Affiliation:
1. 11-9-302 Yumoto-cho , Takarazuka , Hyogo, 665-0003 , Japan
2. Department of Mathematics, Faculty of Education, Kagawa University , Saiwaicho 1-1, Takamatsu , Kagawa, 760-8522 , Japan
Abstract
AbstractLetXXbe a simple normal crossing (SNC) compact complex surface with trivial canonical bundle which includes triple intersections. We prove that ifXXisdd-semistable, then there exists a family of smoothings in a differential geometric sense. This can be interpreted as a differential geometric analogue of the smoothability results due to Friedman, Kawamata-Namikawa, Felten-Filip-Ruddat, Chan-Leung-Ma, and others in algebraic geometry. The proof is based on an explicit construction of local smoothings around the singular locus ofXX, and the first author’s existence result of holomorphic volume forms on global smoothings ofXX. In particular, these volume forms are given as solutions of a nonlinear elliptic partial differential equation. As an application, we provide several examples ofdd-semistable SNC complex surfaces with trivial canonical bundle including double curves, which are smoothable to complex tori, primary Kodaira surfaces, andK3K3surfaces. We also provide several examples of such complex surfaces including triple points, which are smoothable toK3K3surfaces.
Reference25 articles.
1. W. Barth, K. Hulek, C. Peters, and A. Van de Ven, Compact complex surfaces, Second edition, A Series of Modern Surveys in Mathematics, vol. 4, Springer-Verlag, Berlin, 2004.
2. R. Clancy, New examples of compact manifolds with holonomy Spin, Ann. Glob. Anal. Geom. 40 (2011), 203–222.
3. K. Chan, N.C. Leung, Z.N. Ma, Geometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties, J. Differential Geom. arXiv: 1902.11174v5.
4. M. Doi, Gluing construction of compact complex surfaces with trivial canonical bundle, J. Math. Soc. Japan 61 (2009), 853–884.
5. M. Doi and N. Yotsutani, Doubling construction of Calabi-Yau threefolds, New York J. Math. 20, (2014), 1203–1235.