The Grothendieck–Riemann–Roch theorem

 
 

Time and place

Sommersemester 2018
Mondays 16-18, M311 (SFB room)

Description

This lecture course will be centred around the celebrated Grothendieck–Riemann–Roch theorem, proven by A. Grothendieck in 1957. Along the way, we will see how it can be naturally generalized to the setting of derived algebraic geometry. Finally, we will also discuss how the derived Grothendieck-Riemann-Roch formula gives rise to formulas for the virtual fundamental class originally predicted by M. Kontsevich.

Prerequisites: a working knowledge of algebraic geometry and category theory. More advanced topics (including perfect complexes, derived schemes, and the language of higher category theory) will be reviewed as we need them.

Lecture notes

Lecture 0: Overview (Khan)
Lecture 1: Higher categories and simplicial commutative rings (Khan)
Lecture 2: Derived schemes I (Binda)
Lecture 3: Derived schemes II (Binda)
Lecture 4: K-theory of derived schemes I (Khan)
Lecture 5: K-theory of derived schemes II (Khan)
Lecture 6: Derived blow-ups and virtual Cartier divisors (Binda)
Lecture 7: The projective bundle formula (Khan)
Lecture 8: The blow-up formula (Khan)
Lecture 9: Derived exterior powers and lambda-operations (Binda)

These are preliminary notes; read with caution.

Exercises

Sheet 1 (pdf)
Sheet 2 (pdf)
Sheet 3 (pdf)

References

  • P. Berthelot, A. Grothendieck, L. Illusie, Théorie des intersections et théorème de Riemann-Roch (SGA 6).
  • W. Fulton, Intersection theory.
  • A. A. Khan, Virtual Cartier divisors and blow-ups.
  • J. Lurie, Spectral algebraic geometry.

Contact

adenullel.khanull[email protected]nullhematik.uni-renullgensburg.dnulle
federnullico.binull[email protected]nullhematik.uni-renullgensburg.dnulle

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