Compactification of linear manifolds and applications

Linear submanifolds of strata of holomorphic abelian differentials are rare objects. Up to today, only few series in low dimension are known. The classification of linear submanifolds is an open problem, and the question about the existence of examples in higher dimension is still unanswered.

One approach to their study and a possible classification is to analyze their closure inside as suitable compactification of the ambient stratum. I will describe the phenomena encountered at the boundary for the example of the Gothic locus, one of the few exceptional linear submanifolds recently discovered.

As an application of the compactification, I will describe the intersection theoretic framework to compute invariants of linear submanifolds (for example the Masur-Veech volume and Siegel-Veech constants) that we implemented as a SageMath package.