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Linear differential equations modeled after hyperquadrics. (English) Zbl 0661.35014

Let \(M=H_ 2\) be the Siegel upper half space of degree 2 and \(\Gamma\) (2) be the Siegel modular group of level 2. The space of regular orbits of \(H_ 2\) modulo \(\Gamma\) (2) is known to be the space \[ \Lambda =\{(x^ 1,x^ 2,x^ 3)\in {\mathbb{C}}^ 3| \quad x^ i\neq 0,1,x^ j\quad (i\neq j)\}. \] Let \(\pi\) : \(H_ 2\to \Lambda\) be the natural projection. The space \(\Lambda\) can be thought of as the parameter space of a family of curves of genus 2: \[ C(x):\quad v^ 2=\quad u(u-1)(u- x^ 1)(u-x^ 2)(u-x^ 3). \] The periods of C(x) give the (multi- valued) inverse map \(\pi^{-1}\). The system of differential equations on \(P^ 3\supset \Lambda\) which is satisfied by \(\pi^{-1}\) (the uniformizing equation of \(\Lambda)\) is obtained explicitly.
To do so, holomorphic mapping z of complex manifold \(M^ n\) of dimension n into \(P^{n+1}\) is studied in general. There is a system (EQ) of linear differential equations on M of rank \((=\) the dimension of the solution space) \(n+2\) whose \(n+2\) linearly independent solutions give the map z. Any such system can be written in the form \[ (EQ)\quad \partial^ 2w/(\partial x^ i\partial x^ j)=g_{ij}\partial^ 2w/(\partial x^ 1\partial x^ n)+\sum^{n}_{k=1}A^ k_{ij}\partial w/\partial x^ k+A^ 0_{ij}w\quad (1\leq i,j\leq n). \] On the other hand, the hypersurface z(M) is determined by the holomorphic conformal connection h and the cubic invariant form \(\tau\). The coefficients of (EQ) are explicitly described in terms of h and \(\tau\) : the coefficients \(g_{ij}\) represent the induced conformal metric II associated with h and the coefficients \(A^ k_{ij}\) and \(A^ 0_{ij}\) are expressed in terms of II and \(\tau\). As a corollary, it is shown that \(\tau =0\) if and only if z(M) is a piece of a non- degenerate quadric \(Q_ n\) in \(P^{n+1}\), and that in this case if \(n\geq 3\), all the coefficients of (EQ) is determined by h.
Since the Siegel upper half space \(H_ 2\) is a part of \(Q_ 3\) (noncompact form of \(Q_ 3\) i.e. the bounded symmetric domain of type IV), the problem of finding the system satisfied by \(\pi^{-1}\) reduces to obtain the conformal structure on \(\Lambda\) induced from the natural conformal structure on \(H_ 2\). The uniformizing equation is also given.
Reviewer: T.Sasaki

MSC:

35G05 Linear higher-order PDEs
35A20 Analyticity in context of PDEs
32A10 Holomorphic functions of several complex variables
32Q99 Complex manifolds
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