On Families of Projective Manifolds

Professor Kang Zuo

The Chinese University of Hong Kong, Hong Kong


In this lecture I shall give a general overview on my recent joint work with Eckart Viehweg on the analogous questions of the following Shafarevich's conjecture for families of curves, solved by A. Parshin and A. Arakelov.

Let Y denote a projective curve, and let S Y be a finite set of points. Then it is true:

(I) There are only finitely many isomorphism classes of smooth non-isotrivial families of curves over Y\S.

(II) If 2g(Y) 2 + # S 0 then there are no such families.

(I) and (II) essentially follow from the observation, that if f: X Z is a family of semi-stable curves over a projective manifold Z which is smooth over Z \ S and is non-constant in any direction, then the sheaf Image20 of the differential 1-forms with logarithmic poles along S is ample with respect to a Zariski open subset.

For the higher dimensional case one of the main difficulties is that a priori there is no Torelli theorem and that one has to consider iterated Kodaira-Spencer maps. To overcome this difficulty we proceed follows. Using Viehweg's theorem on the positivity of direct image sheaves and taking suitably branched coverings of the original family one gets a Variation of Hodge Structures (VHS) with the property that the Kodaira-Spencer class of the original family still operates on a certain non-trivial part of the Hodge bundles corresponding to this VHS. Consequently, applying the semi-negativity of the kernel of the Kodaira-Spencer map we obtain a non-trivial map from an ample sheaf into some symmetric power of Image21 (up to a finite covering). II) and the boundedness part in (I) follow from the existence of this map and Viehweg's theorem on the explicit positivity of direct image sheaves.

Many interesting problems related to I) and II) in the higher dimensional case are still left unsolved. For examples: the rigidity problem on a family of projective manifolds f: X Y the positivity properties of Image21 and the complex hyperbolicity for a higher dimensional base manifold Y\ S.

We hope through a better understanding on the geometric meaning of this VHS and its iterated Kodaira-Spencer map to get some new approachs to these open problems.


October 20, 2000 (Friday) 


4:00 - 5:00pm


Room 517, Meng Wah Complex


All are welcome