Yoonjoo Kim, Two results on Lagrangian fibrations

I would like to report two ongoing results on Lagrangian fibrations of smooth symplectic varieties. The first is the construction of a delta-regular smooth group scheme that acts on a given Lagrangian fibration. It is a generalization of the result of Arinkin-Fedorov, who proved the result under the assumption that every fiber is integral. The second is the computation of the degree of the discriminant divisor of a Lagrangian fibration, assuming the total space is a (projective) hyper-Kähler manifold. This resolves a question raised by Sawon, who first discovered the formula and proved it under the assumption that every general singular fiber is integral. The first result is joint work with Mark de Cataldo and Christian Schnell.