Journal of the Mathematical Society of Japan

Gluing construction of compact complex surfaces with trivial canonical bundle

Mamoru DOI

Full-text: Open access

Abstract

We obtain a new construction of compact complex surfaces with trivial canonical bundle. In our construction we glue together two compact complex surfaces with an anticanonical divisor under suitable conditions. Then we show that the resulting compact manifold admits a complex structure with trivial canonical bundle by solving an elliptic partial differential equation. We generalize this result to cases where we have other than two components to glue together. With this generalization, we construct examples of complex tori, Kodaira surfaces and K3 surfaces. Lastly we deal with the smoothing problem of a normal crossing complex surface $X$ with at most double curves. We prove that we still have a family of smoothings of $X$ in a weak sense even when $X$ is not Kählerian or $H^{1}(X,\mathcal{O}_{X})\neq 0$, in which cases the smoothability result of Friedman [Fr] is not applicable.

Article information

Source
J. Math. Soc. Japan Volume 61, Number 3 (2009), 853-884.

Dates
First available: 30 July 2009

Permanent link to this document
http://projecteuclid.org/euclid.jmsj/1248961480

Digital Object Identifier
doi:10.2969/jmsj/06130853

Zentralblatt MATH identifier
05603964

Mathematical Reviews number (MathSciNet)
MR2552917

Subjects
Primary: 58J37: Perturbations; asymptotics
Secondary: 14J28: $K3$ surfaces and Enriques surfaces 32J15: Compact surfaces 53C56: Other complex differential geometry [See also 32Cxx]

Keywords
gluing complex surfaces with trivial canonical bundle smoothing

Citation

DOI, Mamoru. Gluing construction of compact complex surfaces with trivial canonical bundle. Journal of the Mathematical Society of Japan 61 (2009), no. 3, 853--884. doi:10.2969/jmsj/06130853. http://projecteuclid.org/euclid.jmsj/1248961480.


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