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401-4146-DRL 2 Credits DR D-MATH

Derived Algebraic Geometry

Lecturers & Examiners: Dr. Arkadij Bojko
Only for ETH D-MATH doctoral students and for doctoral students from the Institute of Mathematics at UZH. The latter need to send an email to Jessica Bolsinger ( ) with the course number. The email should have the subject „Graduate course registration (ETH)“.
VVZ CR n/a

Last Updated: 2026-02-05 16:22:19

Abstract

Classical algebraic geometry, taught in graduate-level courses, is only a shadow of the complete framework offered by derived algebraic geometry. I will describe the new insights and applications that are offered by the latter while avoiding being as technical as the standard literature cited below.

Objective

A keen listener should understand by the end of the course why derived algebraic geometry is useful and have an idea of where to begin in applying it to problems in enumerative questions.

Content

Starting from the primary building blocks called cdga's, I will first develop some intuition behind derived algebraic geometry by explaining the hidden smoothness phenomenon - the main benefit of working with derived algebraic geometry. Moving on to the global picture, I will motivate the definition of derived stacks, and shifted symplectic structures while describing their natural origin coming from Calabi-Yau categories. I will end by discussing dg-quivers and their moduli stacks of dg-representations as a natural source of examples.

Resources

Literature

A. Bojko, Derived algebraic geometry (A guide to local models for shifted symplectic structures), https://shorturl.at/epvZ4 . B. Toën, Derived Algebraic Geometry, arXiv:1401.1044, 2014. J. Lurie. Higher topos theory, Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2009. J. Lurie, On Infinity Topoi, arXiv:math/0306109, 2003. J. Lurie, Derived Algebraic Geometry, Ph.D. thesis, Massachusetts Institute of Technology, Dept. of Mathematics, 2004. B. Toën and G. Vezzosi. Homotopical algebraic geometry I: Topos theory”, Advances in mathematics, 2005. B. Toën and G. Vezzosi, From HAG to DAG: Derived Moduli Stacks: Axiomatic, Enriched and Motivic Homotopy Theory, 2004. B. Toën and M. Vaquié, Moduli of objects in dg-categories, Annales scien-tifiques de l’Ecole normale supérieure, 2007. C. Brav, V. Bussi, and D. Joyce, A Darboux theorem for derived schemes with shifted symplectic structure, Journal of the American Mathematical Society, 2019. D. Joyce , P. Safronov, A Lagrangian Neighbourhood Theorem for shifted symplectic derived schemes, In Annales de la Faculté des sciences de Toulouse: Mathématiques, 2019. D. Borisov, and D. Joyce, Virtual fundamental classes for moduli spaces of sheaves on Calabi–Yau four-folds, Geometry & Topology, 2017. Y.T. Lam, PhD thesis, https://people.maths.ox.ac.uk/joyce/theses/LamDPhil.pdf .

Learning Materials (Links)

General Information

Language
English
Levels
DR

Examination

Type
ungraded semester performance
The exam is only offered in the examination session immediately following the course. In particular, no repetition exam can be offered.

Registration & Places

Priority: Registration for the course unit is only possible for the primary target group

Course Components

Type Title Time & Place Hours
lecture Derived Algebraic Geometry
  • Mon 12:15-14:00 (HG E 22)
  • 14.03 Date 16:15-18:00 (HG E 1.1)
  • 25.04 Date 16:15-18:00 (HG E 1.1)
  • 02.05 Date 16:15-18:00 (HG E 1.1)
  • 30.05 Date 12:15-14:00 (HG E 22)
2 h weekly

Offered In