Very ampleness of the bicanonical line bundle on compact complex 2-ball quotients.(English)Zbl 1408.14058

The author studies the bicanonical map of a complex 2-ball quotient, a smooth compact complex surface of the form $$M=B^2_{\mathbb C}/\Pi$$ where $$B^2_{\mathbb C}=\{(z_1,z_2)\in {\mathbb C}^2 | |z_i|^2+|z_2|^2<1\}$$ is the open complex ball of dimension $$2$$ and $$\Pi$$ is a cocompact torsion free lattice in $$PU(2,1)$$.
This included the fake projective planes, the 50 pairs of conjugated complex manifolds with the same Betti numbers of the complex projective plane not isomorphic to it, classified by D. I. Cartwright and T. Steger [C. R., Math., Acad. Sci. Paris 348, No. 1–2, 11–13 (2010; Zbl 1180.14039)]. Among the fake projective planes, four pairs are said to be of minimal type, as the latice involved is not contained properly as a sub-lattice in another lattice of $$PU(2,1)$$.
The author claims to have proven the following:
Theorem. The bicanonical line bundle $$2K_M$$ is very ample for any smooth compact complex 2-ball quotient $$M$$ with $$c_2(M)=3$$, a part from fake projective planes of minimal type.
Unfortunately, the proof is not correct. Several gaps in the proof have been pointed out by F. Catanese and Y. Keum in the last section of their paper [“The bicanonical map of fake projective planes with an automorphism”, Int. Math. Res. Not. (to appear), doi:10.1093/imrn/rny214 ].

MSC:

 14E25 Embeddings in algebraic geometry 14J29 Surfaces of general type

Keywords:

ball quotients; embedding

Zbl 1180.14039
Full Text:

References:

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