Differential Function Fields and Moduli of Algebraic Varieties

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1226

Alexandru Buium

Differential Function Fields and Moduli of Algebraic Varieties

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo

Author

Alexandru Buium Department of Mathematics, National Institute for Scientific and Technical Creation B-dul Pacii 220, 79622 Bucharest, Romania

Mathematics Subject Classification (1980): 13N05, 14020, 14L30 ISBN 3-540-17194-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-17194-0 Springer-Verlag New York Berlin Heidelberg

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.

© Springer-Verlag Berlin Heidelberg 1986 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210

INTROI1UCTION,

Our background consists of two theories each having quite classical roots namely: A) The theory of algebraic differential equations (AOE's) with no movable singularity and 8) The Galois theory of ADE's, The first theory was initiated by Fuchs, PoincarJ, Painleve and has been given modern treatments through the work of several people (for a foliation-theoretic approach see Gerard-Sec end Jouanolou

[Jl ]

[GS1

while for a differential algebraic approach in

the one dimensional case see t-1atsuda fr'ltdJ> , The second theory goes back to Picard and Vessiot and reached a very elegant and general form through the work of Kolchin (Koln] 1

n .. 3,

The primary goal of this research monograph is to relate the two theories above; this will turn out to be profitable for both of them, To establish the link between A) and B) the first step is to develop a higher dimensional differential algebraic version of A), None of the methods used in purpose:

[GS] and

[Jl ]

f9s], [J11, [:1tdJ

seems suitable for thiS

are too "an aLv t Lc " while [MtdJ is too related

to the one-dimensional case, Our approach will be quite different end will lead us beyond our "primary goal", to what we called a "differential descent theory", This theory has an interest in itself and should be viewed flS an "infinitesimal" analog of Shimura-Matsusaka theory of fields of moduli [Sh 21,[Mtk], Our proofs in this step will be combinfltions of moduli-theoretic methods (deformations of polarized algebraiC varieties and compact analytic spaces) and differential algebraic methods (logarithmic derivatives on algebraiC

IV

groups). The second step in our approach will be Galois-theoretic. We shall use results proved in the first step plus Kolchin's differential Galois theory to describe in detail the

R).

and

between A)

Proofs will also involve an analysis of

K'/K-forms of quasi-

homogenous projective varieties and some geometry of automorphisms of surfaces and abelian varieties. The book is organ