Sommersemester 2021: Seminar Heisenberg algebras and Hilbert schemes of points on surfaces

Prof. Dr. Katrin Wendland
Dr. Mara Ungureanu

Wann und wo: Di 14 - 16, HSII, Albertstrasse 23b

Vorbesprechung: Dienstag, 09.02.2021, 14:00 - 16:00 im virtuellen bbb-Raum SR125.
Bei Interesse bitte Voranmeldung bis 05.02.2021 per e-Mail an Dr. Mara Ungureanu.

Topic:
Moduli spaces parametrising objects associated with a given variety X are a rich source of spaces with interesting structures. Not only do they inherit various properties from X, but they may also bring to light hidden structures that cannot be accessed via the original variety.
The purpose of this seminar is to explore an example of this phenomenon. We shall introduce the Hilbert scheme X^[n] of a smooth surface X parametrising n-tuples of points of X. We shall study its geometric properties and establish, among other things, the non-trivial fact that X^[n] is itself a smooth variety. One of the new and surprising features of X^[n] is that one can construct a representation of the so-called Heisenberg algebra on its cohomology groups. We shall see that this is not just a pretty result, but it is also very useful in understanding how characteristic classes of line bundles on X relate to those of certain interesting vector bundles on X^[n]. Moreover, this construction also provides a connection to mathematical physics as it gives a geometric realisation of a formula for the Euler characteristic of the moduli space of N=4 Yang-Mills instantons, though this will be beyond the scope of our seminar.

Literatur:
Die Links führen auf Webseiten, von denen aus dem Universitätsnetz die jeweiligen Referenzen zugänglich sind. Falls kein Link gesetzt ist, finden Sie die Referenz in der Bibliothek des Mathematischen Institutes Freiburg.

• W. Fulton, J. Harris, Representation Theory, New York, NY: Springer, 2004
• J. Harris, Algebraic Geometry, New York, NY: Springer, 1992
• D. Huybrechts, Complex Geometry, Springer 2005
• J. Kock, I. Vainsencher, An invitation to quantum cohomology, Birkäuser, 2007
• M. Lehn, Chern classes of tautological sheaves on Hilbert schemes of points on surfaces, Inventiones Mathematicae 136 (1999) 157--207; arXiv:math/9803091
• E. Miller, B. Sturmfels, Combinatorial Commutative Algebra, Springer 2005
• H. Nakajima, Lectures on Hilbert schemes of points on surfaces, AMS 1999
• F. Rota, Symmetric products and the Hilbert scheme of points
• K. Smith, L. Kahanpää, P. Kekäläinen, W. Traves, An Invitation to Algebraic Geometry, New York, NY: Springer, 2000

Vortragsprogamm:
Das Vortragsprogramm finden Sie hier.
Die Vorträge können auf Deutsch oder auf Englisch präsentiert werden.

Tutorium: Dr. Mara Ungureanu