3rd IST Lecture Series in Algebraic Geometry & Physics
Vector Bundles on Algebraic Curves
Lisbon, I.S.T., 7/01/2002-11/01/2002
1 - Quotients in algebraic geometry,Monday, 7th:
2 - Vector bundles on algebraic curves: an overview,Tuesday,
3 - Construction of moduli spaces of vector bundles,Wednesday,
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
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
known about the moduli spaces used to classify bundles (more details
Lectures 3 and 4).
3. Construction of moduli spaces of vector bundles.
This lecture will describe the construction of moduli spaces using
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
obtain this information. This aspect of moduli spaces has been of
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
less thoroughly than the topology. This lecture will describe some
of these geometrical aspects, especially the Segre stratification
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
No specific references; see under Lectures 3, 4, 5.
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
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),
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
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 lectures will be held at
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
For any questions contact us:
João Pimentel Nunes