Graduate lecture course, 14 lectures, Summer Term 2022.

Professor Joyce

Derived Algebraic Geometry

Fridays 2pm - 4pm on MS Teams, starting on 29th April and finishing on 17th June. No lecture on Friday 10th June.

This course is put on by the Taught Course Centre, and will be transmitted to the Universities of Bath, Bristol, Oxford, Warwick and Imperial College London.

I am sorry, but if you are not at one of these five universities, you can't come to the course. I have no control over this. Even if someone gave you the Teams link, you can only get into the Teams meeting if your e-mail address has been authorized by the TCC staff.

Synopses:

Derived Algebraic Geometry is the study of "derived" schemes and "derived" stacks. These are enhanced versions of classical schemes and (Artin) stacks which have a richer geometric structure. This richer structure is particularly important in problems involving virtual classes and enumerative invariants, and in moduli spaces of objects in Calabi-Yau categories. As an example, let X be a smooth projective scheme of dimension m. Then we can form the classical moduli stack M and the derived moduli stack M of coherent sheaves on X. A coherent sheaf E in coh(X) corresponds to points [E] in M and M. The classical cotangent complex LM knows about the Ext groups Exti(E,E) for i=0,1, but the derived cotangent complex LM knows about the Ext groups Exti(E,E) for all i=0,1,...,m. That is, the richer geometric structure on the derived stack M remembers all of the deformation theory Ext*(E,E).

Derived Algebraic Geometry is famously hard to learn -- the foundational papers, due to Lurie and ToŽn-Vezzosi, run to 1000's of pages. Part of the difficulty is that the theory must be set in the world of ∞-categories.

After a brief discussion of derived geometry in the first week, I will spend several weeks talking about other stuff which is interesting for its own sake: classical stacks, triangulated categories and derived categories (derived categories are a necessary prequel to derived stacks and derived algebraic geometry, which were developed much earlier than DAG), and ∞-categories, before trying to explain derived schemes and derived stacks at the end of the course.

Disclaimer.

I am not an expert in DAG. I've mostly picked the subject up by osmosis, without actually spending two years reading Lurie and ToŽn-Vezzosi. So if you ask me technical questions about the fppf topology, etc, I won't know the answer. I can probably teach you how to bluff about DAG at parties, though.

Lecture 1: Different kinds of spaces in algebraic geometry: schemes, stacks, higher stacks, derived stacks. Basics of category theory, categories, functors. The Yoneda Lemma. Schemes as functors AlgKSets, and stacks as functors AlgKGroupoids. Grothendieck’s approach to moduli spaces as ‘representable functors’.

Lecture 2: What is derived geometry? Derived schemes and stacks. Commutative differential graded algebras, examples. Bťzout’s Theorem and derived Bťzout’s Theorem. Patching together local models in derived geometry. Fibre products in ordinary categories; why we need higher categories in derived geometry.

Lecture 3: Introduction to 2-categories. Fibre products in 2-categories. First look at ∞-categories.

Lecture 4: Global quotient stacks, as a way of introducing Deligne-Mumford and Artin stacks before giving the formal definition, and emphasizing the 2-categorical aspects. 1- and 2-morphisms of quotient stacks. The classifying stack of principal bundles. Fibre products. Sheaves and stacks on topological spaces.

Lecture 5: Classical stacks. Moduli functors, moduli schemes and moduli stacks. Grothendieck topologies. Sheaf-type properties of functors. Definition of stacks. Deligne-Mumford and Artin stacks.

Lecture 6: Abelian categories. Coherent sheaves coh(X) and quasicoherent sheaves qcoh(X). Examples.

Lecture 7: Introduction to derived categories and triangulated categories.

Lecture 8: More on triangulated categories and derived categories. Derived categories of coherent sheaves Dbcoh(X). Relation to ∞-categories.

Lecture 9: Informal introduction to ∞-categories. Model categories. Simplicial sets. Simplicial categories.

Lecture 10: More about simplicial sets. Simplicial objects. Kan complexes and weak Kan complexes. Quasicategories, a model for ∞-categories.

Lecture 11: Stable ∞-categories as the "correct" definition of triangulated category. Dg-categories. Segal categories. Foundational theories of DAG in the literature. Higher stacks.

Lecture 12: The definitions of derived schemes and derived stacks, following ToŽn-Vezzosi. Cotangent complexes of derived schemes and stacks. Quasi-smooth derived stacks. Examples of derived schemes and stacks.

Lecture 13: Enumerative invariants in Algebraic Geometry (e.g. Gromov-Witten invariants, algebraic Donaldson invariants). Obstruction theories and virtual classes on classical schemes and stacks, following Behrend-Fantechi. How they are used to define enumerative invariants. Relation to Derived Algebraic Geometry: a classical stack with obstruction theory is the semi-classical shadow of a quasi-smooth derived stack. Enumerative invariants usually count only 'semistable' objects. Stability and semistability, in the sense of stability conditions on abelian categories, and in Geometric Invariant Theory, and the relation between these.

Lecture 14: Shifted symplectic derived stacks, following Pantev-ToŽn-Vaquiť-Vezzosi, and applications to Donaldson-Thomas theory of Calabi-Yau 3- and 4-folds.

Bibliography:

Useful survey papers on DAG:

[1] B. ToŽn, ‘Derived Algebraic Geometry’, arXiv:1401.1044.

[2] B. ToŽn, 'Higher and derived stacks: a global overview', math.AG/0604504.

Foundational papers on DAG:

[3] J. Lurie, ‘Derived Algebric Geometry V: Structured spaces’, arXiv:0905.0459.

[4] B. ToŽn and G. Vezzosi, ‘Homotopical Algebraic Geometry II: Geometric Stacks and Applications’, Mem. Amer. Math. Soc. 193 (2008), no. 902.† math.AG/0404373.

[5] B. ToŽn and G. Vezzosi, 'From HAG to DAG: derived moduli stacks', math.AG/0210407.

[6] T. Pantev, B. ToŽn, M. Vaquiť, and G. Vezzosi, 'Shifted Symplectic Structures', arXiv:1111.3209.

Classical schemes, coherent sheaves:

[7] R. Hartshorne, 'Algebraic Geometry', Graduate Texts in Math. 52, Springer, New York, 1977.

Classical stacks:

[8] T.L. Gomez, 'Algebraic stacks', Proc. Indian Acad. Sci. Math. Sci. 111 (2001), 1-31. math.AG/9911199.

[9] M. Olsson, 'Algebraic Spaces and Stacks', A.M.S. Colloquium Publications 62, A.M.S., Providence, RI, 2016.

[10] G. Laumon and L. Moret-Bailly, 'Champs algebriques', Ergeb. der Math. und ihrer Grenzgebiete 39, Springer-Verlag, Berlin, 2000.

Triangulated and derived categories:

[11] R.P. Thomas, 'Derived categories for the working mathematician', math.AG/0001045.

[12] S.I. Gelfand and Y.I. Manin, 'Methods of Homological Algebra', second edition, Springer, 2002.

[13] D. Huybrechts, 'Fourier-Mukai transforms in Algebraic Geometry', Oxford University Press, Oxford, 2006.

PDF files to download:

Slides for lectures 1 and 2

Slides for lectures 3 and 4

Slides for lectures 5 and 6

Slides for lectures 7 and 8

Slides for lectures 9 and 10

Slides for lectures 11 and 12

Slides for lectures 13 and 14

Miniprojects for assessment as a broadening course