UC Berkeley

Let k be an algebraically closed field and Z an effective Cartier divisor in the projective over k with complement U. When k = C, a local system on the analytification of U is said to be physically rigid when it is determined by the conjugacy classes of its monodromy operators around the points of Z. Katz proves a convenient cohomological characterization of irreducible physically rigid local systems. Roughly, it arises from the observation that irreducible physically rigid local systems are smooth isolated points in the moduli of local systems on U with fixed local monodromy data along Z.

In this dissertation, we consider the situation where char(k) > 0 and local systems are replaced with overconvergent isocrystals on U. The "moduli of overconvergent isocrystals" is an elusive object, but we establish some results about the formal deformation theory of overconvergent isocrystals with fixed "local monodromy" along Z. These results bear strong resemblances to facts about the infinitesimal structure of the moduli of local systems with fixed monodromy.

En route, we establish a general result which shows that a Hochschild cochain complex governs deformations of a module over an arbitrary associate algebra. We also relate this Hochschild cochain complex to a de Rham complex in order to understand the deformations of a differential module over a differential ring.