Moduli Space of Irregular Singular Parabolic Connections of Generic Ramified Type on a Smooth Projective Curve

We give an algebraic construction of the moduli space of irregular singular connections of generic ramified type on a smooth projective curve. We prove that the moduli space is smooth and give its dimension. Under the assumption that the exponent of ramified type is generic, we give an algebraic symplectic form on the moduli space. Introduction Moduli space of irregular singular connections are expected to be obtained as a canonical degeneration of the moduli spaces of regular singular connections. Indeed Painlevé VI equations can be obtained as the isomonodromic deformation of rank two regular singular connections on P with four poles and other types of Painlevé equations are known to be obtained as generalized isomonodromic deformations of rank two connections on P with irregular singularities. The space of initial conditions of Painlevé equations are constructed by K. Okamoto in [14] for all types whose compactifications are classified by H. Sakai in [16] and these spaces are related to one another by canonical degenerations. K. Iwasaki, M.-H. Saito and the author constructed in [11], [12] and [10] the moduli space of regular singular parabolic connections on smooth projective curves and recovered the space of initial conditions of Painlevé VI equations constructed by K. Okamoto as a special case. We can obtain the geometric Painlevé property of the isomonodromic deformation on the moduli spaces in general and then we can say that the constructed moduli space is the space of initial conditions of the isomonodromic deformation. Here see [10], Definition 2.4 for the definition of the geometric Painlevé property. The author’s hope is to extend this procedure to good moduli spaces of irregular singular connections. Notice that we cannot assume that the underlying vector bundle is trivial, when the base curve is P and the degree of the underlying bundle is zero, for the purpose of obtaining the geometric Painlevé property. In [2], O. Biquard and P. Boalch gave an analytic construction of the moduli space of irregular singular connections whose leading term at each singular point is semi-simple without the assumption of the triviality of the underlying bundle. A typical case of semi-simple leading term is a generic unramified connection. M.-H. Saito and the author gave in [13] an algebraic construction of the moduli space of unramified irregular singular connections on a smooth projective curve and they do not assume that the underlying bundle is trivial even if the base curve is P and the degree of the bundle is zero. Here notice that usual ramified connections do not have semi-simple leading term at each ramified point. Compared with unramified case, the algebraic construction of the moduli space of irregular singular connections of ramified type seems to be difficult. P. Boalch gave in [3] an algebraic construction of the moduli space of irregular singular connections on P. In fact he constructed the moduli space of irregular singular connections with generic generalized eigenvalues which is locally framed at singular points and called it the extended moduli space. He gave in [3] a definition of the moduli space of irregular singular connections without local framing, which is obtained as a symplectic reduction of the extended moduli space. He gave in [4] an explicit description of the relations between two moduli spaces. In [8], C. L. Bremer and D. S. Sage generalized the result [3] by P. Boalch to ramified connections and gave an algebraic construction of the moduli space of 2010 Mathematics Subject Classification. 14D20, 53D30, 32G34, 34M55. 1 2 MICHI-AKI INABA irregular singular connections on P which is locally framed with the formal type admitting ramified type. The definition of the moduli space without local framing was also given in [8]. K. Hiroe and D. Yamakawa gave in [9] a clear summary of the construction of the moduli space as a symplectic reduction of the extended moduli space. Here we remark that in all these results, the underlying vector bundles on P are assumed to be trivial. For the character variety side, the moduli spaces were constructed in [4], [5], [6], [7] and [15], which are expected to be related with appropriate moduli spaces of irregular singular connections via the generalized Riemann-Hilbert correspondence. The difficulty of the construction of the moduli space of irregular singular connections seems to be in the control of the formal data of the irregular singular connections. Indeed the automorphisms of a formal connection effect on the global automorphisms of the underlying bundle and this effect makes it hard to construct the global moduli space. Boalch’s idea is to conquer this difficulty by considering local framing and he defined the extended moduli space. The moduli space of irregular singular connections on P without local framing is obtained as a symplectic reduction of the extended moduli space but here the underlying bundles are assumed to be trivial. In this paper, we construct a moduli space of ramified connections on a smooth projective curve whose generalized eigenvalues are generic in ramified types. A typical one is the moduli space of pairs (E,∇E) of an algebraic vector bundle E on a smooth projective curve and a connection ∇E on E such that the leading coefficients of the generalized eigenvalues of ∇E at each unramified point are mutually distinct and that the formalization of ∇E at each ramified point is isomorphic to ∇ν : C[[w]] r ∋ f 7→ df + νf ∈ C[[w]] ⊗ dz zm , where z is a uniformizing parameter of the base curve at the ramified point, w 7→ w = z is a ramified covering and ν = ∑mr−r l=0 cl wdw wmr−r+1 is generic, especially c1 6= 0. We want to give the moduli space which is a canonical degeneration of the moduli space of regular singular connections of full dimensional type. So we use the Hukuhara-Turrittin invariants as a formal data. In fact we paraphrase the Hukuhara-Turrittin invariants of the ramified irregular singular connections to a certain data obtained by restriction of the connection to a divisor on the curve. A key idea is to introduce a data of filtration on the restriction of the underlying bundle to the divisor of poles. The idea of considering this filtration is similar to that in [8]. Once the formulation of the objects representing ramified connections are established, a construction of the moduli space becomes a standard task. For the construction of the moduli space, we use a locally closed embedding of the moduli space to the moduli space of parabolic ΛD-triples constructed in [11], which is essentially obtained by a similar method to the Quot-scheme construction of the moduli space of vector bundles. The basic idea of the moduli space construction was inspired by Simpson’s works in [18] and [19]. The smoothness and the dimension counting of the moduli space follow from a standard deformation theory. We can construct a canonical two form on the moduli space and we can prove its d-closedness and non-degeneracy in the case of generic exponent by construction of deformations of the moduli space to the moduli spaces of unramified connections and regular singular connections. Acknowledgments. This work is partially supported by JSPS Grant-in-Aid for Scientific Research (C) 26400043. 1. Formal data of a ramified connection and its paraphrase In this section we first recall elementary properties of formal connections of ramified type. Consider the formal power series ring C[[z]] and denote by C((z)) its quotient field. We say that (W,∇) is a formal connection over C[[z]] if V is a free C[[z]]-module of finite rank and ∇ : W −→ W ⊗ C[[z]] · dz zm is a C-linear map satisfying ∇(fa) = a ⊗ df + f∇(a) for f ∈ C[[z]] and a ∈ W . Thanks to the Hukuhara-Turrittin theorem ([1], Proposition 1.4.1 or [17], Theorem 6.8.1), we can see that there is a positive integer l and for a variable w with w = z, there are ν1, . . . , νs ∈ MODULI SPACE OF RAMIFIED CONNECTIONS 3 ∑mr−r k=0 C · wdw wml−l+1 , positive integers r1, . . . , rs such that (W⊗C((w)),∇⊗C((w))) is isomorphic to (C((w))1 , d+J(ν1, r1))⊕· · ·⊕(C((w))s , d+J(νs, rs)) where d+J(νi, ri) : C((w))i −→ C((w))i · dz zm is the connection given by