Séminaire de Géométrie Complexe

Measuring holes of hypersurfaces

by Dr Lionel Lang (University of Gävle)

salle 207 bâtiment 1R2 (Institut de Mathématiques de Toulouse)

salle 207 bâtiment 1R2

Institut de Mathématiques de Toulouse

In 2000, Mikhalkin introduced a class of real algebraic planar curves now known as simple Harnack curves. Among their many nice properties, these curves appear as spectral curves of planar dimers. In this context, Kenyon and Okounkov showed that any simple Harnack curve is determined by the logarithmic area of some well chosen membranes bounded on the curve (plus some boundary conditions). This is a very special situation since, in general,  the areas of these membranes only provide local coordinates on the space of curves under consideration.


In this talk, I'd like to discuss a generalization of this fact to arbitrary dimension, namely how logarithmic volumes of  well chosen membranes provide local coordinates on linear systems of hypersurfaces. Moreover, these local coordinates have an obvious tropicalization that gives rise to global coordinates on the corresponding linear system of tropical hypersurfaces. Eventually, if time permits,  I'd like to discuss potential applications to deformation of real algebraic hypersurfaces.