Differential Function Fields and Moduli of Algebraic Varieties
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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
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© 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
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