Derived algebraic/differential geometry literature

Basic setting of derived geometry

The general framework is given in:

  1. B.Toën, G.Vezzosi. Homotopical algebraic geometry I: topos theory. Advances in Mathematics 193 (2005)
  2. B.Toën, G.Vezzosi. Homotopical algebraic geometry II: geometric stacks and applications. Memoirs of the AMS 902 (2008)

However a more accessible source would be:

  1. B.Toën. Simplicial presheaves and derived algebraic geometry. In Sim- plicial methods for operads and algebraic geometry. Birkhäuser               (2010).

To deal with infinitesimal geometry another useful source will be:

  1. J.P.Pridham. Presenting higher stacks as simplicial schemes. Ad- vances in Mathematics 238 (2013)

To cover the C∞-side we might need to look:

  1. O.Ben-Bassat, K.Kremnizer. Non-Archimedean analytic geometry as relative algebraic geometry. arXiv:1312.0338
  2. D.Borisov, K.Kremnizer. Quasi-coherent sheaves in differential geometry. arXiv:1707.01145 [math.DG]
  3. D.Borisov, J.Noel. Simplicial approach to derived differential man- ifolds. (2011) arXiv:1112.0033v1 [math.DG]

Loop spaces and differential forms

The building blocks are:

  1. B.Toën, G.Vezzosi. Algèbres simpliciales S1-équivariantes, théories de de Rham et théorèmes HKR multiplicatifs. Composition Mathematica 147/06 (2011)
  2. B.Toën, G.Vezzosi. Caractères de Chern, traces ëquivariantes et géométries algébriques dérivée. Selecta Mathemtica 21/2 (2014)
  3. D.Ben-Zvi, D.Nadler. Loop spaces and connections. J. of Topology 5 (2012)

culminating in:

  1. T.Pantev, B.Toën, M.Vaquié, G.Vezzosi. Shifted symplectic structures. Publ. math. de l’IHÉS 117/1 (2013)
  2. J-L.Loday. Cyclic homology. Springer (1992)

Shifted symplectic structures

  1. T.Pantev, B.Toën, M.Vaquié, G.Vezzosi. Shifted symplectic struc- tures. Publ. math. de l’IHÉS 117/1 (2013)
  2. Ch.Brav, V.Bussi, D.Joyce. A Darboux theorem for derived schemes with shifted symplectic structure. J. of the AMS 910 (2018)
  3. D.Joyce, P.Safronov. A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes. arXiv 1506.04024 [math.AG]
  4. D.Borisov, D.Joyce. Virtual fundamental classes for moduli spaces of sheaves on Calabi–Yau four-folds. Geometry and Topology 21             (2017).

Uhlenbeck–Yau construction and correspondence

TBA

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