Algebraic Geometry references for shifted structures

Question

I come from symplectic/poisson geometry and I am curious about shifted symplectic/poisson structure on stacks. From what I've saw, the general context there is derived schemes/stacks, so, Algebraic Geometry.

What would be the shortest (and most useful) reference for learning AG for that? Hartshorne?

Answer 1

Going from Hartshorne to stacks and higher algebra to DAG to derived symplectic geometry would take several years at least. This area arguably began with the paper Shifted symplectic structures by T. Pantev, B. Toën, M. Vaquié & G. Vezzosi, so I'd start with the references there and try to backfill any knowledge gaps. Some helpful surveys:

1. Derived Algebraic Geometry by Bertrand Toën
2. Homotopical algebraic geometry II: geometric stacks and applications by B. Toën & G. Vezzosi
3. Derived stacks in symplectic geometry by Damien Calaque. In fact almost everything by Calaque and his students will be useful.

You may also want to take a look at the Stacks project.