SF3612 Complex algebraic geometry
(Spring 2020)

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

Now on Wednesdays, 10:15–12:00, see schedule below, on Zoom (655 6200 8973).

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

(The lecture notes from lectures 1—6 have kindly been contributed by Francesca Tombari.)

#	Date	Place	Topic	Ref	HW
1	Jan 28, 13–15	KTH F11	Introduction. Holomorphic functions. Analytic vs algebraic subsets. Manifolds & sheaves.	[N, 1–2]
2	Feb 4, 10–12*	KTH 3418*	Manifolds & sheaves. Meromorphic functions. Identity theorem.	[H, 2.1, 1.1]	HW2
3	Feb 11, 13–15	KTH F11	Riemann extension theorem. Algebraic dimension. Projective spaces. Complex tori.	[H, 2.1, 1.1]	HW3
4	Feb 18, 13–15	KTH F11	Inverse/implicit function theorem. Affine and projective hypersurfaces. Complete intersections.	[H, 2.1, 1.1]
5	Feb 25, 13–15	KTH F11	Vector bundles	[H, 2.2]
—	Mar 3	—	No lecture
6	Mar 10, 13–15	KTH F11	Line bundles on projective space: tautological and canonical	[H, 2.2, 2.4]	HW6
7	Mar 17, 13–15	Zoom	Divisors	[H, 2.3]
8	Mar 24, 13–15	Zoom	Divisors II	[H, 2.3]
—	Mar 31	—	No lecture
—	Apr 7	—	No lecture
—	Apr 14	—	No lecture
9	Apr 21, 13–15	Zoom	Sections of line bundles	[H, 2.3]
2021
10	Jan 27, 10–12	Zoom	Meromorphic maps, bimeromorphic maps and blow-ups	[H, 2.5]	HW10
11	Feb 3, 10–12	Zoom	Blow-ups	[H, 2.5]
12	Feb 10, 10–12	Zoom	Projectivity
—	Feb 17	—	No lecture
—	Feb 24	—	No lecture
13	Mar 3, 10–12	Zoom	Hermitian metrics and positive line bundles	[H, 3.1, 3.2, 4.1]
14	Mar 10, 10–12	Zoom	On Homework and Kodaira vanishing and Kodaira embedding theorem	[H, 5.2, 5.3]

Department of Mathematics — Royal Institute of Technology (KTH)