This is a broad graduate level course on complex algebraic geometry on 7.5 credits. The course is primarily intended for PhD students in analysis and other non-algebraic subjects. We will also almost exclusively take an analytic viewpoint: that is, work with holomorphic functions and complex manifolds rather than commutative algebra. A secondary goal is that PhD students specializing in algebra meet an analytic side of their subject.

Contents (tentative)

  • Complex manifolds & analytic subvarieties
  • Examples: tori, hypersurfaces, quotients, Grassmannians, ...
  • Projective space & blow-ups
  • Holomorphic vector bundles & divisors
  • Hodge theory, Serre duality & Lefschetz (1,1)-theorem
  • Kodaira's Embedding theorem, Chow's theorem & Kodaira's Vanishing theorem.

Prerequisites

A general background in mathematics (as obtained by a master degree in mathematics). A course in commutative algebra or algebraic geometry is not required. Basic courses in complex analysis, topology and differential geometry would be useful but I'll try to recall the necessary background.

Examiner

David Rydh

Time and place

Mostly Tuesdays, 13:15–15:00, KTH F11, see schedule below. Starting from March 17, the lectures are online on Zoom.

Examination

Home-work assignments, see schedule below.

Literature

Main text-book: (available electronically on SpringLink)
  • [H] D. Huybrechts, Complex geometry: an introduction, Springer, 2005 (Springer Link)
Other references:
  • [GH] P. Griffiths and J. Harris, Principles of algebraic geometry, John Wiley & Sons, 1978.
  • [N] A. Neeman, Algebraic and Analytic Geometry, Cambridge University Press, 2007.

Preliminary schedule

# Date Place Topic Ref HW
1Jan 28, 13–15KTH F11Introduction. Holomorphic functions. Analytic vs algebraic subsets. Manifolds & sheaves.[N, 1–2]
2Feb 4, 10–12*KTH 3418*Manifolds & sheaves. Meromorphic functions. Identity theorem.[H, 2.1, 1.1]HW2
3Feb 11, 13–15KTH F11Riemann extension theorem. Algebraic dimension. Projective spaces. Complex tori.[H, 2.1, 1.1]HW3
4Feb 18, 13–15KTH F11Inverse/implicit function theorem. Affine and projective hypersurfaces. Complete intersections.[H, 2.1, 1.1]
5Feb 25, 13–15KTH F11Vector bundles[H, 2.2]
Mar 3No lecture
6Mar 10, 13–15KTH F11Line bundles on projective space: tautological and canonical[H, 2.2, 2.4]HW6
7Mar 17, 13–15ZoomDivisors[H, 2.3]
8Mar 24, 13–15ZoomDivisors II[H, 2.3]
9Mar 31, 13–15Zoom
10Apr 7, 13–15Zoom
11
12
13
14
...