# zbMATH — the first resource for mathematics

On Ricci flat 3-fold. (English) Zbl 0649.14024
The authors study a particular class of algebraic 3-folds with trivial canonical bundle, obtained as follows. Let G be a cyclic group of order d of automorphisms of a 3-dimensional torus T, such that 0 is an isolated fixed point of G, G acts freely near 0 and $$\det (dg)_ 0=1$$ for all $$g\in G$$. In this case the authors show that the minimal desingularization $$T^{\wedge}/G$$ of T/G is a projective 3-fold with trivial canonical bundle, moreover they give a complete classification of such T/G’s proving that they can be obtained as quotients from three standard examples.
The proof is based on an explicit construction of a minimal desingularization of $${\mathbb{C}}$$ 3/G (where $$G=<\psi >$$, $$\psi$$ linear transformation of order d) by the method of the toroidal embeddings.
Reviewer: L.Picco Botta

##### MSC:
 14J30 $$3$$-folds 14L30 Group actions on varieties or schemes (quotients) 14E15 Global theory and resolution of singularities (algebro-geometric aspects)
Full Text:
##### References:
 [1] Delique, P., Equations Differentielle á points singuliers réguliers, Lecture Notes in Math., 1963, Springer-Verlag, 1970. [2] Hironaka H.: Resolution of singularities of an algebraic variety over a field of characteristic zero,Ann. of Math.,79 (1964), 109–326. · Zbl 0122.38603 · doi:10.2307/1970486 [3] Kempt, G., Knudsen, F., Mumford, D., Saint-Donat, B., Toroidal embedding 1, Springer-Verlag Lecture Notes 339. [4] Laufer, H. B., Normal two-dimensional singularities,Ann. of Math. Studies, Princeton Univ. Press. · Zbl 0245.32005 [5] Mumford, D., Algebraic Geometry 1, complex projective varieties, Springer-Verlag. · Zbl 0356.14002 [6] Satake, I., On a generalization of the notion of manifolds,Proc. Natl. Acad. Sci. U. S. A.,42 (1956), 359–363. · Zbl 0074.18103 · doi:10.1073/pnas.42.6.359 [7] Shimura, G., Introduction to the arithmetic theory of automorphic functions, Publication of the Math. Soc. of Japan 11. · Zbl 0221.10029 [8] Spanier, E. H., Algebraic topology, McGraw Hill, 1966. · Zbl 0145.43303 [9] Yau Shing-Tung, Compact three dimensional Kähler manifolds with zero Ricci curvature. · Zbl 0643.53050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.