|
I will report on joint work with M. Franciosi, and the characterizations we gave of surfaces
whose universal cover is a product of curves.
And especially on joint work and work in progress with Antoniom Di Scala.
The work done gives the characterization of varieties whose universal cover is a polydisk, respectively a bounded symmetric domain of tube type. One defines, for a projective manifold of dimension n, a special tensor as a (non zero) section of the n-th symmetric power of the cotangent bundle twisted by the anticanonical divisor, a semispecial tensor as a (non zero) section of the n-th symmetric power of the cotangent bundle twisted by the anticanonical divisor and by a 2-torsion line bundle, and a slope zero tensor \eta as a (non zero) section of H^0 (S^{mn} (\Omega_X^1)(- mKX )). Theorem 1. The universal cover of X is the polydisk iff X has ample canonical bundle and admits a semispecial tensor such that at some point p the corresponding tangential hypersurface is reduced. Theorem 2. If X has ample canonical bundle, then it admits a semispecial tensor iff the universal cover of X is a special bounded symmetric domain D of tube type. The domain D is determined by the multiplicities of the irreducible components of the corresponding tangential hypersurface. Theorem 3. If X has ample canonical bundle, then it admits a slope zero tensor iff the universal cover of X is a bounded symmetric domain D of tube type. The domain D is determined by the multiplicities of the irreducible components of the corresponding tangential hypersurface. We have then a corollary which extends previous results by Kazhdan. Corollary. Assume that the universal covering of X is a bounded symmetric domain D of tube type. Let X^s be a Galois conjugate of X. Then also the universal cover of X^s is biholomorphic to D. I will also give a general introduction concerning the current knowledge on projective varieties whose universal cover is a bounded domain in C^n. I will in the end explain the general plan we have for the case of branched coverings and logarithmic bundles. |