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Thesis for the Degree of Philosophy

Multipoint Okounkov bodies,

strong topology of

ω-plurisubharmonic functions and Kähler-Einstein

metrics with prescribed

singularities

Antonio Trusiani

Department of Mathematical Sciences Chalmers University of Technology

& University of Rome Tor Vergata

Göteborg, 2020

Supervisors: prof. Stefano Trapani (University of Rome Tor Vergata), prof. David Witt Nyström (Chalmers University of Technology).

Final Defence: 29th September 2020, a.a. 2019/2020. Tor Vergata: XXXII◦ ciclo, tutor prof. F. Bracci, coordinatore prof. A. Braides.

Chalmers: assistant supervisor prof. B. Berndtsson, examiner prof. R. Berman.

Multipoint Okounkov bodies, strong topology of ω-plurisubharmonic functions and Kähler-Einstein metrics with prescribed singularities

Antonio Trusiani Göteborg, 2020 ISBN 978-91-7905-372-7

© Antonio Trusiani, 2020

Doktorsavhandlingar vid Chalmers tekniska högskola Ny series nr 4839 ISSN 0346-718X

Division of Algebra and Geometry Department of Mathematical Sciences Chalmers University of Technology SE-412 96 Göteborg Sweden

E-mail: [email protected]

Department of Mathematical Sciences University of Rome Tor Vergata IT-00133 Roma Italy

E-mail: [email protected]

Typeset with LATEX. Printed in Gothenburg, Sweden 2020

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Multipoint Okounkov bodies, strong topology of ω-plurisubharmonic functions and

Kähler-Einstein metrics with prescribed singularities

Antonio Trusiani

Abstract

The most classical topic in Kähler Geometry is the study of Kähler-Einstein metrics as solution of complex Monge-Ampère equations. This thesis principally regards the investigation of a strong topology for ω-plurisubharmonic functions on a �xed com- pact Kähler manifold (X,ω), its connection with complex Monge-Ampère equations with prescribed singularities and the consequent study of singular Kähler-Einstein metrics. However the �rst part of the thesis, Paper I, provides a generalization of Okounkov bodies starting from a big line bundle over a projective manifold and a bunch of distints points. These bodies encode renowned global and local invariants as the volume and the multipoint Seshadri constant. In Paper II the set of all ω-psh functions slightly more singular than a �xed singular- ity type are endowed with a complete metric topology whose distance represents the analog of the L1 Finsler distance on the space of Kähler potentials. These spaces can be also glued together to form a bigger complete metric space when the sin- gularity types are totally ordered. Then Paper III shows that the corresponding metric topology is actually a strong topology given as coarsest re�nement of the usual topology for ω-psh functions such that the relative Monge-Ampère energy be- comes continuous. Moreover the main result of Paper III proves that the extended Monge-Ampère operator produces homeomorphisms between these complete metric spaces and natural sets of singular volume forms endowed their strong topologies. Such homeomorphisms extend Yau's famous solution to the Calabi's conjecture and the strong topology becomes a signi�cant tool to study the stability of solutions of complex Monge-Ampère equations with prescribed singularities. Indeed Paper IV introduces a new continuity method with movable singularities for classical families of complex Monge-Ampère equations typically attached to the search of log Kähler- Einstein metrics. The idea is to perturb the prescribed singularities together with the Lebesgue densities and asking for the strong continuity of the solutions. The results heavily depend on the sign of the so-called cosmological constant and the most di�cult and interesting case is related to the search of Kähler-Einstein metrics on a Fano manifold. Thus Paper V contains a �rst analytic characterization of the existence of Kähler-Einstein metrics with prescribed singularities on a Fano man- ifold in terms of the relative Ding and Mabuchi functionals. Then extending the Tian's α-invariant into a function on the set of all singularity types, a �rst study of the relationships between the existence of singular Kähler-Einstein metrics and

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genuine Kähler-Einstein metrics is provided, giving a further motivation to study these singular special metrics since the existence of a genuine Kähler-Einstein metric is equivalent to an algebrico-geometric stability notion called K-stability which in the last decade turned out to be very important in Algebraic Geometry.

Keywords: Kähler Geometry, Complex Monge-Ampère equations, Pluripotential theory, Kähler-Einstein metrics, Canonical metrics, Fano manifolds, Okounkov bod- ies, Seshadri constant, Kähler packing.

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Preface

This thesis consists of the following papers:

Paper I Antonio Trusiani, �Multipoint Okounkov bodies� , arxiv preprint, https://arxiv.org/abs/1804.02306;

Paper II Antonio Trusiani, �L1 metric geometry of potentials with prescribed singularities on compact Kähler manifolds� , arxiv preprint, https://arxiv.org/abs/1909.03897;

Paper III Antonio Trusiani, �The strong topology of ω-plurisubharmonic functions� , arxiv preprint, https://arxiv.org/abs/2002.00665;

Paper IV Antonio Trusiani, �Continuity method with movable singularities for classical complex Monge-Ampère equations� , arxiv preprint, https://arxiv.org/abs/2006.09120;

Paper V Antonio Trusiani, �Kähler-Einstein metrics with prescribed singularities on Fano manifolds� , arxiv preprint, https://arxiv.org/abs/2006.09130.

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https://arxiv.org/abs/1804.02306 https://arxiv.org/abs/1909.03897 https://arxiv.org/abs/2002.00665 https://arxiv.org/abs/2006.09120 https://arxiv.org/abs/2006.09130

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Acknowledgments

A PhD thesis does not re�ect the real journey of a PhD student. Mine was not easy at all, although I can bet almost all PhD students usually claim the same. However you all I am going to thank in the sequel only know tips of icebergs of the feelings, problems, doubts, grati�cations and fears I have experienced during the last four years. Anyway I know it would have been impossible for me to conclude this jour- ney without your support and your presence. You all were helping me bringing my burdens during the the steepest climbing and getting closer during the darkest silent nights. So, thank you.

Grazie a Stefano Trapani. Il merito (o la colpa) del mio dottorato di ricerca è in gran parte il tuo. Mi hai supportato �n dall'ultimo anno della magistrale e sei stato prima di tutto un advisor con i contro�occhi, che mi ha contattato più di una volta durante i miei periodi di silenzio. Non è una cosa che gli advisors solitamente fanno. Inoltre sei sempre stato presente nel darmi consigli matematici e non. Un ottimo amico. Spero di poter lavorare insieme in futuro come tu ben sai. �Tack� to David Witt Nyström. I am sorry to not have learnt Swedish to thank you in your �rst language but you know the reasons. Thank you to have welcomed me in Chalmers as your PhD student although we basically did not know each other and there were a lot of bureaucracy problems. Thank you to have been also a patient advisor full of suggestions and advice, and to have spent a lot of time trying to teach me how to write a good introduction of a paper and to set its structure. Thanks to prof. S. Kolodziej for accepting to be the opponent of this thesis, as to the members of committee: prof. C. Arezzo, prof. L. Turowska, prof. A. Sola and prof. C. Spotti. Grazie ai miei fratelli matematici di Roma, Simone Diverio ed Eleonora Di Nezza, per essere stati presenti per domande matematiche e per consigli sul mondo della ricerca sia prima dell'inizio del mio dottorato che durante. Riguardo Tor Vergata, vorrei ringraziare i dottorandi, i postdocs e i professori in- contrati nelle (poche) mie presenze in dipartimento. In particolare grazie ai compo- nenti del gruppo whatsapp DenteTor per piacevoli pranzi insieme, ai miei compagni di u�cio Gianluca, Antonio e Davide per avermi sopportato, a Matteo e Josias per

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interessanti discussioni matematiche ed a Daniele per le conversazioni random. About Chalmers, I would like to thank the group of complex geometry and the oth- ers PhD students, postdocs and professors I met during these years: in particular thank to (my mathematical brother) Jakob, (my o�cemates) Jimmy and Mingchen, Mattias, Richard, Hossein, Bo, Elizabeth, Martin, Lucas, Zakarias, Mats, Håkan, Jincao, Jonathan, Valentina, Kristian, Umberto and Simone. You made me feel relaxed at the department, and with some of you I spent jovial afterworks and other beautiful moments. A special thank to my examiner Robert Berman to have con- tributed to make this joint PhD possible.

Grazie a voi, Amici di sempre. Quelli del paesino: Stefano, Alessio, Marco e Andrea. Quelli del liceo: Fabio e Michele. E il corridore Daniele. Siete stati, siete e sarete sempre presenti nella mia vita. Voi mi conoscete meglio di quanto io potrò mai... e mi siete ancora amici! Grazie anche per essere stati degli autisti impeccabili nei miei frequenti viaggi in aereoporto ad orari assurdi. Grazie ad Alighieri e al gruppo di Atletica per avermi aiutato a scaricare l'inevitabile stress tramite sudore e crampi. Thank to all my new friends I have met in Göteborg, who made me feel happy a lot of times (maybe too many times). Specially thank to Iris & Suraj, catalanasss Berta, cocoach Antonio, Milo Normale, orsetto/