Curriculum Vitæ

Daniele Angella

(Last updated on August 7, 2014)

Personal informations

Name and surname Daniele Angella

Date and place of birth October 17, 1985, Pontremoli (MS), ItalyItalian tax id no. NGL DNL 85R17 G870 RCitizenship ItalianGender male

Residence via iv Novembre, 18 – 54027 Pontremoli (MS) – ItaliaTel. (+39) 0187 832182, mobile (+39) 338 8738311

Actual position research fellowInstitution Istituto Nazionale di Alta Matematicaat Dipartimento di Matematica e Informatica, Universita di Parma

Professional address Dipartimento di Matematica e Informatica, Universita di Parma– Parco Area delle Scienze 53/A – 43124 Parma – Italytel. (+39) 338 8738311, tel. (+39) 0521 906900, fax (+39) 0521906950

E-mail daniele.angella@math.unipr.it

angella@mail.dm.unipi.it

daniele.angella@gmail.com

URL http://www.dm.unipi.it/~angella

http://people.math.unipr.it/daniele.angella

Positions

Actual position

01/12/2013-today

Research fellow (Assegno di Ricerca) by Istituto Nazionale di Alta Matematica, atDipartimento di Matematica e Informatica of Universita di Parma, under the adviceof prof. Adriano Tomassini.

Future positions

01/12/2014-30/11/2016

Junior Visiting Position at Centro di Ricerca Matematica “E. de Giorgi” in Pisa.

Past positions

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01/09/2013-30/11/2013

Grant for abroad research activity by Istituto Nazionale di Alta Matematica, at De-partamento de Matematicas of the Universidad de Zaragoza.

01/01/2010-21/01/2013

PhD student in Mathematics, Dipartimento di Matematica, Universita di Pisa; thesison Cohomological aspects of non-Kahler manifolds, advisor prof. Adriano Tomassini(Universita di Parma), date January 21, 2013, grade eccellente (excellent).

Education

21/01/2013 PhD in Mathematics, at Dipartimento di Matematica of Universita di Pisa, Scuoladi Dottorato “Galileo Galilei”, xxv ciclo; thesis on Cohomological aspects of non-Kahler manifolds, advisor prof. Adriano Tomassini (Universita di Parma), researchdeveloped at Universita di Pisa and during a period at Institut Fourier of UniversiteJoseph Fourier in Grenoble, invited by prof. J.-P. Demailly; date January 21, 2013,grade eccellente (excellent).

14/07/2009 Master degree in Pure and Applied Mathematics (Matematica Pura e Appli-cata, classe 45/S), Universita degli Studi di Parma; research thesis on Proprietacoomologiche di varieta quasi-complesse e deformazioni (Cohomological properties ofalmost-complex manifolds and deformations), advisor prof. Adriano Tomassini, grade110/110 e lode (110/110 with honour, highest possible grade in the Italian system).

21/09/2007 Bachelor degree in Mathematics (Matematica, classe 32), Universita degli Studi diParma; thesis on Varieta simplettiche speciali e loro coomologia (Special symplecticmanifolds and their cohomology), advisor prof. Adriano Tomassini, grade 110/110 elode (110/110 with honour, highest possible grade in the Italian system).

2004 High school diploma, Liceo Scientifico a indirizzo pni (high school focused mainlyon scientific subjects and computer science) “Leonardo da Vinci” in Villafranca inLunigiana (MS), grade 100/100 (highest possible grade in the Italian system).

Awards and grants

2014 Winner of Junior Visiting Position at Centro di Ricerca Matematica “E. de Giorgi”in Pisa.

2013 “Michele Cuozzo” Prize 2013 for PhD thesis, Universita di Roma Tor Vergata.2013 Winner of grants for research activity abroad (three months at Departamento de

Matematicas of the Universidad de Zaragoza) and in Italy (one year at Dipartimentodi Matematica e Informatica of the Universita di Parma, director prof. AdrianoTomassini) by Istituto Nazionale di Alta Matematica.

2009 Winner of grants for PhD studies at: Universita di Pisa (2nd rank); Universita diRoma “La Sapienza” (2nd rank); Universita di Roma “Tor Vergata” (1st rank exæquo).

2004 Winner (19th rank) of a grant as undergraduate student of Science (Chemistry) atScuola Normale Superiore in Pisa.

2004 Winner (17th rank) of a scholarship by Istituto Nazionale di Alta Matematica forattending a bachelor degree in Mathematics.

Teaching

2013 High school teacher of Mathematics and Physics at Liceo Classico “Giacomo Leop-ardi” (high school focused mainly on humanities) in Aulla (MS, Italy), from January17, 2013 to February 10, 2013.

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2012 Tutor for the Geometry and Linear Algebra course (Geometria, modulodell’insegnamento Geometria e Algebra Lineare, Cod. 177AA), teacher prof. CarloPetronio, Bachelor degree in Civil, Environmental, and Construction Engineering (In-gegneria Civile Ambientale e Edile), Universita di Pisa.

2012 Advanced course in Mathematics for high school students at Liceo Scientifico (highschool focused mainly on scientific subjects) “Leonardo Da Vinci” in Villafranca inLunigiana (MS, Italy): Problemi e tecniche dimostrative (Problems and proof tech-niques), April 27, 2012; Un’introduzione alla Geometria: la caratteristica di Euleroed applicazioni (An introduction to Geometry: Euler characteristic and applications),April 13, 2012; La regina della Matematica nel Mondo reale: teoria dei numeri ecrittografia - seconda parte (The Queen of Mathematics in the real World: numbertheory and cryptography, second part), February 24, 2012; La regina della Matem-atica nel Mondo reale: teoria dei numeri e crittografia (The Queen of Mathematicsin the real World: number theory and cryptography), January 13, 2012; Il giocodelle costruzioni con riga e compasso: tre problemi classici irrisolvibili (The game ofcompass and straightedge constructions: three unsolvable problems), December 02,2012.

Other experiences

2014-today System administration and maintenance of the website “Geometria Complessae Geometria Differenziale”, http://gecogedi.dimai.unifi.it, and of the mail-ing list “Geometria globale”, https://lists.math.unifi.it/mailman/listinfo/

geometriaglobale.2014-today Referee.2014-today Reviewer for Zentralblatt MATH, http://www.zbmath.org/authors/?q=ai:

angella.daniele.2013-today Reviewer for Mathematical Reviews, http://www.ams.org/mathscinet/search/

author.html?mrauthid=943911.2014 Organizer, jointly with Simone Calamai and David Petrecca, of the seminars Com-

plex Geometry and Analysis at Dini, Dipartimento di Matematica e Informat-ica “Ulisse Dini”, Universita di Firenze (http://www.dm.unipi.it/~petrecca/CGAlDini.html).

2012-2013 Organizer, jointly with Simone Calamai, of the informal seminars Seminari dei baby-geometri 2012/2013 (Baby-geometry seminars), Dipartimento di Matematica, Uni-versita di Pisa (http://www.dm.unipi.it/~angella/baby-geometri/).

2012 Representative, jointly with Serena Guarino Lo Bianco, of the Mathematics PhDstudents, Universita di Pisa.

2011 Organizer, jointly with Laura Cremaschi, of the seminars Seminari degli ex-studenti diParma – seconda edizione (Parma alumnae’s seminars – second edition), Dipartimentodi Matematica, Universita di Parma, January 09-12, 2012 (http://www.dm.unipi.it/~angella/exstudenti2011/).

2010 Organizer, jointly with Laura Cremaschi, of the seminars Seminari degli ex-studentidi Parma (Parma alumnae’s seminars), Dipartimento di Matematica, Universita diParma, December 21-22, 2010 and January 10, 2011 (http://www.dm.unipi.it/

~angella/exstudenti2010/).

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Research

Projects and Research groups

2012-today Research program PRIN 2010/2011 (Research Project of National Interest) “Varietareali e complesse: geometria, topologia e analisi armonica” (Real and complex mani-folds: geometry, topology, and harmonic analysis), national scientific coordinator prof.Fulvio Ricci, location Pisa, scientific organizer prof. Riccardo Benedetti, duration 36months.

2012-today Research program FIRB 2012 (Hedge Fund for Basic Research) “Geometria Differen-ziale e Teoria Geometrica delle Funzioni” (Differential Geometry and Geometric Func-tion Theory), national scientific coordinator dr. Caterina Stoppato, location Pisa,scientific organizer dr. Jasmin Raissy, duration 48 months.

2010-today GNSAGA (National Group on Algebraic and Geometric Structures and their Appli-cations), section Geometria complessa (Complex Geometry and Topology), of IstitutoNazionale di Alta Matematica.

Collaborators

Adriano Tomassini Dipartimento di Matematica e Informatica, Universita di Parma, Italy.Simone Calamai Dipartimento di Matematica e Informatica “U. Dini”, Universita di

Firenze, Italy.Hisashi Kasuya Graduate School of Mathematical Sciences, University of Tokyo, Japan;

Department of Mathematics, Tokyo Institute of Technology, Japan.Weiyi Zhang Department of Mathematics, University of Michigan, USA;

current address: Mathematics Insitute, University of Warwick, UnitedKingdom.

Federico Alberto Rossi Dipartimento di Matematica e Applicazioni, Universita di Milano Bic-occa, Italy.

Maria Giovanna Franzini Dipartimento di Matematica, Universita di Parma, Italy.Georges Dloussky Institut de Mathematiques, Aix-Marseille Universite, France.Adela Latorre Instituto Universitario de Matematicas y Aplicaciones, Universidad de

Zaragoza, Spain.

Research period abroad

2014 University of the Basque Country, Bilbao, Spain, from June 30, 2014 to July 04, 2014,at First Joint International Meeting RSME-SCM-SEMA-SIMAI-UMI.

2014 Fakultat fur Mathematik of the Universitat Bielefeld, Germany, from May 06, 2014to May 09, 2014, invited by prof. Sonke Rollenske and by dr. Giovanni Bazzoni.

2014 Institut de Mathematiques de Toulouse of the Universite Paul Sabatier Toulouse 3,France, from March 30, 2014 to April 05, 2014, invited by prof.ssa Eveline Legendre.

2014 Centre International de Rencontres Mathematiques, Luminy, Marseille, France, fromMarch 24, 2014 to March 29, 2014, at Komplex Analysis Winter School and Workshop.

2014 Institut de Mathematiques de Marseille of the Aix-Marseille Universite, France, fromFebruary 02, 2014 to February 08, 2014, invited by prof. Georges Dloussky.

2013 Departamento de Matematicas, Universidad de Zaragoza, Spain, from September 01,2013 to today, with a grant by Istituto Nazionale di Alta Matematica.

2012 Institut Fourier, Laboratoire de Mathematiques, Universite de Grenoble i, France,from June 17, 2012 to July 07, 2012, for the summer school Feuilletages, CourbesPseudoholomorphes, Aypplications (Foliations, Pseudoholomorphic curves, Applica-tions).

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2011 Institut Fourier, Laboratoire de Mathematiques, Universite de Grenoble i, France,from January 16, 2011 to February 13, 2011, from March 16, 2011 to March 31, 2011,and from May 18, 2011 to June 09, 2011, invited by prof. Jean-Pierre Demailly.

2011 Centre International de Rencontres Mathematiques, from February 13, 2011 to Febru-ary 26, 2011, Francia, for the thematic month Complex and Riemann Geometry.

Conferences

2015 Joint meeting Nordic Complex Analysis Meeting (NORDAN) and Komplex AnalysisWinter school and workshop (KAWA), from March 23, 2015 to March 28, 2015, CRM“de Giorgi”, Pisa, Italy.

2014 Invited at the conference Progressi Recenti in Geometria Reale e Complessa - IX, fromOctober 19, 2014 to October 23, 2014, CIRM-FBK, Levico Terme (Trento), Italy.

2014 Conference Complex Geometry, Analysis and Foliations, from September 29, 2014 toOctober 03, 2014, ICTP, Trieste, Italy.

2014 Workshop New trends in Differential Geometry 2014, from September 18, 2014 toSemptember 20, 2014, Villasimius (CA), Italy.

2014 Summer school An introduction to Calabi’s extremal Kaehler metrics by Paul Gaudu-chon, from July 21, 2014 to August 01, 2014, SMI, Cortona, Italy.

2014 Invited speaker at First Joint International Meeting RSME-SCM-SEMA-SIMAI-UMI, from June 30, 2014 to July 04, 2014, Bilbao, Spain.

2014 Summer school Asymptotic aspects of complex and algebraic geometry, from June 23,2014 to June 28, 2014, Dipartimento di Matematica e Applicazioni, Universita diMilano Bicocca, Milano, Italy.

2014 Workshop Complex Geometry and Lie Groups, from June 16, 2014 to June 20, 2014,Dipartimento di Matematica “G. Peano”, Universita di Torino, Italy.

2014 Workshop Isoperimetric Problems Between Analysis and Geometry of the ERC Re-search Period on Calculus of Variations and Analysis in Metric Spaces, from June 16,2014 to June 20, 2014, Scuola Normale Superiore, Pisa, Italy.

2014 Workshop Analysis and Geometry on Singular Spaces of the ERC Research Period onCalculus of Variations and Analysis in Metric Spaces, from June 09, 2014 to June 13,2014, Scuola Normale Superiore, Pisa, Italy.

2014 Invited speaker at Conference on Complex Analysis and Geometry in honor of PierreDolbeault, from June 02, 2014 to June 04, 2014, Universite Pierre et Maire Curie,Paris, France.

2014 Speaker at Joint meeting Nordic Complex Analysis Meeting (NORDAN) and KomplexAnalysis Winter school and workshop (KAWA), from March 24, 2014 to March 29,2014, CIRM, Luminy (Marseille), France.

2014 Speaker at School Carnival Differential Geometry, from February 24, 2014 to February27, 2014, Dipartimento di Matematica G. Peano, Universita di Torino, Italy.

2014 Invited speaker at Workshop Secondo workshop PRIN 2010-2011 Varieta reali e com-plesse: geometria, topologia e analisi armonica (Second Workshop PRIN 2010-2011Real and complex manifolds: geometry, topology, and harmonic analysis), from Febru-ary 20, 2014 to February 22, 2014, Scuola Normale Superiore, Pisa, Italy.

2014 Workshop Partial Differential Equations and applications, from February 20, 2014 toFebruary 21, 2014, Dipartimento di Matematica, Universita di Pisa, Italy.

2013 Invited speaker at Workshop New Entries’s Day 2013 and Prize Michele Cuozzo 2013,December 20, 2013, Dipartimento di Matematica, Universita di Roma Tor Vergata,Roma, Italy.

2013 Workshop Fundamental Groups in Arithmetic and Algebraic Geometry, from Decem-ber 16, 2013 to December 20, 2013, Centro di Ricerca Matematica Ennio de Giorgi,Pisa, Italy.

2013 Invited speaker at Workshop Incontro del progetto FIRB 2012 “Geometria Differen-ziale e Teoria Geometrica delle Funzioni” (FIRB workshop “Differential Geometryand Geometric Function Theory”), from October 23, 2013 to October 25, 2013, Di-partimento di Matematica e Informatica “Ulisse Dini”, Universita di Firenze, Italy.

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2013 Conference Complex Analysis and Geometry - XXI, from June 02, 2013 to June 07,2013, CIRM Fondazione Bruno Kessler, Levico Terme (TN), Italy.

2013 Workshop Varieta reali e complesse: geometria, topologia e analisi armonica (Realand complex manifolds: geometry, topology, and harmonic analysis), from February28, 2013 to March 03, 2013, Scuola Normale Superiore, Pisa, Italy.

2012 Invited speaker at Workshop Progressi Recenti in Geometria Reale e Complessa (Re-cent Advances in Real and Complex Geometry), from October 14, 2012 to October19, 2012, CIRM Fondazione Bruno Kessler, Levico Terme (TN), Italy.

2012 Seminars The locally free geometry seminar at Florence, September 14, 2012, Dipar-timento di Matematica ”Ulisse Dini”, Universita di Firenze, Italy.

2012 Conference New Trends in Holomorphic Dynamics, from September 03, 2012 toSeptember 07, 2012, INdAM, Palazzone della Scuola Normale Superiore, Cortona(AR), Italy.

2012 Summer school Feuilletages, Courbes Pseudoholomorphes, Applications (Foliations,Pseudoholomorphic curves, Applications), from June 18, 2012 to July 06, 2012, Insti-tut Fourier, Universite Joseph Fourier, Grenoble, France.

2012 Conference Giornate di Geometria Algebrica ed Argomenti Correlati XI (Days onAlgebraic Geometry and Related Subjects), from May 23, 2012 to May 26, 2012,Centro di Ricerca Matematica Ennio De Giorgi, Pisa, Italy.

2012 Workshop Geometria in Bicocca 2012 (Geometry at Bicocca 2012), from May 10, 2012to May 11, 2012, Dipartimento di Matematica e Applicazioni, Universita di MilanoBicocca, Italy.

2012 Speaker at Seminars Seminario degli ex-studenti di Parma, seconda edizione (Parmaalumnae’s seminars, second edition), from January 09, 2012 to January 12, 2012,Dipartimento di Matematica, Universita di Parma, Italy.

2011 Conference XIX Congresso dell’Unione Matematica Italiana (19th conference of Ital-ian Mathematical Union), from September 12, 2011 to September 17, 2011, UMI,Bologna, Italy.

2011 Summer school Pluripotential Theory, from July 11, 2011 to July 16, 2011, CIME,Cetraro (CS), Italy.

2011 Conference Complex Analysis and Geometry xx, from June 13, 2011 to June 17, 2011,CIRM, Levico Terme (TN), Italy.

2011 Workshop Geometria in Bicocca 2011 (Geometry at Bicocca 2011), from May 12, 2011to May 13, 2011, Dipartimento di Matematica e Applicazioni, Universita di MilanoBicocca, Italy.

2011 Workshop Ricci Solitons Days in Pisa 2011, from April 04, 2011 to April 08, 2011,Centro di Ricerca Matematica Ennio de Giorgi, Pisa, Italy.

2011 Invited speaker at Workshop Non-Kahlerian aspects of complex geometry, fourth weekof the thematic month Complex and Riemann Geometry, from February 21, 2011 toFebruary 25, 2011, CIRM, Luminy campus, Marseille, France.

2011 Workshop Analytic aspects of complex algebraic geometry, third week of the thematicmonth Complex and Riemann Geometry, from February 14, 2011 to February 18,2011, CIRM, Luminy campus, Marseille, France.

2010 Speaker at Seminars Seminario degli ex-studenti di Parma (Parma alumnae’s semi-nars), from December 21, 2010 to December 22, 2010 and January 10, 2011, Diparti-mento di Matematica, Universita di Parma, Italy.

2010 Invited speaker at Conference Progressi Recenti in Geometria Reale e Complessa (Re-cent Advances in Real and Complex Geometry), from October 17, 2010 to October22, 2010, CIRM, Levico Terme (TN), Italy.

2010 Workshop Workshop in Deformation Theory ii, from August 30, 2010 to September03, 2010, Dipartimento di Matematica, Universita la Sapienza di Roma, Italy.

2010 Summer school Ricci Flow and Geometric Applications, from June 28, 2010 to July03, 2010, CIME, Cetraro (CS), Italy.

2010 Conference Complex Geometry, from May 31, 2010 to June 04, 2010, INdAM, LevicoTerme (TN), Italy.

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2010 Workshop Geometria in Bicocca (Geometry at Bicocca), May 7, 2010, Dipartimentodi Matematica e Applicazioni, Universita di Milano Bicocca, Italy.

2010 Lectures An introduction to Topos Theory and its unifying role in Mathematics byOlivia Caramello, from February 22, 2010 to February 26, 2010, Centro di RicercaMatematica Ennio de Giorgi, Pisa, Italy.

Invited talks

2014 June/July 2014 (TBC), Bilbao: Cohomological properties of symplectic manifolds, atthe First Joint International Meeting RSME-SCM-SEMA-SIMAI-UMI, Bilbao, Spain.

2014 June 2014 (TBA), Paris: Cohomologies of complex manifolds, at the Conference onComplex Analysis and Geometry in honor of Pierre Dolbeault, Universite Pierre etMaire Curie Paris 6, France.

2014 May 10, 2014, Pontremoli: joint with Anna Rita Giammetta, Matematica eDemocrazia, at Circolo ARCI Babel – la casa dei Popoli.

2014 May 07, 2014, Bielefeld: Cohomologies of complex manifolds in non-Kahler geome-try, at the seminars Oberseminar Gruppen und Geometrie, Faculty of Mathematics,Universitat Bielefeld, Bielefeld, Germany.

2014 April 04, 2014, Toulouse: Cohomological aspects of non-Kahler manifolds, at theseminars on Analyse et geometrie complexe, Institut de Mathematiques de Toulouse,Universite Paul Sabatier, Toulouse, France.

2014 March 28, 2014, Marseille: On Bott-Chern cohomology for complex manifolds, at theJoint meeting NORDAN and KAWA, CIRM Luminy (Marseille), France.

2014 March 04, 2014, Pisa: Un’introduzione alla Geometria non-Kahleriana (verso il LatoOscuro della Geometria Complessa) (An introduction to non-Kahler geometry), atSeminari dei baby-geometri (Baby-geometry seminars), Dipartimento di Matematica,Universita di Pisa, Italy.

2014 February 25, 2014, Torino: Cohomological aspects of non-Kahler manifolds, at theCarnival Differential Geometry school, Dipartimento di Matematica “G. Peano”, Uni-versita di Torino, Italy.

2014 February 22, 2014, Pisa: Coomologie di varieta complesse (Cohomologies of com-plex manifolds), at the second workshop PRIN 2010-2011 Varieta reali e complesse:geometria, topologia e analisi armonica, Scuola Normale Superiore di Pisa, Italy.

2014 January 22, 2014, and February 13, 2014, and February 19, 2014, and May 13, 2014,and May 20, 2014, Parma: Formalita di varieta complesse, I-II-IV-V-VI (Formalityof complex manifolds, 1-2-4-5-6), at the Seminario di Geometria 2013/2014, Diparti-mento di Matematica e Informatica, Universita di Parma, Italy.

2014 January 03, 2014, Firenze: CMA in Geometria Complessa (Complex Monge Ampereequations in Complex Geometry), at CGAlDini, Dipartimento di Matematica e In-formatica “Ulisse Dini”, Universita di Firenze, Italy.

2013 December 20, 2013, Roma: Cohomological aspects in complex non-Kahler geometry,at workshop New Entries’ Day 2013 Cuozzo Prize, Dipartimento di Matematica, Uni-versita di Roma Tor Vergata, Italy.

2013 November 27, 2013, Torino: Un’introduzione alla Geometria non-Kahleriana (An in-troduction to non-Kahler geometry), at Mathematics Ph.D. Seminars, Scuola di Dot-torato in Scienze della Natura e Tecnologie Innovative, Dottorato in Matematica,Universita di Torino, Italy.

2013 November 21, 2013, Madrid: An introduction to non-Kahler geometry , at SeminarioDoctorandos Matematicas, Facultad de Ciencias Matematicas, Universidad Com-plutense de Madrid, Spain.

2013 November 19, 2013, Madrid: Cohomological aspects in complex non-Kahler geometry,at Seminario de Geometrıa y Topologıa, Departamento de Geometrıa y Topologıa,Universidad Complutense de Madrid, Spain.

2013 November 12, 2013, Zaragoza: Cohomological aspects in complex geometry, workingseminar at Instituto Universitario de Matematicas y Aplicaciones, Departamento deMatematicas, Universidad de Zaragoza, Spain.

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2013 October 24, 2013, Firenze: Relazioni coomologiche e modelli per varieta non-Kahleri-ane (Cohomological relations and models for non-Kahler manifolds), at the workshop“Incontro del progetto FIRB 2012 Geometria Differenziale e Teoria Geometrica delleFunzioni”, Dipartimento di Matematica e Informatica “Ulisse Dini”, Universita diFirenze, Italy.

2013 October 01, 2013, Zaragoza: Cohomologically-computable manifolds, working seminarat Departamento de Matematicas of the Universidad de Zaragoza, Spain.

2013 June 06, 2013, Levico Terme: Cohomologically-computable non-Kahler manifolds, atthe conference “Complex Analysis and Geometry - XXI”, CIRM Fondazione BrunoKessler, Italy.

2012 December 12, 2012, Parma: Seminario di Geometria, IV, seminar at the workinggroup “Seminario di Geometria 2012/2013”, Dipartimento di Matematica e Infor-matica, Universita di Parma, Italy.

2012 December 10, 2012, Firenze: Non-Kahler Geometry: towards the Dark Side of Com-plex Geometry, at “Seminari dei Dottorandi” (PhD student seminars), Dipartimentodi Matematica “Ulisse Dini”, Universita di Firenze, Italy.

2012 November 21, 2012, Pisa: Cohomological aspects of non-Kaehler manifolds, at “Semi-nari di Geometria” (Geometry seminars), Dipartimento di Matematica, Universita diPisa, Italy.

2012 October 19, 2012, Levico Terme: Aspetti coomologici in Geometria non-Kahleriana(Cohomological aspects in non-Kahler geometry), at the workshop “Progressi Recentiin Geometria Reale e Complessa” (Recent Advances in Real and Complex Geometry),CIRM Fondazione Bruno Kessler, Italy.

2012 January 11, 2012, Parma: Introduzione alla Geometria Generalizzata, II (Introductionto Generalized Geometry, second part), at “Seminari degli ex-studenti di Parma, sec-onda edizione” (Parma alumnae’s seminars, second edition ), Dipartimento di Matem-atica, Universita di Parma, Italy.

2011 December 12, 2011, Pisa: Coomologie di varieta complesse e varieta-esempio (Coho-mologies of complex manifolds and toys-manifolds), informal seminar at “SeminariInformali di Matematica dei Dottorandi, Seminari di Algebra e Geometria” (Infor-mal seminars of PhD students, Algebra and Geometry Seminars), Scuola NormaleSuperiore di Pisa, Italy.

2011 October 18, 2011, and November 15, 2011, and November 29, 2011, Parma: Coomolo-gia di Bott-Chern e deformazioni, i-iii-iv (Bott-Chern cohomology and deformations,i-iii-iv), seminar at the working group “Seminario di Geometria 2011/2012”, Dipar-timento di Matematica, Universita di Parma, Italy.

2011 September 16, 2011, Bologna: Scomposizione in coomologia per varieta quasi-complesse (Cohomology decomposition on almost-complex manifolds), communica-tion at “XIX Congresso dell’Unione Matematica Italiana” (19th Conference of Ital-ian Mathematical Union), section Geometria Complessa (Complex Geometry), UMI,Bologna, Italy.

2011 March 21, 2011, Grenoble: On cohomological properties of non-Kahler manifolds,at “Seminaires d’Algebre et geometries” (Algebra and Geometry Seminars), InstitutFourier, Laboratoire de Mathematiques, Universite de Grenoble i, France.

2011 March 18, 2011, Grenoble: Non-Kahler complex varieties and Hodge decomposition,at the workgroup “Pathologies in Institut Fourier” (Pathologies at Institut Fourier),Institut Fourier, Universite de Grenoble i, France.

2011 February 22, 2011, Marseille: On cohomological properties of non-Kahler manifolds,at the workshop “Analytic aspects of complex algebraic geometry”, CIRM, Luminycampus, Marseille, France.

2010 December 22, 2010, Parma: Coomologia di varieta (quasi-)complesse (Cohomology of(almost-)complex manifolds), at the seminars “Seminari degli ex-studenti di Parma”(Parma alumnae’s seminars), Dipartimento di Matematica, Universita di Parma,Italy.

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2010 October 18, 2010, Levico Terme: Scomposizione della coomologia di de Rham peruna classe di varieta quasi-complesse (Decomposition of de Rham cohomology for aclass of almost-complex manifolds), at the conference “Progressi Recenti in GeometriaReale e Complessa” (Recent Advances in Real and Complex Geometry), CIRM, Italy.

2010 April 26, 2010, Pisa: Il ∂∂-lemma: un’introduzione alla geometria Kahleriana (The∂∂-Lemma: an introduction to non-Kahler geometry), informal seminar at “Seminaridei baby-geometri” (Baby-geometry seminars), Dipartimento di Matematica “LeonidaTonelli”, Universita di Pisa, Italy.

Papers and preprints

Papers

[1] Daniele Angella, Adriano Tomassini,On cohomological decomposition of almost-complex manifolds and deformations,J. Symplectic Geom. 9 (2011), no. 3, 403–428,

doi: 10.4310/JSG.2011.v9.n3.a5,http://projecteuclid.org/euclid.jsg/1310388902,(preprint arXiv:0909.3604 [math.DG].)

[2] Daniele Angella, Adriano Tomassini,On the cohomology of almost-complex manifolds,Internat. J. Math. 23 (2012), no. 2, 1250019 (25 pages),

doi: 10.1142/S0129167X11007604,http://www.worldscientific.com/doi/abs/10.1142/S0129167X11007604,(preprint arXiv:1211.6276 [math.DG].)

[3] Daniele Angella,The cohomologies of the Iwasawa manifold and of its small deformations,J. Geom. Anal. 23 (2013), no. 3, 1355–1378,

doi: 10.1007/s12220-011-9291-z,http://www.springerlink.com/content/m7547621g7722265/,(preprint arXiv:1212.4351 [math.DG].)

[4] Daniele Angella, Adriano Tomassini,On the ∂∂-Lemma and Bott-Chern cohomology,Invent. Math. 192 (2013), no. 1, 71–81,

doi: 10.1007/s00222-012-0406-3,http://link.springer.com/article/10.1007%2Fs00222-012-0406-3,(preprint arXiv:1402.1954 [math.DG].)

[5] Daniele Angella, Federico Alberto Rossi,Cohomology of D-complex manifolds,Differ. Geom. Appl. 30 (2012), no. 5, 530–547,

doi: 10.1016/j.difgeo.2012.07.003,http://www.sciencedirect.com/science/article/pii/S0926224512000617,(preprint arXiv:1201.2503 [math.DG].)

[6] Daniele Angella, Simone Calamai,A vanishing result for strictly p-convex domains,Ann. Mat. Pura Appl. (4) 193 (2014), no. 4, 1069–1084,

doi: 10.1007/s10231-012-0315-5,http://link.springer.com/article/10.1007/s10231-012-0315-5,(preprint arXiv:1211.2564 [math.DG].)

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[7] Daniele Angella, Adriano Tomassini, Weiyi Zhang,On cohomological decomposability of almost-Kahler structures,Proc. Amer. Math. Soc. 142 (2014), no. 10, 3615–3630,

doi: 10.1090/S0002-9939-2014-12049-1,http://www.ams.org/journals/proc/2014-142-10/S0002-9939-2014-12049-1/home.html,

(preprint arXiv:1211.2928v1 [math.DG].)

[8] Daniele Angella, Adriano Tomassini,Symplectic manifolds and cohomological decomposition,J. Symplectic Geom. 12 (2014), no. 2, 1–22,

doi: 10.4310/JSG.2014.v12.n2.a1,http://intlpress.com/site/pub/pages/journals/items/jsg/content/vols/0012/0002/a001/

index.html,

(preprint arXiv:1211.2565 [math.SG].)

[9] Daniele Angella,Cohomologies of certain orbifolds,J. Geom. Phys. 171 (2013), 117-126,

doi: 10.1016/j.geomphys.2013.04.008,http://www.sciencedirect.com/science/article/pii/S0393044013000909,(preprint arXiv:1211.2561 [math.DG].)

[10] Daniele Angella, Simone Calamai,Bott-Chern cohomology and q-complete domains,C. R. Math. Acad. Sci. Paris 151 (2013), no. 9–10, 343–348,

doi: 10.1016/j.crma.2013.05.006,http://www.sciencedirect.com/science/article/pii/S1631073X13001118,(preprint arXiv:1305.4011 [math.CV].)

[11] Daniele Angella, Adriano Tomassini,Inequalities a la Frolicher and cohomological decompositions,to appear in J. Noncommut. Geom.,

(preprint arXiv:1403.2298 [math.DG].)

Preprints

[12] Daniele Angella, Maria Giovanna Franzini, Federico Alberto Rossi,Degree of non-Kahlerianity for 6-dimensional nilmanifolds,submitted,

(preprint arXiv:1210.0406 [math.DG].)

[13] Daniele Angella, Hisashi Kasuya,Bott-Chern cohomology of solvmanifolds,submitted,

(preprint arXiv:1212.5708 [math.DG].)

[14] Daniele Angella, Hisashi Kasuya,Cohomologies of deformations of solvmanifolds and closedness of some properties,submitted,

(preprint arXiv:1305.6709 [math.CV].)

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[15] Daniele Angella, Hisashi Kasuya,Symplectic Bott-Chern cohomology of solvmanifolds,submitted,

(preprint arXiv:1308.4258 [math.SG].)

[16] Daniele Angella, Georges Dloussky, Adriano Tomassini,On Bott-Chern cohomology of compact complex surfaces,submitted,

(preprint arXiv:1402.2408 [math.DG].)

[17] Daniele Angella, Adriano Tomassini,Cohomological models and formalities of complex manifolds,submitted.

[18] Daniele Angella, Adriano Tomassini,Stability of holomorphically-parallelizable manifolds,submitted.

[19] Daniele Angella, Simone Calamai,Cohomologies of generalized-complex manifolds and nilmanifolds,submitted,

(preprint arXiv:1405.0981 [math.DG].)

[20] Daniele Angella, Simone Calamai, Adela Latorre,On cohomological decomposition of generalized-complex structures,submitted,

(preprint arXiv:1406.2101 [math.DG].)

[21] Daniele Angella, Hisashi Kasuya,Hodge theory for twisted differentials,preprint.

Books

[17] Daniele Angella,Cohomological aspects in complex non-Kahler geometry,Lecture Notes in Mathematics 2095 (2014), ISBN 978-3-319-02440-0,

http://www.springer.com/mathematics/geometry/book/978-3-319-02440-0.

Communications

[18] Daniele Angella, Adriano Tomassini,Scomposizione in coomologia per varieta quasi-complesse (Cohomology decomposition for almost-complex manifolds),communication at the conference xix Congresso dell’Unione Matematica Italiana (19th confer-ence of Italian Mathematical Union), section Geometria Complessa (Complex Geometry), UMI,Bologna, from September 12, 2011 to September 17, 2011, page 327.

Thesis

[19] Daniele Angella,Cohomological aspects of non-Kahler manifolds,Ph.D. Thesis, Universita di Pisa, academic year 2011/2012,date January 21, 2013, advisor prof. Adriano Tomassini (Universita di Parma),

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http://etd.adm.unipi.it/theses/available/etd-01072013-231626/,(arXiv:1302.0524 [math.DG].)

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Research program

Research interests

Complex non-Kahler Geometry. Symplectic Geometry. Cohomological properties of non-Kahler mani-folds. Functions of complex and hypercomplex variables.

• Several complex variables and analytic spaces, and Symplectic geometry.

– Complex geometry:(Almost-)complex manifolds; Hermitian and Kahlerian manifolds; Cohomology of complexmanifolds, cohomological decomposition and ∂∂-Lemma; Topological aspects of complex man-ifolds.

– Symplectic geometry:Symplectic manifolds; Generalized geometries (a la Hitchin); Cohomology of compact sym-plectic manifolds.

– Deformations of analytic structures:Deformations of complex structures; Deformations of special structures.

– Geometric convexity:q-completeness; Other notions of convexity; Analytical consequences of geometric convexity(vanishing theorems, etc.).

– Complex variables:Holomorphic foliations and vector fields; Hyperbolic and Kobayashi hyperbolic manifolds;Special classes of entire functions and growth estimates; Quasi-analytic and other classes offunctions; Functions of hypercomplex variables; Geometric theory of slice regular functionsof one quaternionic variable; Geometry and dynamics in the open unit ball of quaternions;Quaternionic Hardy spaces.

• Global analysis, analysis on manifolds.

– General theory of differentiable manifolds:de Rham theory; Hodge theory; Elliptic equations on manifolds; and Relations with specialmanifold structures; Kahler-Einstein manifolds; Calabi-Yau theory.

• Topological groups, Lie groups.

– Lie groups:General properties and structure of complex and real Lie groups; Nilpotent and solvable Liegroups.

– Homology and homotopy of topological groups and related structures:Homology and cohomology of Lie groups; Homology and cohomology of homogeneous spacesof Lie groups.

Introduction

A Kahler manifold is a complex manifold endowed with a Hermitian metric whose fundamental formis a symplectic form; one can easily prove that this implies that the complex structure is parallel withrespect to the Levi Civita connection. An important class of examples is provided by projective mani-folds. In a sense, Kahler manifolds can be studied under three different points of view: the complex one,the symplectic one, and the Riemannian one. The existence of these three different structures, whichare closely related each other, allows to prove several interesting results, mainly concerning cohomology,see, e.g., [DGMS75]. Some of these results do not hold when one of the conditions in the definitionof Kahlerness is weakened: for examples, compact Kahler manifolds satisfy the Hodge DecompositionTheorem, stating that the de Rham cohomology decompose by means of the Dolbeault cohomology, butin general on a compact complex non-Kahler manifold this is no more true, while on almost-complexmanifolds the notion of Dolbeault cohomology makes no sense.

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In non-Kahler geometry, the class of nilmanifolds (i.e., compact quotient of a connected simply-connected nilpotent Lie group by a co-compact discrete subgroup) provides important examples andcounter-examples: indeed, on the one hand, their geometry can be often reduced to the study of theassociated Lie algebra, see, e.g., [Nom54, CF01]; on the other hand, nilmanifolds usually do not admittoo strong structures (for example, non-tori nilmanifolds do not admit any Kahler structure, see, e.g.,[Has06]).

My research activity, started during my PhD studies at Dipartimento di Matematica of Universita diPisa, was supervised by prof. Adriano Tomassini (Dipartimento di Matematica e Informatica, Universitadi Parma), and was developed also thanks to a visiting period at Institut Fourier of Universite de Greno-ble I, where I was invited by prof. Jean-Pierre Demailly. My research focuses on non-Kahler geometry,and in particular on cohomological properties of complex, almost-complex, symplectic, or generalized-complex manifolds, on cones of special metric structures on (almost-)complex manifolds, on deformations,on convexity in Riemannian and complex geometry, [Ang13c, Ang13b]. Some of the results have beenobtained in a joint work with Adriano Tomassini (Dipartimento di Matematica e Informatica, Universitadi Parma), Federico Alberto Rossi (Dipartimento di Matematica e Applicazioni, Universita di MilanoBicocca), Simone Calamai (Classe di Scienze, Scuola Normale Superiore di Pisa), Weiyi Zhang (Depart-ment of Mathematics, University of Michigan), Maria Giovanna Franzini (Dipartimento di Matematica,Universita di Parma), Hisashi Kasuya (Graduate School of Mathematical Sciences, The University ofTokyo), Georges Dloussky (Institut de Mathematiques de Marseille, Aix-Marseille Universite).

The interest of the scientific community for the construction of non-Kahler manifolds and for thestudy of their properties is also related to their role in Theoretical Physics, see, e.g., [TY11].

Results

Cohomologies of complex non-Kahler manifolds Other than de Rham cohomology, H•dR(X;C),and Dolbeault cohomology, H•,•

∂(X), the Bott-Chern cohomology and its dual, the Aeppli cohomology,

H•,•BC(X) :=ker ∂ ∩ ker ∂

imm ∂∂, H•,•A (X) :=

ker ∂∂

imm ∂ + imm ∂,

provide further tools for the study of the geometry of compact complex manifolds X. Such cohomologies,initially introduced in 1960s, [BC65, Aep65], have been recently studied by several authors in differentcontexts: see, e.g., [Aep65, BC65, Big69, DGMS75, Var86, AB90, Sch07, Koo11, Bis11, TY11].

In [Ang11, AT13, AFR12, AK12, AK13a, ADT14], we have studied properties and cohomologicaldecomposition on compact complex manifolds; see also the monograph [Ang13b].

The Hodge decomposition theorem states that, on a compact Kahler manifold, the de Rham coho-mology splits into the direct sum of the Dolbeault cohomology groups. More precisely, the property ofsatisfying the ∂∂-Lemma in the sense of P. Deligne, Ph. A. Griffiths, J. Morgan, e D. P. Sullivan, thatis, the property that every ∂-closed ∂-closed d-exact differential form is also ∂∂-exact, is satisfied ona class of compact complex manifolds containing compact Kahler manifolds, compact manifold in theFujiki class C, [Fuj78], and compact Moishezon manifolds, [Moı66]. The ∂∂-Lemma yields the formalityof the de Rham complex of the underlying differentiable manifold, and therefore it gives topologicalobstructions to the existence of Kahler structures on differentiable manifolds. For a compact complexmanifold, the Frolicher inequality states that, for every k ∈ Z, it holds∑

p+q=k

dimCHp,q

∂(X) ≥ dimCH

kdR(X;C) .

In [AT13] (Invent. Math., 2013), we have proven an inequality a la Frolicher relating the dimensions ofthe Bott-Chern cohomology groups of a compact complex manifold to its Betti numbers: more precisely,for every k ∈ Z, it holds ∑

p+q=k

(dimCHp,qBC(X) + dimCH

p,qA (X)) ≥ 2 bk , (1)

where bk := dimCHkdR(X;C) is the k-th Betti number of X. Furthermore, this inequality allows to

characterize the ∂∂-Lemma on compact complex manifolds: in fact, we have proven that the equality in

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(1) holds for every k ∈ Z if and only if the manifold satisfies the ∂∂-Lemma. As a corollary, we havegiven another proof of the stability of the ∂∂-Lemma under small deformations of the complex structure,already proven by C. Voisin, [Voi02], and by C.-C. Wu, [Wu06], and by A. Tomasiello, [Tom08].

In [Ang11] (J. Geom. Anal., 2013), we studied the Bott-Chern cohomology for nilmanifolds, namely,compact quotient of connected simply-connected nilpotent Lie groups by cocompact discrete subgroups.More precisely, we proved a result a la Nomizu for the Bott-Chern cohomology, that is, we provedthat, for suitable left-invariant complex structures on nilmanifolds, the Bott-Chern cohomology can becomputed just by means of the Lie algebra, i.e., just by means of left-invariant differential forms. Asan application, we explicitly studied the Bott-Chern cohomology of the Iwasawa manifold and of itssmall deformations, in order to complete the results by M. Schweitzer in [Sch07], and to give a finercohomological classification of the space of small deformations of the Iwasawa manifold, which wasstudied by I. Nakamura in [Nak75].

In [AFR12], we used the results in [Ang11] to investigate the Bott-Chern cohomology of 6-dimensionalnilmanifolds endowed with the left-invariant complex structures classified by M. Ceballos, A. Otal, L.Ugarte, and R. Villacampa in [COUV12], and to study their degree of non-Kahlerianity, see also [LUV12];we also studied pluriclosed metrics (also called skt metrics) on such nilmanifolds.

In [Ang13a] (J. Geom. Phys., 2013), we studied properties of the Bott-Chern cohomology for orbifoldsgiven by quotients of a complex manifold by a finite group of biholomorphisms.

In [AK12, AK13a], we generalized the results in [Ang11] to an algebraic and more general context,and we got, in particular, thanks to the results in [Kas10, Kas11b, Kas12a] (see also [Kas11a, Kas12c,Kas12b]), some new results concerning the Bott-Chern cohomology of certain classes of solvmanifolds.Such results allow to provide interesting examples on the behaviour of cohomological properties underdeformations of the complex structure. In particular, it is provided an example of a family of complexmanifolds satisfying the ∂∂-Lemma but whose limit does not.

In [ADT14], we studied Bott-Chern cohomology on compact complex surfaces. In particular, wecomputed such a cohomology for surfaces in class VII and for compact complex surfaces diffeomorphicto solvmanifolds.

Cohomological properties of almost-complex manifolds and of symplectic manifolds Coho-mological properties of almost-complex manifolds have been introduced and studied by several authors,see, e.g., [LZ09, DLZ10, DLZ11, FT10, AT11, AT12b, Zha11, ATZ12, DZ11, TWZZ11, HMT11, LT12,DLZ12]. In particular, in [LZ09], T.-J. Li and W. Zhang introduced and studied certain subgroups ofthe de Rham cohomology of compact almost-complex manifolds, and they investigated when such sub-groups provide a decomposition of the de Rham cohomology of an almost-complex manifold similar tothe decomposition of differential forms. Such subgroups have applications in the study of compatibleand tamed symplectic cones, in connection with a conjecture by S. K. Donaldson, [Don06]. In such acontext, in [DLZ10], it has been proved that, on every compact 4-dimensional almost-complex manifold,the above subgroups always provide a decomposition of the real second de Rham cohomology group.

In [AT11, AT12b, ATZ12], we continued the study concerning the connection between de Rhamcohomology of an almost-complex manifold and the action induced by the almost-complex structure onthe space of differential forms; see also the monograph [Ang13b].

More precisely, in [AT11] (J. Symplectic Geom., 2011), we studied cohomological decompositionproperties of the de Rham cohomology of (almost-)complex manifolds, and we showed their instabilityunder small deformations of the complex structure, by explicit investigating the small deformations ofthe Iwasawa manifold. Furthermore, we proved that, on 6-dimensional nilmanifolds endowed with left-invariant complex structure, there is no tamed symplectic structure; this is connected with a question byT.-J. Li and W. Zhang, [LZ09], and by J. Streets and G. Tian, [ST10], which is similar to the Donaldsonconjecture for 4-dimensional almost-complex manifolds.

In [AT12b] (Internat. J. Math., 2012), we continued the study of the subgroups of de Rham co-homology introduced by T.-J. Li and W. Zhang, by studying both their decomposition properties onalmost-complex manifolds, and their behaviour along curves of almost-complex structures and theirsemi-continuity properties. Furthermore, by using the methods by D. P. Sullivan, [Sul76], we found aquantitative relation between the cone of balanced metrics ([Mic82]) and the cone of strongly-Gauduchonmetrics ([Pop13]) on compact complex manifolds.

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In [ATZ12] (Proc. Amer. Math. Soc., to appear), we have studied the decomposition properties ofthe de Rham cohomology of compact almost-Kahler manifolds, also providing an example of an almost-Kahler structure on the Iwasawa manifold for which the decomposition of differential 2-forms does notinduce a decomposition of the corresponding de Rham cohomology group; this shows a different behaviourbetween the integrable complex case of Kahler manifolds, and the non-integrable almost-complex caseof almost-Kahler manifolds. Furthermore, we have studied a Lefschetz-type property in connection withcohomological decomposition properties.

An analogue of the Hodge theory on compact symplectic manifold has been developed starting withthe works by J.-L. Koszul, [Kos85], and J.-L. Brylinski, [Bry88], and then studied in, e.g., [Mat95, Yan96,Mer98, Gui01, Cav05, TY12a, TY12b]. In [TY12a, TY12b, TY11], L.-S. Tseng and S.-T. Yau introducedand studied several cohomologies on compact symplectic manifolds. The Generalized Complex Geometrywas born from an idea by N. J. Hitchin, [Hit03], and has been developed, among others, by M. Gualtieri,[Gua04, Gua11], and by G. R. Cavalcanti, [Cav05]. It provides a more general point of view, in whichone can frame both Complex Geometry and Symplectic Geometry, and by means of which one can builda parallel between different properties: indeed, complex structures and symplectic structures appear asspecial and extreme cases of generalized complex structures, and many complex properties (for example,the ∂∂-Lemma) have a well defined symplectic counterpart (in the example, the ddΛ-Lemma, equivalently,the Hard Lefschetz Condition), and conversely.

In [AT14, AT12a, AK13b], we have studied cohomological properties of symplectic manifolds and,more in general, of generalized complex manifolds.

More precisely, in [AT14] (J. Symplectic Geom., 2014), we have studied the cohomology of compactmanifolds X endowed with a symplectic form ω, by studying when the Lefschetz decomposition of forms,∧•X =

⊕r∈N ω

r ∧ ker (−ιω−1) b∧•−2rX , induces a decomposition at the level of de Rham cohomology.In particular, we have proved that this holds true always for the second de Rham cohomology groupof a compact symplectic manifold, and for every de Rham cohomology groups on a compact symplecticmanifold satisfying the Hard Lefschetz Condition. Explicit examples on nilmanifolds and solvmanifoldswere studied, by means of a theorem a la Nomizu.

In [AT12a], we generalized the results in [AT13] to an algebraic and more general context, in orderto get, as corollaries, other than the results already proved in [AT13], further results on the cohomologyof compact symplectic manifolds and of compact generalized complex manifolds. More precisely, aninequality a la Frolicher has been proven also in the symplectic framework: namely, for any k ∈ Z, itholds

dimRker d ∩ ker dΛ

imm ddΛ+ dimR

ker ddΛ

imm d + imm dΛ≥ 2 bk , (2)

where dΛ := [d, −ιω−1 ], and where bk := dimRHkdR(X;R) is the k-th Betti number of X; we further

proved that the equality in (2) holds for any k ∈ Z if and only if X satisfies the ddΛ-Lemma, equiv-alently, the Hard Lefschetz Condition. An analogous inequality has been proven, more in general, inthe generalized complex setting, and, also in this case, it allows to characterize certain cohomologicalproperties.

In [AK13b], we investigated the symplectic cohomologies for solvmanifolds, by using the generalresults in [AK12]. Explicit examples on low-dimensional nilmanifolds and solvmanifolds are provided.

Special structures on manifolds and cohomology In [AR12] (Differ. Geom. Appl., 2012), westudied the cohomology of D-complex manifolds in the sense of F. R. Harvey and H. B. Lawson (see,e.g., [HL10]). An almost-D-complex structure on a manifold X is the datum of an endomorphismK ∈ End(TX) such that K2 = idTX and that the corresponding eigen-bundles T+ and T− have thesame rank; furthermore, if such eigen-bundles are involutive, then the structure is called integrable or alsoa D-complex structure; these structures are, in a sense, the hyperbolic analogue of complex structures.

In considering a D-complex counterpart of Dolbeault cohomology, one gets possibly non-finite dimen-sional cohomology groups, also in the compact D-complex case. Therefore, we studied certain subgroupsof the de Rham cohomology, connected with the action of the D-complex structure on differential forms,and we investigated the cohomological decomposition properties and their behaviour along curves ofD-complex structures. We proved that, on a 4-dimensional nilmanifold, every left-invariant D-complexstructures induces a decomposition of the second de Rham cohomology group by means of the invariant

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and anti-invariant groups for the action of the D-complex structure. Furthermore, we studied several ex-plicit examples, thank to a result a la Nomizu, and then we investigated curves of D-complex structures,showing in particular that the existence of D-Kahler structures is not stable under small deformations:this provides a difference with the Kahler case, where K. Kodaira and D. C. Spencer’s theorem holds,[KS60].

In [AC12] (Ann. Mat. Pura Appl., 2012), inspired by a vanishing result of Dolbeault cohomology forq-complete domains of Cn by A. Andreotti and H. Grauert, [AG62], we studied cohomological propertiesof domains of Rn endowed with exhaustion functions whose Hessian satisfies certain positivity conditions.In [HL12, HL11], F. R. Harvey and H. B. Lawson introduced the notion of p-convexity. In particular,by adapting the L2 techniques developed by L. Hormander, [Hor65], and used, among others, by A.Andreotti and E. Vesentini, [AV65a, AV65b], we gave another proof of a result by J.-P. Sha, [Sha86], andby H. Wu, [Wu87], concerning the vanishing of certain de Rham cohomology groups for strictly p-convexdomains of Rn.

In [AC13] (C. R. Math. Acad. Sci. Paris, 2013), we study Bott-Chern and Aeppli cohomologies ofopen manifolds, especially under assumptions on the vanishing of certain Dolbeault cohomology groups.In particular, we introduce the class of cohomologically Bott-Chern q-complete manifolds.

Further researches

Aeppli and Bott-Chern cohomologies, and generalized cohomologies. We would like to con-tinue to investigate the cohomological properties of complex non-Kahler manifolds, and in particularto deepen the study of the Bott-Chern and Aeppli cohomologies of compact complex manifolds. Wewould like to generalize the study of such cohomologies to the symplectic and generalized complex con-text, in order to provide further tools to investigate the geometry of non-Kahler manifolds. We wouldlike to investigate the cohomological and geometric study of complex manifolds by constructing certainalgebraic models of the double complex of complex-valued differential forms, in particular by studyingnecessary and sufficient conditions assuring formality. In particular, the aim is to deepen the study ofsome special classes of manifolds, among which nilmanifolds and solvmanifolds, manifolds constructedby LdM-V-M-B [LdMV97, Mee00, Bos01], compact complex surfaces, toric varieties, twistor spaces, . . . ,also with the aim to provide classification results.

We would like to deepen the study of the cohomologies of compact complex manifolds endowed withspecial metrics, as, for example, balanced metrics in the sense of M. L. Michelsohn, [Mic82], or pluriclosedmetrics, [Bis89], or strongly-Gauduchon metrics, [Pop13], investigating the possible connection betweencohomological decompositions and the existence of geometric structures. In such a context, geometricproblems often reduce to equations similar to Monge-Ampere equations: it would be useful to investigateand develope further analytic tools.

We would like to study the behaviour of cohomological decomposition properties and of the existenceof special metric under deformations of the complex structure (see, for example, [Pop11]). In thiscontext, nilmanifolds provide useful examples to study small deformations of manifolds endowed withspecial metrics, see, e.g., [LUV12, AFR12]; the results in [AK12] may allow to provide further toolsto investigate small deformations of compact complex manifolds having cohomological decompositionproperties.

We would like to further investigate the generalized complex and symplectic counterpart of Bott-Chern and Aeppli cohomologies, which have been recently introduced and studied by L.-S. Tseng and S.-T. Yau, [TY12a, TY12b, TY11]. In particular, we would like to deepen the study of the relation betweenthese cohomologies and the de Rham cohomology; for example, we would like to find further geometricapplications of the inequality a la Frolicher for the generalized Bott-Chern cohomology. In particular, wewould like to further investigate the study of cohomological properties for compact symplectic manifold,as the Hard Lefschetz Condition.

Cohomologies of nilmanifolds and solvmanifolds. In joint work with H. Kasuya, we would like todeeply investigate the methods to compute the Bott-Chern cohomology, in order to provide new tools forthe study of cohomological properties of solvmanifolds and of their deformations (see [Ang11] and [Kas10,Kas11b, Kas12a, Kas11a, Kas12c, Kas12b]). This may provide further explicit examples of complex non-Kahler manifolds with special cohomological properties, to be compared with the manifolds in C of

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Fujiki, [Fuj78], Moishezon manifolds, [Moı66], twistor-spaces, [AHS78]. In particular, such examplesmay provide negative answers or positive evidences to many still-open questions concerning complexgeometry and deformations, see, for example, [Pop11].

It would be useful to further investigate the relations between the combinatorial and the geometricproperties of nilmanifolds, as in [PT09], and to study results a la Nomizu for the Dolbeault cohomologyof general nilmanifolds, see [Con06, Rol11], or to extend such results for the computation of othercohomologies, for example, cohomologies induced by generalized complex structures on nilmanifolds orsolvmanifolds.

Cohomology and convexity. In joint work with S. Calamai, we would like to continue the study ofcohomology of domains of Rn or Cn endowed with convexity properties, by investigating the relationbetween convexity and vanishing of certain cohomology groups. In particular, we would study therelation between A. Andreotti and H. Grauert’s q-completeness, [AG62, Rot55], and the vanishing ofcertain Bott-Chern cohomology groups, with the aim to characterize the latter class. Furthermore, wewould investigate the geometry of q-complete domains also by studying their Dolbeault homotopy groups,as introduced by J. Neisendorfer and L. Taylor in [NT78].

Construction of complex non-Kahler manifolds by means of dynamical systems. In[LdMV97, Mee00, Bos01], following an idea by S. Lopez de Medrano and A. Verjovsky, a construc-tion of complex manifolds by means of singular holomorphic foliation of Cn has been developed andinvestigated. Such a method generalizes the construction of the Hopf manifolds, [Hop48], and of the Cal-abi and Eckmann manifolds, [CE53]). In a sense, the class of Lopez de Medrano, Verjovsky, Meersseman,and Bosio’s manifolds is similar to the class of nilmanifolds: on the one hand, such construction allowsto reduce some of the geometric properties of the above manifolds to combinatorial properties; on theother hand, but for tori, such manifolds are non-Kahler (in fact, many of the known non-Kahler examplesarise in this way). Furthermore, the cohomology of the manifolds constructed in this way is often trivialin many degrees. A conjecture by F. Bogomolov states that every compact complex manifold can beconstructed generalizing this construction. It would be interesting to investigate whether LdM-V-M-Bmanifolds are endowed with special metrics.

Cohomology of almost-complex manifolds. It would be interesting to further study properties ofcohomological decomposition on almost-complex manifolds: for example, one would characterize sucha decomposition property inside certain classes of almost-complex manifolds, such as compact almost-Kahler manifolds. Moreover, it may be interesting to study more in details the behaviour of such adecomposition property along curves of almost-complex structures.

Cones of symplectic structures, and special and canonical metric structures on complexmanifolds. No example of complex non-Kahler manifold admitting tamed symplectic structure is know,at now, see [LZ09, ST10]; such a question relates to a question by S. K. Donaldson concerning compact4-dimensional almost-complex manifolds, [Don06].

Recently, D. Popovici introduced and studied the notion of strongly-Gauduchon metric in [Pop13].Such metrics satisfy both the stability under deformations as the Kahler metrics (see [KS60]) and thestability under modifications as balanced metrics (see [AB90]). It would be interesting to study furtherthe existence of strongly-Gauduchon metrics, or the existence of balanced metrics, particularly in con-nection with cohomological properties, such as the ∂∂-Lemma. In particular, it would be interestingto obtain results concerning the limit of complex manifolds endowed with strongly-Gauduchon metrics,with the aim to extend the results in [Pop13] on the degeneracy of projective manifolds.

It would be interesting to introduce and investigate a notion of canonical metrics on complex mani-folds.

References

[AB90] L. Alessandrini and G. Bassanelli, Small deformations of a class of compact non-Kahler manifolds, Proc. Amer.Math. Soc. 109 (1990), no. 4, 1059–1062.

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[AC12] D. Angella and S. Calamai, A vanishing result for strictly p-convex domains, Ann. Mat. Pura Appl.Online First, http://link.springer.com/article/10.1007/s10231-012-0315-5 (2012), 16 pages, DOI:10.1007/s00222-012-0406-3.

[AC13] , Bott-Chern cohomology and q-complete domains, arXiv:1305.4011v1 [math.CV], to appear in C. R.Math. Acad. Sci. Paris, 2013.

[ADT14] D. Angella, G. Dloussky, and A. Tomassini, On bott-chern cohomology of compact complex surfaces, 2014.

[Aep65] A. Aeppli, On the cohomology structure of Stein manifolds, Proc. Conf. Complex Analysis (Minneapolis, Minn.,1964), Springer, Berlin, 1965, pp. 58–70.

[AFR12] D. Angella, M. G. Franzini, and F. A. Rossi, Degree of non-kahlerianity for 6-dimensional nilmanifolds,arXiv:1210.0406v1 [math.DG], 2012.

[AG62] A. Andreotti and H. Grauert, Theoreme de finitude pour la cohomologie des espaces complexes, Bull. Soc.Math. France 90 (1962), 193–259.

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