Differential function fields and moduli of algebraic varieties
The aim of this monograph is to establish a link between the theory of algebraic differential equations (ADE) with no movable singularity and the Galois theory of ADE's. In the one-dimensional case a modern treatment of the first mentioned theory was given by \textit{M. Matsuda} [First order algebraic differential equations (Lect. Notes Math. 804) (1980; Zbl 0447.12014)], while the second mentioned theory was given its final form in papers of Kolchin [e.g. \textit{E. R. Kolchin}, ''Differential algebra and algebraic groups'' (1973; Zbl 0264.12102)]. After giving same background in chapter I, in chapter II the author generalizes most of Matsuda's results to the higher dimensional case. Let K be a field of characteristic zero, \(\Delta\) a - not necessarily finite - set of differential operators of K, \(F| K\) a function field. If \(\Delta\) is finite then \(F| K\) has a \(\Delta\)-model; in general a \(\Delta\)-function field in said to have no movable singularity if it has a projective \(\Delta\)-model. \(F| K\) is called split if \(F=K(F^{\Delta})\). Such function fields have no movable singularity. The two important resuls of chapter II are the following: Let K be an algebraically closed \(\Delta\)-field and V a projective variety over K. Then \(K^{\Delta}\) is a field of definition of V (theorem 1.1). From Kolchin's theory the author uses the concepts of weak and strong normality for \(\Delta\)-fields \(F| K:\) \(F| K\) is called weakly normal if \(F^{Gal_{\Delta}(F| K)}=K\) and strongly normal if in addition for any K-isomorphism \(\sigma\) of F into a \(\Delta\)-extension E of F both extensions \(F\cdot \sigma F/F\) and \(F\cdot \sigma F/\sigma F\) are split. Using theorem 1.1 the author shows (theorem 2.1 and corollary 2.2): If \(F| K\) has no movable singularity then there exists a \(\Delta\)-field extension \(K_ 1| K\) with \(K_ 1^{\Delta}| K^{\Delta}\) algebraic such that \(Q(F\otimes_ K K_ 1)/K_ 1\) is split. If moreover \(\Delta\) is finite and the operators in \(\Delta\) are pairwise commuting and K is algebraically closed then \(K_ 1| K\) can be chosen as a strongly normal extension. In chapter III there are given the main results concerning the connections between the properties of being weakly normal (WN), strongly normal (SN) and having no movable singularity (NMS). Let \(\Delta\) be finite and the operators in \(\Delta\) pairwise commuting. Suppose K is algebraically closed and \(F^{\Delta}=K^{\Delta}\). Then \(F| K\) has no movable singularity iff there exists a Picard-Vessiot extension \(E| F\) such that \(E| K\) is strongly normal (theorem 2.1). Using this result, in theorem 3.1 it is proven that \(F| K\) is (SN) iff it is (WN) and (NMS). Here the (NMS)-condition can be removed in the following cases: \((1)\quad tr.\deg. F| K=1;\) (2) \(tr.\deg. F| K=2\) and \(\kappa(F| K)\geq 0\) (\(\kappa\) is the Kodaira dimension) (3) \(tr.\deg. F| K=q(F| K)\) and \(\kappa(F| K)\geq 0\) \((q(F| K)\) is the irregularity of \(K_ aF/K_ a\) where \(K_ a\) is an algebraic closure of K). In chapter IV the author gives many comments showing that Kolchin's theory and the theory developed in this monograph have applications to classical situations. Strongly normal function fields lead to differential systems which can be linearized by means of abelian functions and \(\theta\)-functions - so this linearization property will also hold for function fields with no movable singularity. The author states that the algebraically complete integrable Hamiltonian systems - they can be integrated by \(\theta\)-functions - fit into his theory of function fields having (NMS). He shows this in the case of the Euler equations describing the movement of a rigid body with fixed gravity center.
scientific article; zbMATH DE number 3991636