Publication: GIT characterizations of Harder-Narasimhan filtrations.
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2013-07-23
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Universidad Complutense de Madrid
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Abstract
Esta tesis estudia la relación entre la filtración de Harder-Narasimhan y cierta noción de máxima inestabilidad GIT. En un problema de móduli cuya construcción del espacio de móduli se realice a través de la Teoría Geométrica de Invariantes (GIT), se suele imponer una condición de estabilidad en los objetos, para poder obtener un espacio que los parametrice. Diremos que un objeto es inestable si contradice esta condición de estabilidad, en cuyo caso tendrá asociado, de forma natural, una filtración canónica, llamada de Harder-Narasimhan, que proporciona una noción de máxima forma de desestabilizar un objeto inestable.
Por otra parte, al construir el espacio de móduli como un cociente GIT, aparece de forma natural una noción de estabilidad para las órbitas. Un objeto inestable produce un punto GIT inestable y usando el criterio de Hilbert-Mumford, existen subgrupos uniparámetricos que satisfacen cierta condición numérica. A estos los llamamos subgrupos desestabilizantes. La cuestión natural es si hay un subgrupo distinguido entre todos ellos. Kempf da una noción de subgrupo máximamente desestabilizante, y encuentra que es único (salvo conjugación por cierto parabólico). Este producirá una filtración del objeto original, y la pregunta natural es si esta filtración coincide con la de Harder-Narasimhan.
En esta tesis se responde afirmativamente a esta pregunta para diferentes problemas de móduli: haces coherentes sin torsión, pares holomorfos, haces de Higgs, tensores de rango 2 y representaciones de un carcaj.
El interés de este trabajo es que muestra que podemos considerar que la filtración de Harder-Narasimhan viene de la teoría GIT. Por lo tanto, en los problemas de móduli donde no hay ya una noción de filtración de Harder-Narasimhan, se puede tomar por tal a la filtración de Kempf, como ocurre para los tensores de rango 2. Esto puede tener aplicaciones muy interesantes, pues la filtración de Harder-Narasimhan es una herramienta muy potente para estudiar propiedades de los espacios de móduli.
This thesis studies the relation between the Harder-Narasimhan filtration and certain notion of GIT maximal unstability.
In a moduli problem where the construction of the moduli space is made through the Geometric Invariant Theory (GIT), one usually imposes a stability condition in the objects, in order to obtain a space which parametrizes them. We say that an object is unstable if it contradicts the stability condition, in which case it will have naturally associated a canonical filtration, called the Harder-Narasimhan filtration, providing the maximal way of destabilizing an unstable object. On the other hand, when constructing the moduli space as a GIT quotient, it appears in a natural way a notion of stability for the orbits. An unstable object produces a GIT unstable point and using the Hilbert-Mumford criterion, there exist 1-parameter subgroups which satisfy certain numerical condition. We call them destabilizing subgroups. The natural question is whether there exists a distinguish subgroup among all of them. Kempf gives a notion of maximally destabilizing subgroup, and finds that it is unique (up to conjugation by a parabolic subgroup). This one produces a filtration of the original object, and the natural question is whether this filtration coincides with the Harder-Narasimhan one. This thesis gives a positive answer to this question in different moduli problems: torsion free coherent sheaves, holomorphic pairs, Higgs sheaves, rank 2 tensors and quiver representations. Out of this work, we can consider that the Harder-Narasimhan filtration comes from GIT theory. Hence, in moduli problems where there is no notion of Harder-Narasimhan filtration, we can define the Harder-Narasimhan filtration to be the Kempf one, as it happens in the rank 2 tensors case. This can have interesting applications, given that the Harder-Narasimhan filtration is a strong tool to study properties of moduli spaces.
In a moduli problem where the construction of the moduli space is made through the Geometric Invariant Theory (GIT), one usually imposes a stability condition in the objects, in order to obtain a space which parametrizes them. We say that an object is unstable if it contradicts the stability condition, in which case it will have naturally associated a canonical filtration, called the Harder-Narasimhan filtration, providing the maximal way of destabilizing an unstable object. On the other hand, when constructing the moduli space as a GIT quotient, it appears in a natural way a notion of stability for the orbits. An unstable object produces a GIT unstable point and using the Hilbert-Mumford criterion, there exist 1-parameter subgroups which satisfy certain numerical condition. We call them destabilizing subgroups. The natural question is whether there exists a distinguish subgroup among all of them. Kempf gives a notion of maximally destabilizing subgroup, and finds that it is unique (up to conjugation by a parabolic subgroup). This one produces a filtration of the original object, and the natural question is whether this filtration coincides with the Harder-Narasimhan one. This thesis gives a positive answer to this question in different moduli problems: torsion free coherent sheaves, holomorphic pairs, Higgs sheaves, rank 2 tensors and quiver representations. Out of this work, we can consider that the Harder-Narasimhan filtration comes from GIT theory. Hence, in moduli problems where there is no notion of Harder-Narasimhan filtration, we can define the Harder-Narasimhan filtration to be the Kempf one, as it happens in the rank 2 tensors case. This can have interesting applications, given that the Harder-Narasimhan filtration is a strong tool to study properties of moduli spaces.
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Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, Departamento de Álgebra, leída el 07/06/2013