Moduli of semistable logarithmic connections

Under the Riemann Hilbert correspondence, which is an equivalence of categories, local systems on a nonsingular complex projective variety X correspond to pairs E = (s, V) where ' is a locally free sheaf on X, and V: 8 Q1 @ is an integrable connection on F. Generalizing this correspondence to noncomplete quasi-projective varieties, Deligne (see [D]) showed that the correct algebro-geometric objects which correspond to local systems on such a variety Y are the so-called regular connections on Y. The condition of regularity on algebraic connections on Y can be expressed as follows. Fix a Hironaka completion X of Y, that is, nonsingular projective variety X which contains Y as an open subvariety, such that the complement S = X Y is a smooth divisor with normal crossings. A logarithmic connection F = (Y, V) on X with singularities over S is by definition a torsion-free coherent sheaf Y on X together with a map V: 9 -* 21[log S]? 9 which satisfies C-linearity and the Leibniz rule, where i14[logS] is the sheaf of rational 1-forms on X with logarithmic poles on S. The curvature of the connection V is assumed to be zero. Finally, an algebraic connection E = (F, V) on Y is regular if and only if there exists a logarithmic connection F on X with singularities over S which extends E. If such an extension F exists, the underlying sheaf F can be chosen to be locally free, and will be called a logarithmic lattice for E. A logarithmic lattice for a given regular connection is of course not unique, but there is a canonical way of choosing it called Deligne's construction, which only depends on the choice of a set-theoretic section of the map exp: C -. C* . As such a section is not even continuous, the Deligne lattice does not behave well in families of regular connections. Simpson has given in [S] a construction of a moduli scheme for (nonsingular) connections on a projective variety X. However, a simple example shows that it would be unreasonable to expect in general the existence of a moduli scheme for regular connections on a quasi-projective variety Y. For this, let Y be the affine line minus the origin, with coordinate x. The differential equation dy/dx tylx = 0 with parameter t defines a family of regular connections on the sheaf Y, parametrized by the affine line T with coordinate t. We can see that t1 and t2 parametrize isomorphic connections if and only if t1 t2