Derived Algebraic/Differential Geometry

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.

Lecturer

Date

8th June ~ 10th October, 2022

Prerequisite

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

Syllabus

Section I:

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:

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:

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:

Chapter 16: Examples and applications 2 (Part III)

Chapter 17: Uhlenbeck–Yau construction and correspondence Examples (Part 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

Audience

Graduate

Video Public

Yes

Notes Public

Yes

Language

English

Lecturer Intro

Artan Sheshmani is a Professor of pure Mathematics, specialized in Algebraic geometry, Enumerative and Derived Geometry, and Mathematics of String Theory. He is a Professor at Beijing Institute of Mathematical Sciences and Applications in Beijing, and a senior personnel (Professor) at Simons Collaboration Program on Homological Mirror Symmetry ( Harvard University Center for Mathematical Sciences and Applications), and an Affiliate Faculty Member at Harvard University- MIT IAiFi (Institute for Artificial Intelligence and Fundamental Interactions). Between 2020 and 2023, he jointly held the visiting professor position at Institute for the Mathematical Sciences of the Americas at University of Miami, where he was part of the research collaboration program on "Hodge theory and its applications". During the past 5 years while at Harvard CMSA he was also a visiting professor at Harvard Physics department (2020-2022), and an Associate Professor of Mathematics at Institut for Mathematik (formerly the Center for Quantum Geometry of Moduli Spaces) at Aarhus University in Denmark (2016-2022). His work is mainly focused on Gromov Witten theory, Donaldson Thomas theory, Calabi-Yau geometries, and mathematical aspects of String theory. He studies geometry of moduli spaces of sheaves and curves on Calabi Yau spaces, some of which arise in the study of mathematics of string theory. In his research he has worked on understanding dualities between geometry of such moduli spaces over complex varieties of dimension 2,3,4 and currently he is working on extension of these projects from derived geometry and geometric representation theory point of view. In joint work with Shing-Tung Yau (BIMSA, YMSC, Tsinghua, Harvard Math, Harvard CMSA, and Harvard Physics departments), Cody Long (Harvard Physics), and Cumrun Vafa (Harvard Math and Physics departments) he worked on geometry moduli spaces of sheaves with non-homolomorphic support and their associated non-BPS (non-holomorphic) counting invariants. In 2019 he was one of the 30 winners of the IRFD "Research Leader" grant (approx 1M USD) on his project "Embedded surfaces, dualities and quantum number theory". The project was additionally co-financed by Harvard University CMSA and Aarhus University (Approx total. 400K USD). Detail of IRFD "Research Leader" grant: https://dff.dk/en/grants/research-leaders-2018.