Moduli of unramified irregular singular parabolic connections on a smooth projective curve

Abstract

In this paper we construct a coarse moduli scheme of stable unramified irregular singular parabolic connections on a smooth projective curve and prove that the constructed moduli space is smooth and has a symplectic structure. Moreover, we will construct the moduli space of generalized monodromy data coming from topological monodromies, formal monodromies, links, and Stokes data associated to the generic irregular connections. We will prove that for a generic choice of generalized local exponents, the generalized Riemann-Hilbert correspondence from the moduli space of the connections to the moduli space of the associated generalized monodromy data gives an analytic isomorphism. This shows that differential systems arising from (generalized) isomonodromic deformations of corresponding unramified irregular singular parabolic connections admit the geometric Painlevé property as in the regular singular cases proved generally. IntroductionLet m, l be positive integers, and let ν be an element of C dw/w lm−l+1 + · · · + C dw/w. We denote the C [[w]]-module C[[w]] ⊕r with the connectionHere e 1 , . . . , e r is the canonical basis of C[[w]] ⊕r and e 0 = 0.We have the following fundamental theorem. THEOREM 0.1 (HUKUHARA AND TURRITTIN) Let V be a free C[[z]]-module of rank r, and let ∇ : V → V ⊗ dz/z m be a connection. Then there are positive integers l, s, r 1 , .. . , r s such that for a variable w which is the family of moduli spaces of α-stable parabolic connections over a certain space T •,s νres of parameters including generic, simple exponents ν with the fixed residue part ν res (see (26)). The isomonodromic flows define an isomonodromic foliation or an isomonodromic differential system on the phase space, and its geometric Painlevé property follows easily from the definition based on Theorem 5.1. The geometric Painlevé property gives a complete and clear proof of the analytic Painlevé property for the isomonodromic differential systems with nonresonant and irreducible exponents ν res or p = e(ν res ).As explained in [9], it is important to construct the fibers of the phase space of the isomonodromic differential system as smooth algebraic schemes. One can use affine algebraic coordinates of the fibers over an open set of parameter spaces to write down the differential systems explicitly. Then the differential systems satisfy the analytic Painlevé property, which easily follows from the geometric Painlevé property.We should mention that Malgrange [14] and [15] and Miwa [16] gave proofs of the analytic Painlevé property for isomonodromic differential systems for irregular connections on P 1 . However, to give a complete proof of the geometric Painlevé property, we believe that our algebro-geometric construction of the family of the moduli spaces of connections is indispensable (see also [10], [11], [8] for the regular singular cases).Bremer and Sage [2] studied the moduli space of irregular singular connections on P 1 . They considered also the ramified case. However, they assumed that the bundle V is trivial, whi...

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“…2. The case α = β −1 = ±1 can be handled as in (1). One finds (with a similar notation) a geometric quotient R(α, α −1 ) * of T (α, α −1 ) * and a bijectionS(α, The locus of the points in T (1, 1) which describe the monodromy data for modules in S(1, 1) which are direct sums is the union of the two closed sets a 1 = a 2 = 2 = 3 = 0 and a 1 = a 2 = 34 = 0.…”