### Gluing construction of compact complex surfaces with trivial canonical bundle

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

#### 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.

First Page:
Primary Subjects: 58J37
Secondary Subjects: 14J28, 32J15, 53C56
Full-text: Open access

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

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