Orbifold-uniformizing differential equations. III: Arrangements defined by 3-dimensional primitive unitary reflection groups
[For part I see ibid. 267, 125-142 (1984; Zbl 0521.58052; for part II see the preceding review Zbl 0579.58024] Let G be a 3-dimensional primitive unitary reflection group and A be the line arrangement on the complex projective plane X defined by the group G. For each group G, G-invariant Fuchsian systems E of linear differential equations of rank \((=the\) dimension of the solution space) three with regular singularities along A are constructed. In general, a system of rank 3 on X (with inhomogeneous coordinates \(y^ 1, y^ 2)\) is represented by a tensor \[ \omega =\sum P_{ijk} dy^ i\otimes dy^ j\otimes dy^ k(dy^ 1\wedge dy^ 2)^{-1}. \] If \(p: {\mathbb{C}}^ 3-\{0\}\to X\) denotes the natural projection \((x^ 0,x^ 1,x^ 2)\to (y^ 1=x^ 1/x^ 0\), \(y^ 2=x^ 2/x^ 0)\), the associate form \(p^*\omega\) to E is explicitly expressed as G-invariant symmetric 3-form.
G-invariant Fuchsian systems E of linear differential equations
scientific article; zbMATH DE number 3927954
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