Deformation theory of complex structures

Research seminar organised by Richard Cleyton and Simon Chiossi in the winter semester 2006/07

This series of lectures/seminars are intended as an introduction to the topic (what are deformations, why are they relevant, what's all the fuss about &c). The course will be based on one hour seminars held by participants, and is open to anybody interested, experts, non-experts and students: thus if you wish to prepare a seminar, please contact Richard or Simon. Lecturers are NOT supposed to be necessarily experts though, our aim with this whole thing only being to learn about the subject. For our own selfish sake we would like lectures to be presented in english (but by no means be scared away by this).

Students who wish to use this course for credits (Leistungsschein) are required to present some material. All speakers should in any case talk to Richard/Simon in order to coherently follow the plan and not set off in other directions (which, given the borderline topic, is all the more easy).
The course consists of 8-9 two-hour lectures, every fortnight more or less. Each lecture should be divided into two seminars of 45 minutes each, where possible, for a total of 16-18 hours altogether.

Where: RUD 25, 1'315 (Seminarraum)

Schedule:

Kick-off: 18 October 2006, from 11.00 to 13.00. Meetings are always on Wednesday, 11.15-12.00, 12.15-13.00, as shown below.

Weds. 18 October
- Simon Chiossi: Complex manifolds, families of compact complex mfd.s (M_t)_B=(M,B,π), Examples (complex tori). Notes
- Richard Cleyton: Local triviality, infinitesimal deformations, H¹(M,Θ), Examples (tori, CP¹). Notes
Weds. 1 November
- Ivan Minchev: Thm: h_t¹:=dimH¹(M_t,Θ_t) indep. of t & dM_t/dt=0 ⇒ loc. triviality ([2]: Thms 4.1,4.2,4.3, pp.195-200, 352-354). Continues on 15 November!
- Svatopluk Krysl: Thm: The map t → h_t¹ is upper semi-continuous ([2]: Thm 4.4 pp.352)
Weds. 15 November
- Ivan Minchev: Part 2
- Stefan Murygin: h_t¹=0 implies rigidity ([Frölicher-Nijenhuis], [1]: Thm 4.5) + an example: Hopf surfaces ([1]: pp. 49-50, 69-71, 208)
Weds. 6 December
- Christian Merdon: Another application: the rigidity of CPⁿ ([2]: p.216 + [5]: the Euler sequence)
- Radoslaw Balcer: The obstruction space H²(M,Θ) and necessary conditions for existence ([2]: pp.210-214) Notes
Weds. 10 January 2007: (Review lectures)
- Tobias Florek: Cech sheaves and exact sequences in cohomology + examples
- Volker Schloßhauer: Dolbeault cohomology I: d-bar operator δ and (p,q)-forms, Dolbeault complex and cohomology;
Weds. 24 January: (Review lecture)
- Volker Schloßhauer: Dolbeault cohomology II: harmonic representatives and the isomorphism H^q(M,&Theta^p)=H^p,q_δ(M) Notes
Weds. 31 January:
- Simon Chiossi: complete families and the completeness thm ([2]: pp. 228, 230, 285-304)
- Simon Chiossi: The form φ(t) ∈ T⊗ &Lambda^0,1 defining the complex structure M_t ([1]: pp.148-155)
Weds. 7 February:
- Richard Cleyton: Existence thm. ([Kodaira-Nirenberg-Spencer], [1]: pp.155-165)
Weds. 14 February
- Richard Cleyton: Explicit examples with obstructions (Kodaira surface)
- Simon Chiossi: Summary

Richard and I wish to thank everybody for actively participating and putting up with our deficiencies.
Please feel free to come and see us if you need extra input on some material or background topic, like those marked ♣.

Synopsis:

Complex analytic families of compact complex manifolds and infinitesimal deformations
Issues of rigidity and local triviality of families (Kodaira-Spencer, Frölicher-Nijenhuis)
Unobstructed existence (Kodaira-Nirenberg-Spencer)
Completeness
Obstructed existence (Kuranishi)
(?) Stability, applications to nilmanifolds and other recent results...

Literature:

[*] S.G.Chiossi, R.Cleyton: Deformation theory of complex structures, notes of the lectures, contain big section with the necessary backgrounds (on their way).

[1] K.Kodaira, J.Morrow: Complex manifolds. Reprint of the 1971 edition with errata. AMS Chelsea Publishing, Providence, RI, 2006. ISBN 0-8218-4055-X (very concise - almost no proofs - but has the outline of the course)
[2] K.Kodaira: Complex manifolds and deformation of complex structures. Translated from the 1981 Japanese original by Kazuo Akao. Reprint of the 1986 English edition. Classics in Mathematics. Springer-Verlag, Berlin, 2005. ISBN 3-540-22614-1 (very detailed, exposition unfortunately rather disorientating)
@ ZB-Naturwissenschaften: SK 350
[3] M.Kuranishi: Deformations of compact complex manifolds. Séminaire de Mathématiques Supérieures, No. 39 (Été 1969). Les Presses de l'Université de Montral, Montreal, Que., 1971.
[4] M.Manetti: Lectures on deformations of complex manifolds (click on 'Notes') (algebraic approach, but rather self-contained)

Backgrounds:

Though the lectures are open to anybody, we strongly recommend knowledge of:
   ♣ several complex variables.
   ♣ complex manifolds.
   ♣ some sheaf theory and cohomology.

[5] P.Griffiths-J.Harris: Principles of algebraic geometry, Reprint of the 1978 original. Wiley Classics Library. Wiley & Sons, New York, 1994, ISBN: 0-471-05059-8.
[6] R.O.Wells: Differential Analysis on Complex Manifolds, Second edition. Graduate Texts in Mathematics 65. Springer-Verlag, New York-Berlin, 1980. x+260 pp. ISBN: 0-387-90419-0.

Other material

Anybody highly encouraged to post relevant stuff here. Please make suggestions.

Notes on the correspondence between complex tori and elliptic curves, via Weierstraß' ℘-function (by SGC)
Andrei Moroianu: Kähler Geometry LMS Student Texts 69, 2007.

We welcome suggestions and constructive criticism

Questions: please e-mail

Richard : cleyton@math.hu-berlin.de or Simon : sgc@math.hu-berlin.de