Topological Classification of Lagrangian Fibrations

We define topological invariants of regular Lagrangian fibrations using the integral affine structure on the base space and we show that these coincide with the classes known in the literature. We also classify all symplectic types of Lagrangian fibrations with base $\rpr$ and fixed monodromy representation, generalising a construction due to Bates.

💡 Research Summary

The paper addresses the topological classification of regular Lagrangian fibrations by exploiting the integral affine structure naturally induced on the base space. A regular Lagrangian fibration consists of a symplectic manifold ((M^{2n},\omega)) together with a smooth surjective map (\pi:M\to B) whose fibres are Lagrangian submanifolds. The key observation is that the base (B) inherits an integral affine structure (\mathcal A): locally there exist coordinates whose transition functions lie in the semi‑direct product (\mathrm{GL}(n,\mathbb Z)\ltimes\mathbb R^{n}). This affine data encodes the monodromy representation (\rho:\pi_{1}(B)\to\mathrm{GL}(n,\mathbb Z)) and a cohomology class measuring the failure of a global Lagrangian section.

The authors first define two topological invariants directly from (\mathcal A). The affine monodromy class (