Hauptseminar Geometrie

Alfonso Zamora-Saiz, PhD, Instituto Superior Técnico, Lisabon

Abstract. We will discuss constructions of moduli spaces in algebraic geometry by using Geometric Invariant Theory (GIT). When performing such constructions we usually impose a notion of stability for the objects we want to classify and another notion of GIT stability appears, then it is shown that both notions coincide. For an object which is unstable (i.e. contradicting the stability condition) there exists a unique canonical filtration, called the Harder-Narasimhan filtration. On the other hand, GIT stability is checked by 1-parameter subgroups by the classical Hilbert-Mumford criterion, and it turns out that there exists a unique 1-parameter subgroup giving a notion of maximal unstability in the GIT sense. We show how to prove that this special 1-parameter subgroup can be converted into a filtration of the object and coincides with the Harder-Narasimhan filtration, hence both notions of maximal unstability are the same. We will present the correspondence for the moduli problem of classifying coherent sheaves on a smooth complex projective variety (joint work with Tomás Gómez and Ignacio Sols). A similar treatment can be used to prove the analogous result for other moduli problems: holomorphic pairs, Higgs sheaves, rank 2 tensors, finite dimensional quiver representations. Also, similar ideas allow to study stability for (G,h)-constellations (joint work with Ronan Terpereau).

Dienstag, den 2. Juni 2015 um 13:00 Uhr, in INF288, HS5 Dienstag, den 2. Juni 2015 at 13:00, in INF288, HS5

Der Vortrag folgt der Einladung von The lecture takes place at invitation by Professor Anna Wienhard