levi form of any boundary component is zero. If $K$ has $l\ge 1$ pseudoconvex boundary components and at least one nonparabolic end, then the first Betti number of $K$ is at least $l$. In either case, any boundary component has a nonvanishing first Betti number. If $K$ has one pseudoconvex boundary component with vanishing first Betti number, then the first Betti number of $K$ is also zero. Especially significant are applications to Kähler ALE manifolds and to Kähler 4-manifolds. This significantly extends prior results in this direction (e.g., those of Kohn and Rossi) and uses substantially simpler methods.

Article information

Source
Michigan Math. J., Advance publication (2019), 16 pages.

Dates
Revised: 4 April 2019
First available in Project Euclid: 23 July 2019

https://projecteuclid.org/euclid.mmj/1563847454

Digital Object Identifier
doi:10.1307/mmj/1563847454

Subjects
Primary: 53C55, 53C23, 31C12, 31C10, 58J99

Citation

Weber, Brian. Topology of Kähler Manifolds with Weakly Pseudoconvex Boundary. Michigan Math. J., advance publication, 23 July 2019. doi:10.1307/mmj/1563847454. https://projecteuclid.org/euclid.mmj/1563847454

References

• [1] A. Andreotti and E. Vesentini, Carleman estimates for the Laplace–Beltrami equation in complex manifolds, Publ. Math. Inst. Hautes Études Sci. 25 (1965), no. 1, 81–130.
• [2] H. Grauert, On Levi’s problem and the imbedding of real-analytic manifolds, Ann. of Math. 68 (1958), no. 2, 460–472.
• [3] I. Holopainen, Nonlinear potential theory and quasilinear mappings on Riemannian manifolds, Ann. Acad. Sci. Fenn. Math. Diss. 74 (1990), 1–45.
• [4] I. Holopainen and P. Koskela, Volume growth and parabolicity, Proc. Amer. Math. Soc. 129 (2001), no. 11, 3425–3435.
• [5] L. Hörmander, $L^{2}$ estimates and existence theorems for the $\bar{\partial }$ operator, Acta Math. 113 (1965), 89–152.
• [6] J. J. Kohn, Harmonic integrals on strongly pseudo-convex manifolds, I, Ann. of Math. (2) 78 (1963), no. 1, 112–148.
• [7] J. J. Kohn, Harmonic integrals on strongly pseudo-convex manifolds, II, Ann. of Math. 79 (1964), no. 3, 450–472.
• [8] J. J. Kohn and H. Rossi, On the extension of holomorphic functions from the boundary of a complex manifold, Ann. of Math. (2) 81 (1965), no. 2, 451–472.
• [9] P. Li, On the structure of complete Kähler manifolds with nonnegative curvature near infinity, Invent. Math. 99 (1990), 576–600.
• [10] P. Li and L. Tam, Positive harmonic functions on complete manifolds with non-negative curvature outside a compact set, Ann. of Math. 125 (1987), 171–207.
• [11] P. Li and L. Tam, Harmonic functions and the structure of complete manifolds, J. Differential Geom. 35 (1992), 359–383.
• [12] P. Li and L. Tam, Green’s functions, harmonic functions, and volume comparison, J. Differential Geom. 41 (1995), 277–318.
• [13] T. Ohsawa, On complete Kähler domains with $C^{1}$-boundary, Publ. Res. Inst. Math. Sci. 16 (1980), 929–940.