Derived Algebraic/Differential Geometry

Lecturer

Date & Time

[Beijing Time] 08:50 - 10:35, each Mon & Thu, start from Jun 8, 2022

Location

ZOOM: 559 700 6085, PW: BIMSA

Introduction

Derived Algebraic Geometry is a machinery regarded as an extension of algebraic geometry, whose goal is to study exotic geometric settings and situations that might occur in algebraic geometry where algebraic geometry might not be able to rigorously study. Take for instance the example of intersection of two subvarieties, X, Y, in a fixed ambient smooth algebraic variety Z. We have the notion of "nice intersection" (or generic intersection) of X and Y in Z, which is equivalent to transverse intersection of the X and Y. In this case the span of the tangent space generated by tangent spaces of X and Y is equal to the tangent space of Z and their locus of intersection X∩Y will be of expected dimension. Now take for instance a “bad intersection” of X and Y, that is non-generic or non-transverse intersection of X and Y in Z. The latter situation might occur if X and Y intersect over points or loci with higher multiplicities or when their locus of intersection is not of expected dimension. Certainly such situation has been addressed in algebraic geometry, using cohomology theory, that is, one realizes the locus of intersection, X∩Y, as a cohomology class in the ambient cohomology theory of Z (for instance an element in de Rham cohomology of Z, or in complex cobordism ring of Z, or an element of the K-theory in the intersection ring of Z) and studies the intersection of X and Y cohomologically. The drawback of this approach is that X ∩Y is realized only as a cohomology class and not as a geometric object any more. The Derived Algebraic Geometry allows one to construct a geometric object associated to the non-generic locus of intersection of X and Y, which is called the “Derived Scheme”. It is roughly speaking the homotipical perturbation of the naive locus of intersection of X and Y, and contains the data of higher multiplicity components of X∩Y or components with defects of expected dimension. The machinery of homotopy theory in derived algebraic geometry enables one to identify the points in derived intersection of X and Y as points which lie in X and Y respectively, together with certain continuous homotopy maps between them (as opposed to generic intersection of X and Y where points in X∩Y are given by points which lie both in X and Y simultaneously). Similarly taking “bad” quotients of algebraic schemes/varieties by non-proper or non-free actions is yet another example which can be modeled geometrically and rigorously by derived algebraic geometry. In usual setting the points in quotient of a variety X by the action of a free proper group G are the ones that lie in the orbit space of elements of G. However in instances where the group G is acting non-freely on X, the derived algebraic geometry enables one to realize the points in the "bad quotient" as points which lie in the orbit of elements of G up to homotopy, that is two points, a and b, lie in the orbit of an element of G if they are related to each other by a homotopy path or a path with homotopical structure in that orbit. The course sets foundations to such theory of derived schemes, and follows by discussing the derived moduli spaces, and specially the derived structure of the moduli spaces of coherent sheaves, and some applications of derived geometry in enumerative geometry (specially the Donaldson-Thomas theory) of Calabi-Yau 3 folds and 4 folds will be discussed at the end.

Prerequisite

Commutative Algebra (Atiyah McDonald or Rottman), Algebraic Geometry (Hartshorne or Grothendieck’s EGA/SGA)

Syllabus

Section I: Basic setting of derived geometry (Goal: To collect the minimum set of tools needed to do algebraic geometry in the derived context.)
Chapter 1: Differentially Graded Algebras, basic properties
Chapter 2: Differentially Graded Schemes, basic constructions and properties
Chapter 3: Derived Artin stacks
Chapter 4: Cotangent complexes

Section II: Loop spaces and differential forms (Goal: This is the algebraic heart of the course – here we learn the homological techniques that are needed for shifted symplectic forms.)
Chapter 5: De Rham complexes and S1-equivariant schemes (loop spaces)
Chapter 6: Chern character
Chapter 7: Local structure of closed differential forms in the derived sense Part I
Chapter 8: Local structure of closed differential forms in the derived sense Part II
Chapter 9: Cyclic homology

Section III: Shifted symplectic structures (Goal: To see applications of the algebraic techniques from above in the geometric context of the actual moduli spaces.)
Chapter 10: Definition and existence results
Chapter 11: Lagrangians and Lagrangian fibrations
Chapter 12: Lagrangians and Lagrangian fibrations
Chapter 13: Intersections of Lagrangians
Chapter 14: Examples and applications 2 (Part I)
Chapter 15: Examples and applications 2 (Part II)

Section IV: Uhlenbeck–Yau construction and correspondence
Chapter 16: Examples and applications 2 (Part III)
Chapter 17: Uhlenbeck–Yau construction and correspondence Examples (Part I)

Reference

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:
3. 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:
4. 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:
5. O.Ben-Bassat, K.Kremnizer. Non-Archimedean analytic geometry as relative algebraic geometry. arXiv:1312.0338
6. D.Borisov, K.Kremnizer. Quasi-coherent sheaves in differential geometry. arXiv:1707.01145 [math.DG]
7. 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:
8. B.Toën, G.Vezzosi. Algèbres simpliciales S1-équivariantes, théories de de Rham et théorèmes HKR multiplicatifs. Composition Mathematica 147/06 (2011)
9. B.Toën, G.Vezzosi. Caractères de Chern, traces ëquivariantes et géométries algébriques dérivée. Selecta Mathemtica 21/2 (2014)
10. D.Ben-Zvi, D.Nadler. Loop spaces and connections. J. of Topology 5 (2012)
culminating in:
11. T.Pantev, B.Toën, M.Vaquié, G.Vezzosi. Shifted symplectic structures. Publ. math. de l’IHÉS 117/1 (2013)
12. J-L.Loday. Cyclic homology. Springer (1992)

Shifted symplectic structures
13. T.Pantev, B.Toën, M.Vaquié, G.Vezzosi. Shifted symplectic struc- tures. Publ. math. de l’IHÉS 117/1 (2013)
14. Ch.Brav, V.Bussi, D.Joyce. A Darboux theorem for derived schemes with shifted symplectic structure. J. of the AMS 910 (2018)
15. D.Joyce, P.Safronov. A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes. arXiv 1506.04024 [math.AG]
16. 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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