3rd IST Lecture Series in Algebraic Geometry & Physics

Vector Bundles on Algebraic Curves

Lisbon, I.S.T., 7/01/2002-11/01/2002
(Room P3.10)

by

Peter NEWSTEAD
(University of Liverpool)

Schedule

1 - Quotients in algebraic geometry,Monday, 7th: 11-12:30

2 - Vector bundles on algebraic curves: an overview,Tuesday, 8th: 11-12:30

3 - Construction of moduli spaces of vector bundles,Wednesday, 9th: 11-12:30

4 - Topology of the moduli spaces,Thursday, 10th: 9:30 -11

5 - Geometry of the moduli spaces,Friday, 11th: 11-12:30

SHORT DESCRIPTION OF THE LECTURES

1. Quotients in algebraic geometry.

There are various methods of constructing quotients in algebraic

geometry. The lecture will concentrate on Mumford's Geometric Invariant

Theory (GIT).

2. Vector bundles on algebraic curves: an overview.

The basic classification of vector bundles on algebraic curves was

carried out 40 years ago. The lecture will describe this classification

in broad terms (more details in Lecture 3) and also survey what is now

known about the moduli spaces used to classify bundles (more details in

Lectures 3 and 4).

3. Construction of moduli spaces of vector bundles.

This lecture will describe the construction of moduli spaces using GIT.

4. Topology of the moduli spaces.

Much is now known about the topology of the moduli spaces, especially

their cohomology. This lecture will describe some of the methods used to

obtain this information. This aspect of moduli spaces has been of much

interest to theoretical physicists in connection with Seiberg-Witten

invariants and similar computations.

5. Geometry of the moduli spaces.

The geometry of the moduli spaces has also been studied, though perhaps

less thoroughly than the topology. This lecture will describe some

of these geometrical aspects, especially the Segre stratification and

Brill-Noether theory.

REFERENCES

Lecture 1
P. E. Newstead, Introduction to Moduli Problems and Orbit Spaces, Tata Institute Lecture Notes on Mathematics, Vol 51, 1978
I. Dolgachev, Introduction to Geometric Invariant Theory, Lecture Note series 25, Seoul National University, Research Institute of Mathematics Global Analysis Research Centre, Seoul, 1994
D. Mumford, J. Fogarty and F. Kirwan, Geometric Invariant Theory, 3rd. edition, Springer-Verlag, Berlin, 1994

Lecture 2
No specific references; see under Lectures 3, 4, 5.

Lecture 3
M. S. Narasimhan and C. S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. 82 (1965), 540-567
C. S. Seshadri, Fibres vectoriels sur les courbes algebriques, Asterisque 96, 1982
J. Le Potier, Lectures on vector bundles, Cambridge studies in advanced mathematics Vol. 54, CUP, 1997

Lecture 4
P. E. Newstead, Topological properties of some spaces of stable bundles, Topology 6 (1967), 241-262
P. E. Newstead, Characteristic classes of stable bundles of rank 2 over an algebraic curve, Trans. Amer. Math. Soc. 169 (1972), 337-345
M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. London A308 (1982), 523-615.
F. Kirwan, The cohomology ring of moduli spaces of bundles over Riemann surfaces, J. Amer. Math. Soc. 5 (1992), 853-906
M. Thaddeus, Conformal field theory and the cohomology of the moduli space of stable bundles, J. Diff. Geom. 35 (1992), 131-149
D. Zagier, On the cohomology of moduli spaces of rank two vector bundles over curves, Progr. Math. 129 (1995), 533-563
V. Yu. Baranovskii, Cohomology ring of the moduli space of stable vector bundles with odd determinant, Izv. Ross. Akad. Nauk Ser. Mat. 58 (1994), 204-210
A. D. King and P. E. Newstead, On the cohomology ring of the moduli space of rank 2 vector bundles on a curve, Topology 37 (1998), 407-418
B. Siebert and G. Tian, Recursive relations for the cohomology ring of moduli spaces of stable bundles, Turkish J. Math. 19 (1996), 131-144
R. Herrera and S. Salamon, Intersection numbers on moduli spaces and symmetries of a Verlinde formula, Comm. Math. Phys. 188 (1997), 521-534

Lecture 5
A. King and A. Schofield, Rationality of moduli of vector bundles on curves, Indag. Math. (N.S.) 10 (1999), 519-535
U. N. Bhosle, Moduli of orthogonal and spin bundles on hyperelliptic curves, Comp. Math. 51 (1984), 15-40
M. Teixidor i Bigas, Brill-Noether theory for stable vector bundles, Duke Math. J. 62 (1991), 385-400
L. Brambila-Paz, I. Grzegorczyk and P. E. Newstead, Geography of Brill-Noether loci for small slopes, J. Alg. Geom. 6 (1997), 645-669
V. Mercat, Le probleme de Brill-Noether pour les fibres stables de petite pente, J. Reine Angew. Math. 506 (1999), 1-14
L. Brambila-Paz, V. Mercat. P. E. Newstead and F. Ongay, Nonemptiness of Brill-Noether loci, Internat. J. Math. 11 (2000), 737-760
H. Lange and M. S. Narasimhan, Maximal subbundles of rank 2 vector bundles on curves, Math. Ann. 266 (1983), 55-72
L. Brambila-Paz and H. Lange, A stratification of the moduli space of vector bundles on curves, J. Reine Angew. Math. 499 (1998), 173-187
B. Russo and M. Teixidor i Bigas, On a conjecture of Lange, J. Alg. Geom. 8 (1999), 483-496

The VENUE

The lectures will be held at Instituto Superior Técnico , Av. Rovisco Pais, Lisbon, Edifício de Pós-Graduações, ROOM P3.10. (3rd Floor of the Mathematics Department). The mathematics department is located in building #5 in the map of the campus of the IST.

ORGANIZATION and SPONSORSHIP

Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Departamento de Matemática of Instituto Superior Técnico
Integrability, Moduli Spaces and String Theory (FCT-POCTI/33943/MAT/2000) Fundação para a Ciência e Tecnologia
CERN/P/FIS/40108/2000 Non-Perturbative Aspects of Fields and Strings

For any questions contact us:

Carlos Florentino
José Mourão
João Pimentel Nunes