Differential function fields and moduli of algebraic varieties / Alexandru Buium

This research monograph represents a very interesting and broad attempt to link, and simultaneously to carry on, two rather classical topics in algebraic analysis, namely (1) the Fuchs-Poincaré-Painlevé theory of algebraic differential equations without movable singularities, and (2) the Galois theory of algebraic differential equations, introduced and developed by Picard, Vessiot, and Kolchin. That attempt seems to be essentially motivated by the very fact that differential-algebraic methods have already proved advantageous, in the past, in the local and global classification theory of (polarized) algebraic varieties. Thus the author's methods in systematically unifying, generalizing, and extending different aspects in the theory of algebraic differential equations are particularly based upon algebro-geometric techniques and results (such as deformation theory of algebraic varieties and compact analytic spaces, logarithmic derivatives on algebraic groups, fields of definition of algebraic varieties, cycles and Chow schemes, and Grothendieck's theory of descent); and the applications to the classification theory of algebraic varieties, in turn, are very much in the spirit of the Shimura-Matsusaka approach via fields of moduli. ... (MathSciNet)