# Extended period mappings and moduli spaces

#### February 26, 12:00pm - February 26, 1:00pm

##### Speaker(s): Phillip Griffiths (Institute of Advanced Study)
Moduli spaces $\mathcal{M}$ with canonical completions $\overline{\mathcal{M}}$ have been defined and proved to exist for varieties of general type. For algebraic curves there are the classical spaces $\overline{\mathcal{M}}_g$ of Deligne-Mumford, and for general dimensions $n$ the construction is due to Kollar-Shepherd-Barron-Alexeev (KSBA). Both the local and global structures of $\partial \overline{\mathcal{M}}_g$ are much studied and in many ways well understood. For $n = 2$ the local singularity structure of surfaces $X$ corresponding to points of the boundary $\partial\mathcal{M} = \overline{\mathcal{M}}\backslash \mathcal{M}$ is also reasonaby well understood and there is a classification of the singularity types. In contrast, aside from the work of FPR on $I$-surfaces I know of no other examples of the global structure of $\partial \mathcal{M}$.

On the other hand, using Lie theory and analytic methods a good deal is known about the local boundary structure of the space of Hodge structures. Since period mappings from moduli to Hodge structures provide an important invariant, it is natural to use them to study them in the boundary structure of moduli. In this talk I will try to explain for a general audience two canonical completions of the period mapping on $\mathcal{M}$ and discuss how these may be used to help understand the global structure of $\partial \mathcal{M}$ in the case of algebraic surfaces.

Based on joint work either completed or in preparation with Mark Green, Radu Laza and Colleen Robles, and on correspondence and discussions with Marco Franciosi, Rita Pardini and Sonke Rollenske (FPR).

Some references are
arXiv:2020.06720 ([GGR]), arXiv:1708.09523 ([GGLR]) and
M. Franciosi, R. Pardini and S. Rollenske, Gorenstein stable surfaces with $K^2_X = 1$ and $p_g > 0$, Math. Nachr. 290(5-6):794--814, 2017, arXiv:1511.03238 ([FPR17]). CONTACT robles@math.duke.edu for zoom link