Home » Lectures » Fabian Lehmann: Lectures

Fabian Lehmann: Lectures

September 8, 2022
TITLE: An embedding problem for closed 3-forms on 5-manifolds

ABSTRACT: The real part of a holomorphic volume form restricts to a closed 3-form on a 5-dimensional submanifold of a complex 3-fold with a Calabi-Yau structure. Vice versa, in analogy with the embedding problem for abstract CR-structures, this leads to the question which closed 3-forms on a given 5-manifold can be realised by an embedding into a Calabi-Yau 3-fold. We describe the structure induced on the 5-manifold by a closed 3-form and introduce a convexity notion. The main result is that in the “strongly pseudoconvex” case the embedding problem can be solved perturbatively if a finite dimensional vector space of obstructions vanishes. The proof uses the theory of sub-elliptic operators and the Nash-Moser inverse function theorem. The main example is the standard embedding of S^5 in C^3, for which the obstruction space vanishes. This is joint work with Simon Donaldson.

September 13, 2021
TITLE: Non-compact Spin(7)-manifolds

In the non-compact setting, symmetry reduction methods can be used to simplify the condition for Spin(7)-holonomy, which in general is given by a large, non-linear, first order PDE system, to a system of ODEs. I will talk about a particular example with symmetry group SU(3). I will outline a rigorous proof for the existence of two families of complete Spin(7)-metrics, where all members are either asymptotically locally conical (ALC), or asymptotically conical (AC).

These families were conjectured to exist earlier and fit into the landscape of other known families of non-compact G2 and Spin(7) holonomy spaces. Time permitting, I will also discuss the deformation theory of AC Spin(7)-manifolds. The talk is based on arXiv:2012.11758 and arXiv:2101.10310.

Slides of Lecture