Shifted Symplectic Fibrations and Derived Thurston Theorem
In classical symplectic geometry, under mild conditions, Thurston proved that one can construct a compatible symplectic form on the total space of a symplectic fibration with a connected symplectic base. Here we prove a derived symplectic analogue of this result, called the derived Thurston theorem. More precisely, we show that if a morphism $π: X \rightarrow S$ of derived stacks has a shifted symplectic fibration structure and the target stack $S$ admits a shifted symplectic structure, then, under certain conditions, one can construct a shifted symplectic structure on a dense Zariski-open derived substack of the source stack $X$, compatible with $π$ in a sense similar to the classical case. In this derived setting, we also present numerous examples of shifted symplectic fibrations, develop an affine model construction for such fibrations, and provide applications of the derived Thurston theorem.