Connections on principal bundles over curves in positive characteristics. (Q1880003)

Let \(X\) be an irreducible smooth projective curve over an algebraically closed field \(k\) of characteristic \(p>5\). Let \(E_G\) be a principal \(G\)-bundle over \(X\), where \(G\) is a connected reductive algebraic group over \(k\). The aim of the present paper is to provide a criterion for the existence of a connection on the bundle \(E_G\). In the case of \(\text{char}(k)= 0\) this problem was solved by A. Weil as early as in 1938. Now the authors derive a criterion in positive characteristics \((p> 5)\), which states the following: Assume that the group \(G\) does not contain a simple factor of the form \(\text{SL}(n)/\mathbb{Z}\) where \(\mathbb{Z}\) is contained in the center of \(\text{SL}(n)\). Then a \(G\)-bundle \(E_G\) over \(X\) admits a connection if and only if for every pair \((H,\chi)\), where \(\chi\) is a character of the Levi factor \(H\) of some parabolic subgroup of \(G\), the degree of the line bundle \(E\) induced by \((H,\chi)\) is a multiple of the characteristic \(p\). Moreover, the authors give a slight refinement of this main theorem of theirs, and they investigate the case of a simple structure group \(G\) more closely. Finally, they discuss the obstruction to the existence of connections on \(E_G\) for a simple group \(G\) explicitely. In particular, if \(G\) is a classical group not of the form \(\text{SL}(n)/\mathbb{Z}\) as above, then the given criterion for the existence of connections on \(E_G\) remains valid even if \(\text{char}(k)= p\geq 3\).