## On Grauert’s examples of complete Kähler metrics

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- by Sahil Gehlawat and Kaushal Verma PDF
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## Abstract:

Grauert showed that the existence of a complete Kähler metric does not characterize domains of holomorphy by constructing such metrics on the complements of complex analytic sets in a domain of holomorphy. In this note, we study the holomorphic sectional curvatures of such metrics in two prototype cases namely, $\mathbb {C}^n \setminus \{0\}, n \ge 2$ and $\mathbb {B}^N \setminus A$, $N \ge 2$ and $A \subset \mathbb {B}^N$ is a hyperplane of codimension at least two. This is done by computing the Gaussian curvature of the restriction of these metrics to the leaves of a suitable holomorphic foliation in these two examples. We also examine this metric on the punctured plane $\mathbb {C}^{\ast }$ and show that it behaves very differently in this case.## References

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## Additional Information

**Sahil Gehlawat**- Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India
- ORCID: 0000-0002-7321-7820
- Email: sahilg@iisc.ac.in
**Kaushal Verma**- Affiliation: Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India
- MR Author ID: 650937
- Email: kverma@iisc.ac.in
- Received by editor(s): August 6, 2020
- Received by editor(s) in revised form: July 22, 2021
- Published electronically: April 14, 2022
- Additional Notes: The first author was supported by the CSIR SPM Ph.D. fellowship.
- Communicated by: Harold P. Boas
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**150**(2022), 2925-2936 - MSC (2020): Primary 32Q05, 32Q10; Secondary 32Q02
- DOI: https://doi.org/10.1090/proc/15795
- MathSciNet review: 4428878