Betti numbers of real semistable degenerations via real logarithmic geometry

TL;DR

This paper develops a new approach using real logarithmic geometry to bound Betti numbers of real fibers in semistable degenerations, extending previous combinatorial methods to more general degenerations.

Contribution

It introduces real logarithmic geometry techniques to analyze Betti numbers in real degenerations, broadening the scope beyond toric cases.

Findings

Provides bounds for Betti numbers of real fibers near degeneration points.

Generalizes previous combinatorial results to non-toric degenerations.

Utilizes real logarithmic geometry as a new tool in the study of real degenerations.

Abstract

Let $X\rightarrow C$ be a totally real semistable degeneration over a smooth real curve $C$ with degenerate fiber $X_0$. Assuming that the irreducible components of $X_0$ are simple from a cohomological point of view, we give a bound for the individual Betti numbers of a real smooth fiber near $0$ in terms of the complex geometry of the degeneration. This generalizes previous work of Renaudineau-Shaw, obtained via combinatorial techniques, for tropical degenerations of hypersurfaces in smooth toric varieties. The main new ingredient is the use of real logarithmic geometry, which allows to work with not necessarily toric degenerations.