Integrable connections and Galois representations (Q2900280)

The article announces a result of the author in a preprint [``Deformation of \(\ell\)-adic sheaves with undeformed local monodromy'', \url{arxiv:1103.1093}]. The author begins with an explanation of Riemann-Hilbert correspondence. He motivates the study of connections with irregular singularities by some examples.NEWLINENEWLINEThe Riemann-Hilbert correspondence implies that a locally free sheaf with an integrable connection with regular singularity on a smooth quasi-projective variety is determined by the monodromy representation. But this fails for connections with irregular singularity.NEWLINENEWLINEA lisse \(l\)-adic sheaf \(\mathcal{F}\) on a curve \(X\) is called physically rigid if it is determined by its restriction to the completed punctured spectrum \(\eta_P\) at every closed point \(P\) of the smooth completion \(\bar X\) of \(X\). In other words, \(\mathcal{F}\) is determined by the induced Galois representations of the local fields \(\eta_P\) for all closed points \(P\) in \(\bar X\). N. Katz showed that if \(H^1(\bar X, j_{!*}\mathcal{E}nd(\mathcal{F}))=0\) then \(\mathcal{F}\) is physically rigid and he conjectured that the converse should hold too. In [Asian J. Math. 8, No. 4, 587--606 (2004; Zbl 1082.14506)], \textit{S. Bloch} and \textit{H. Esnault} proved the converse in the classical setup, i.e., for a locally free sheaf with integrable connection on a smooth complex curve.NEWLINENEWLINEIn fact Bloch and Esnault showed that the deformation functor of a locally free sheaf with integrable connection \((\mathcal{V}, \nabla)\) with undeformed local formal data is pro-representable by a complete \(\mathbb{C}\)-algebra which is algebraizable. Moreover, the tangent space of this deformation functor is isomorphic to \(H^1(\bar X, j_{!*}\mathcal{E}nd((\mathcal{V},\nabla)))\). The result announced by the author is an \(l\)-adic analogue of this result of \textit{S. Bloch} and \textit{H. Esnault} [Asian J. Math. 8, No.~4, 587--606 (2004; Zbl 1082.14506)]. But the author does not claim that the complete \(\bar \mathbb{Q}_l\)-algebra pro-representing the deformation functor is algebraizable. The deformation functor sends an Arin local ring \(R\) with residue field \(\bar \mathbb{Q}_l\) to the set isomorphism classes of lisse \(R\)-sheaves \(\mathcal{G}\) on \(X\) such that \(\mathcal{G}\otimes_R \bar \mathbb{Q}_l\) is isomorphic to the given \(l\)-adic sheaf \(\mathcal{F}\) and \(\mathcal{G}|_{\eta_P}\cong \mathcal{F}|_{\eta_P}\otimes_{\mathbb{\bar Q}_l} R\) for every \(P\in \bar X \setminus X\).NEWLINENEWLINEFor the entire collection see [Zbl 1235.00045].