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.
- 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.
PrerequisitesA 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.
Time and placeNow on Wednesdays, 10:15–12:00, see schedule below, on Zoom (655 6200 8973).
ExaminationHome-work assignments, see schedule below.
LiteratureMain text-book: (available electronically on SpringLink)
- [H] D. Huybrechts, Complex geometry: an introduction, Springer, 2005 (Springer Link)
- [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.)
|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]|
|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]|