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Moduli of Homomorphisms of Sheaves a Dissertation Submitted in Partial Moduli of homomorphisms of sheaves A Dissertation Submitted in Partial Fulfilment of the Requirements for the Degree of Doctor rerum naturalium to the Department of Mathematics and Computer Science of Freie Universität Berlin by Benedikt Vincent Josias Trageser Berlin, 2019 Supervisor: Prof. Dr. Victoria Hoskins Second examiner: Prof. Dr. Frances Kirwan Date of defense: 16.09.2019 Danksagung Für die exzellente Betreuung meiner Doktorarbeit und Begleitung meiner wissenschaftlichen Tätigkeit an der FU-Berlin bedanke ich mich bei Prof. Dr. Victoria Hoskins, für viele Anregungen, die Zweitkor- rektur und Anreise aus Oxford bei Frau Prof. Dr. Francis Kirwan. Ein besonderer Dank gilt meinen Mitdoktoranden Alejandra Rincon, Dominic Bunett und Hangxing Lin für den spannenden und kritischen Austausch. Unvergessen bleibt die Freude über Cyril, Sylke und Martin, die sich auf den langen Weg nach Berlin gemacht haben, um bei der Verteidigung meiner Doktorarbeit dabei zu sein. Einen herzlichen Dank meinen Eltern Christine und Günter. Abstract Moduli spaces arise in classification problem in algebraic geometry; typically when we try to classify geometric objects we find that they have discrete invariants but these are not sufficient to classify them. Thus we would like to be able to construct moduli spaces whose points correspond to equivalence classes of the objects to be classified (with fixed discrete invariants) and whose geometry reflects the way they can vary in families. This thesis studies moduli problems for homomorphisms of sheaves over a fixed projective variety X indexed by a quiver; that is, we are looking at representations of a quiver in the category of coherent sheaves over X where the sheaves indexed by the vertices of the quiver are fixed and it is only the homomorphisms between them indexed by the arrows of the quiver which vary. More precisely, we define a the moduli functor for homomorphisms of sheaves over a fixed projective variety X and show that the construction of a moduli space for homomorphisms of sheaves over X indexed by a quiver Q can be reduced to the construction of quotients for actions of the product of the automorphism groups of the sheaves over X labelled by the vertices of Q on affine varieties. Additionally we show that the automorphism groups of the sheaves over X are linear algebraic groups. In the case that these are reductive groups, such quotients can be constructed and studied using Mumford’s classical geometric invariant theory (GIT). However in general these automorphism groups are not reductive, so a significant part of this thesis studies ways in which Mumford’s GIT can be extended to actions of non-reductive linear algebraic groups on an affine variety, before applying them to representations of quivers in homomorphisms of semisimple sheaves over X: Abstract Modulräume entstehen bei der Betrachtung von Klassifikationsproblemen in der algebraischen Geometrie. Bei der Klassifizierung von geometrische Objekten ergibt sich gewöhnlich, dass die diskreten Invarianten nicht ausreichen, um diese geometrischen Objekte zu klassifizieren. Das Ziel ist Modulräume zu konstruieren, dessen Punkte mit den Äquivalenzklassen der zu klassifizierenden Objekte korrespondieren, wobei die diskreten Invarianten fixiert werden, und deren Geometrie das Verhalten von Familien der zu klassifizierenden Objekte reflektiert. Diese Arbeit betrachtet Modulprobleme für Homomorphismen von Garben über einer pro- jektiven Varietät X indiziert von einem Köcher; dass heißt, wir betrachten Darstellungen eines Köchers in der Kategorie der kohärenten Garben über X wobei wir die Garben fixieren, die von dem Punkten des Köchers indiziert werden, und nur die Homomorphismen, indiziert durch die Pfeile, variieren. Wir definieren Modulfunktoren für Homomorphismen von Garben über einer projektiven Varietät X indiziert von einem Köcher Q: Für diese Modulfunktoren zeigen wir, dass die Konstruktion von Modulräumen durch Quotienten für die Gruppenwirkung von Produkten der Automorphismengruppen der Garben über X auf affinen Varietäten, die die Homomorphismen parametrisieren realisiert werden kann. Darüber hinaus wird gezeigt, dass die Automorphismengruppen von Garben über X lineare al- gebraische Gruppen sind. Im Falle, dass diese Gruppen reduktiv sind, können diese Quotienten mit Hilfe von Mumfords klassischer geometrischer Invariantentheorie (GIT) studiert und konstruiert werden. Im allgemeinen Fall sind diese Automorphismengruppen nicht reduktiv, so dass ein signifik- anter Teil dieser Arbeit Wege studiert, um Mumfords GIT auf Gruppenwirkungen nicht-reduktiver linear algebraischer Gruppen auf affinen Varietäten anzuwenden. Im letzten Teil der Arbeit neh- men wir an das alle Garben halbeinfach sind. Unter dieser Annahme werden die Ergebnisse zur nicht-reduktiven GIT auf Modulprobleme für Homomorphismen indiziert von einem Köcher Q über einer projektiven Varietät X angewandt. Contents 1 The moduli problem for sheaf homomorphisms 9 1.1 Quiver representations in an abelian category..................... 9 1.1.1 Quivers ...................................... 9 1.1.2 Quiver representations.............................. 11 1.2 The moduli functor.................................... 12 1.3 The geometric invariant theory set up for the moduli problem............ 19 2 Background on Geometric Invariant Theory 21 2.1 Algebraic groups ..................................... 21 2.2 Linear algebraic monoids................................. 24 2.3 Group actions and quotients of group actions ..................... 24 2.4 Reductive geometric invariant theory.......................... 31 2.5 Non-reductive geometric invariant theory ....................... 35 2.5.1 Several notions of (semi)stability for linear algebraic groups......... 35 2.5.2 The U^-Theorem.................................. 37 2.6 Linearised actions on affine varieties .......................... 40 2.7 Geometric invariant theory for affine varieties..................... 42 3 Actions of unipotent groups on affine varieties 47 3.1 Background on Ga-actions................................ 50 3.2 On trivial Ga-quotients of affine varieties........................ 51 3.3 On trivial U-quotients of affine varieties ........................ 54 4 Non-reductive geometric invariant theory for affine varieties 61 4.1 Classical approach to construct quotients........................ 62 4.2 Using quotienting in stages and a certain embedding to construct quotients . 72 4.2.1 The proof of Theorem 4.2.5........................... 75 4.2.2 Towards potential generalisations of Theorem 4.2.5.............. 87 4.3 Examples ......................................... 90 5 Towards moduli spaces for sheaf homomorphisms indexed by a quiver 95 5.1 Description of Aut(E) for a coherent sheaf....................... 96 5.1.1 The description by Drézet and Trautmann................... 96 5.1.2 Automorphism groups of quiver representations in the category of finite dimensional vector spaces............................ 97 5.1.3 Grading subgroups................................ 99 5.2 Moduli spaces for sheaf homomorphisms indexed by a quiver . 101 5.2.1 Several approaches to moduli for sheaf homomorphisms . 101 5.2.2 Moduli for homomorphisms of sheaves indexed by a quiver with loops . 110 5.2.3 Moduli spaces for sheaf homomorphisms indexed by a quiver via reductive Geometric Invariant Theory...........................118 A Some results related to Geometric Invariant Theory 123 A.1 Polyhedral cones .....................................123 A.2 Ga-actions on affine varieties ..............................124 A.3 Białynicki-Birula-decomposition.............................127 A.3.1 Categorical Białynicki-Birula-decomposition for Gm . 128 A.3.2 Categorical Białynicki-Birula-decomposition for certain reductive groups . 129 B The work of Drézet and Trautmann 131 B.1 An explicit description of the automorphism groups..................131 B.2 An extension of the action................................133 B.3 The notion of stability ..................................136 B.4 Examples .........................................138 1 B.5 Comparison with reductive GIT.............................139 Bibliography 143 Selbständigkeitserklärung 147 2 Introduction The aim of this thesis is to study the moduli problem for homomorphisms of sheaves over a project- ive variety indexed by a quiver. This problem has been studied previously in two special cases. If we fix the projective variety Spec k then we are considering quiver representations in the category of finite dimensional vector spaces; this moduli problem has been studied by King (see [44]). Spe- cialising in the other direction by fixing the quiver consisting of an arrow connecting two distinct vertices we obtain the moduli problem of sheaf homomorphisms studied by Drézet and Trautmann (see [25]). In both cases the problem is approached from the perspective of geometric invariant theory. The purpose of Mumford’s reductive geometric invariant theory (see [30]) is to construct quotients for actions of reductive groups on varieties. For moduli of quiver representations the groups are all reductive and reductive geometric invariant theory can be directly applied. The moduli problem of sheaf homomorphisms involves non-reductive groups. The approach of Drézet and Trautmann is to translate this problem of non-reductive geometric invariant theory into a problem of reductive geometric invariant theory. More recently Bérczi, Doran, Hawes and Kirwan (see [9]) developed non-reductive geometric invariant theory for algebraic actions of linear
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