On the Moduli of SL ( 2 ) - bundles with Connections on P 1 \ { x 1 , . . . , x 4 }

The moduli spaces of bundles with connections on algebraic curves have been studied from various points of view (see [6], [10]). Our interest in this subject was motivated by its relation with the Painlevé equations, and also by the important role of bundles with connections in the geometric Langlands program [4] (for more details see the remarks at the end of the introduction). In this work, we consider SL(2)-bundles on P1 with connections. These connections are supposed to have poles of order 1 at fixed n points, and the eigenvalues ±λi of the residues are fixed. We call these bundles (λ1, . . . , λn)-bundles. Our aim is to find all invertible sheaves on the moduli space of (λ1, . . . , λn)-bundles and to compute the cohomology of these sheaves for n = 4. In this work, the ground field is C, that is, ‘space’ means ‘C-space’, P1 means PC, and so on. Let us formulate the main results of this work. Fix x1, . . . , xn ∈ P1(C), n ≥ 4, xi 6= xj for i 6= j, and λ1, . . . , λn ∈ C.