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Catanese and Franciosi defined a semispecial tensor as a (non zero) sectionof the n-th symmetric power of the cotangent bundle twisted by theanticanonical divisor and by a 2-torsion line bundle. A slope zero tensor isinstead a section of the nm-th symmetric power of the cotangent bundle twistedby m times the anticanonical divisor. With these definitions we have: Theorem1. The universal cover of X is the polydisk iff 1) holds. 1) X has amplecanonical bundle and admits a semispecial tensor such that at some point p thecorresponding hypersurface in the projectivized tangent space is reduced.Theorem 2. If X has ample canonical bundle it admits a slope zero tensor if andonly if 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 componentsof the corresponding tangential hypersurface. We have then a corollary whichextends previous results by Kazhdan. Corollary. Assume that the universalcovering of X is a bounded symmetric domain D of tube type. Let X^s be a Galoisconjugate of X : then also the universal cover of X^s is biholomorphic to D.

A Characterization of Varieties whose Universal Cover is the Polydisk or a Tube Domain