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A lower bound for the size of monodromy of systems of ordinary differential equations. (English) Zbl 0813.32020

Algebraic geometry and analytic geometry, Proc. Conf., Tokyo/Jap. 1990, ICM-90 Satell. Conf. Proc., 198-230 (1991).
Let \(X\) be a compact Riemann surface. The article concerns the complex analytic isomorphism between the moduli spaces of rank two determinant one systems of ordinary differential equations on \(X\) and representation of \(\pi_ 1 (X)\) in \(Sl(2,\mathbb{C})\). A system of ODE’s on \(X\) is a pair \((N, \nabla)\) where \(N\) is a locally free \({\mathcal O}_ X\)-module and \(\nabla : N \to \Omega_ X^ 1 \otimes_{\mathcal O}N\) is a morphism of sheaves such that \(\nabla (an)= a \nabla(n) + (da)n\) for \(a \in {\mathcal O}_ X\) and \(n \in N\). Let \(\rho\) be the monodromy representation of \((N, \nabla)\) (i.e. \(\rho : \pi_ 1 (X,p) \to Gl ({\mathcal V}_ p)\) where \({\mathcal V}\) is the sheaf of analytic horizontal sections \({\mathcal V} = (N^{an})^ \nabla)\). The article contains:
(1) An analytic proof of the existence of a line bundle \(L\) such that for any rank two system of ODE’s \((N, \nabla)\) on \(X\), \(N \otimes L\) is generated by global sections and \(H^ 1(X, N \otimes L) = 0\).
(2) A proof of the existence of a parameter space \({\mathcal Z}\) for quadruples \((N, \nabla, \delta, \nu)\) where \((N,\nabla)\) is a rank two system of ODE’s on \(X\), \(\delta : \Lambda^ 2 (N, \nabla) \to ({\mathcal O}_ X,d)\) is an isomorphism and \(\nu = (\nu_ 1, \dots, \nu_ p)\) is a collection of sections of \(H^ 0 (X,N \otimes L)\) which generate \(N\otimes L\). The parameter space \({\mathcal Z}\) is a quasiprojective variety.
(3) The construction of an algebraic exhaustion function \(\psi_{\mathcal Z}\) on \({\mathcal Z}\). Let \(W \subset \mathbb{P}^ n\) a quasiprojective variety and set \(W_ \infty = \overline W \backslash W\). A function \(\psi_ W\) (on \(W)\) is called an algebraic exhaustion function for \(W\) if, for some constants \(C\) and \(k\), one has \(C^{-1} d(w, W_ \infty)^{-1/k} \leq \psi_ W (w) \leq Cd(w,W_ \infty)^{-k}\). This condition is independent of the projective embedding.
The main result of the paper is the following:
Theorem: Fix generators \(\gamma_ i\) of \(\pi_ 1 (X)\). There are constants \(C\) and \(k\) such that: For all semisimple rank two determinant one system of ODE’s \((N, \nabla, \delta)\) with monodromy matrices bounded by \(| \rho (\gamma_ i) | \leq T\), there exists a collection of generating sections \(\nu_ 1, \dots, \nu_ p\) in \(H^ 0(X,N \otimes L)\) such that \(\psi_{\mathcal Z} (N, \nabla, \delta, \nu) \leq C (\log T)^ k\).
The main analytic tool used for obtaining the estimate is the notion of harmonic metric for a local system.
For the entire collection see [Zbl 0744.00034].

MSC:

32G34 Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation)
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
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