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Positivity of direct image sheaves and applications to families of higher dimensional manifolds. (English) Zbl 1092.14044

Demailly, J.P. (ed.) et al., School on vanishing theorems and effective results in algebraic geometry. Lecture notes of the school held in Trieste, Italy, April 25–May 12, 2000. Trieste: The Abdus Salam International Centre for Theoretical Physics (ISBN 92-95003-09-8/pbk). ICTP Lect. Notes 6, 249-284 (2001).
The paper contains a collection of lectures on the theory of families of projective varieties. More precisely, the author explains how one can use positivity results on the direct image of the relative dualizing sheaf, to derive information on non iso-trivial families.
Using such positivity results, Arakelov and Parshin proved that one has only a finite number (up to isomorphisms) of non–isotrivial families of curves \(f:X\to Y\setminus S\) over a fixed curve \(Y\) of minus a finite set \(S\), and none of them exists when \(2g(Y)-2+\#S\leq 0\).
For some families of higher dimensional varieties, one can use similar methods to obtain upper bounds for \(\det(f_*\omega_{X/Y})\). When a good compactification of the moduli space exists, this in turn implies that the induced maps \(Y\setminus S\to M\) are parameterized by some scheme of finite type.
The author presents here, in a simplified way, the main sheaf–theoretical background and the actual achievements of the theory.
For the entire collection see [Zbl 0986.00053].

MSC:

14J10 Families, moduli, classification: algebraic theory
14H10 Families, moduli of curves (algebraic)
14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
14F17 Vanishing theorems in algebraic geometry
14C20 Divisors, linear systems, invertible sheaves
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