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)

**[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] |