The eleven dimensional supergravity equations, resolutions and Lefschetz fiber metrics

This thesis consists of three parts. In the first part, we study the eleven dimensional supergravity equations on B × S considered as an edge manifold. We compute the indicial roots of the linearized system using the Hodge decomposition, and using the edge calculus and scattering theory we prove that the moduli space of solutions, near the Freund–Rubin states, is parameterized by three pairs of data on the bounding 6-sphere. In the second part, we consider the family of constant curvature fiber metrics for a Lefschetz fibration with regular fibers of genus greater than one. A result of Obitsu and Wolpert is refined by showing that on an appropriate resolution of the total space, constructed by iterated blow-up, this family is log-smooth, i.e. polyhomogeneous with integral powers but possible multiplicities, at the preimage of the singular fibers in terms of parameters of size comparable to the length of the shrinking geodesic. This is joint work with Richard Melrose. In the third part, the resolution of a compact group action in the sense described by Albin and Melrose is applied to the conjugation action by the unitary group on self-adjoint matrices. It is shown that the eigenvalues are smooth on the resolved space and that the trivial tautological bundle smoothly decomposes into the direct sum of global one-dimensional eigenspaces. Thesis Supervisor: Richard B. Melrose Title: Simons Professor of Mathematics