Geometry Main

The 11th GTSS


İstanbul Center for Mathematical Sciences - Online (Youtube Channel)
July 17-29, 2023

First Week


TIME              SPEAKER                  TITLE
Jul 17-21
Rosa Sena-Dias Action-angle coordinates and Riemannian geometry on toric manifolds
S : 1a , 1b , 2 , 3 , 4
Lunch Time
Jul 17-21
Nicoletta Tardini
Topics in strong Kähler with torsion (SKT) metrics
S : 1 , 2 , 3 , 4
Jul 17-20
Marcos Miguel Origlia
Left invariant LCK geometry on Lie groups
S : 1

Second Week


TIME              SPEAKER                  TITLE
Jul 25-26
Thomas Murphy
Topics in (Quasi-)Einstein metrics
Postponed to next year
Lunch Time
Jul 24-29
Dan Popovici Special Hermitian Metrics on Compact Complex Manifolds
S : 1 , 2 , 3 , 4
Jul 24-29
Viviana del Barco
Topics in ad-invariant metrics
S : 1 , 2 , 3 , 4

Scientific Commitee


Vicente Cortés University of Hamburg, Germany
Anna Fino University of Torino, Italy
Ljudmila Kamenova Stony Brook University, USA
Lei Ni University of California at San Diego, USA
Tommaso Pacini University of Torino, Italy
Gregory Sankaran University of Bath, UK
Misha Verbitsky IMPA, Brasil

Organizing Commitee


Craig van Coevering Bosphorus University
İzzet Coşkun University of Illinois at Chicago, USA
Mustafa Kalafat Rheinische Friedrich-Wilhelms-Universität Bonn

Registration       Poster       Participants       Arrival


The 11th GTSS will be held at the IMBM, İstanbul Center for Mathematical Sciences (Online)
which is established in the main (South) Campus of Boğaziçi University (Bosphorus).
There will be mini-courses of introductory nature, related to the Geometry-Topology research subjects.


Graduate students, recent Ph.D.s and under-represented minorities are especially encouraged to the summer school.
Daily expenses including bed, breakfast, lunch, dinner is around 20€.
You may stay at cheap hostels around Taksim(Nightlife) square and take the subway to the Campus easily.
We are trying to arrange accomodation on Campus as well.

Please fill out the Registration form in order to attend the "Online" summer school. Registration is free but mandatory.

Airport: İstanbul Airport - IST is the closest one. Take a bus from the airport to the 4. Levent. Then take a taxi or subway to reach to the main/south campus.
Alternatively you can use Sabiha Gökçen Airport - SAW and take a bus from there to 4. Levent. Then take a taxi or subway to reach to the main/south campus.
Navigate the link above for more detailed arrival and venue information.

Visas: Check whether you need a visa beforehand.


Action-angle coordinates and Riemannian geometry on toric manifolds

In this series of lectures my goal is to use the action-angle description of toric manifolds to discuss two problems in Kähler Geometry.

The first problem concerns the search for Einstein metrics. Toric Kähler manifolds have given rise to several explicit examples of Einstein metrics. But since many Kähler manifolds do not carry Kähler-Einstein metrics, it is natural to relax the Kähler condition. Derdziński has obtained a construction for 4d conformally Kähler, Einstein metrics using special extremal Kähler metrics. In some recent joint work with G. Oliveira we set out to apply Derdziński’s results to extremal metrics with U(2) symmetry arising from the so called Calabi ansatz. My goal in the first 3 lectures of the series is to report on our findings.

A second problem I want to discuss is that of finding minimal Lagrangian tori. We will see that in the realm of toric geometry, action-angle coordinates are extremely useful in tackling the problem.

In the first lecture, I will give an overview of toric manifolds and action-angle coordinates on them. I will describe Calabi’s ansatz in these coordinates. I will then use this ansatz to write the first known examples of explicit extremal non constant scalar curvature Kähler metrics on Hirzebruch surfaces. These are due to Calabi.

In the second lecture I will discuss Derdziński’s results. I will define Bach-flat metrics and explain how to find them in our setting. In the third lecture I will move on to explain my joint work with G. Oliveira. In particular I will explain how to interpret the Einstein metrics arising from the Calabi ansatz via Derdziński duality. Some of the metrics we obtain are fillings of S3/Zm. As it turns out, these had been previously written down in different coordinates by Page and Pope. We explain how to “cap them off”. We also discuss the case of CP2#-CP2.

In the fourth lecture I shift gears and use the action-angle setup on toric manifolds to address a different problem in Kähler Geometry namely that of finding minimal Lagrangian tori. I shall explain why compact toric manifolds always carry such objects and how find them. This is joint work with G. Oliveira.

  1. Abreu, Miguel. Kähler geometry of toric manifolds in symplectic coordinates.
    Eliashberg, Yakov (ed.) et al., Symplectic and contact topology: Interactions and perspectives.
    Papers of the workshop on symplectic and contact topology, quantum cohomology,
    and symplectic field theory, Montreal and Toronto, Canada, March–April 2001.
    AMS. Fields Inst. Commun. 35, 1-24 (2003). Available on the ArXiv at

  2. Derdzinski, Andrzej. Self-dual Kaehler manifolds and Einstein manifolds of dimension four.
    Compos. Math. 49, 405-433 (1983). Available at the authors website.

  3. LeBrun, Claude. Bach-flat Kähler surfaces.
    J. Geom. Anal. 30, No. 3, 2491-2514 (2020). Available on the ArXiv at

  4. Oliveira, Gonçalo; Sena-Dias, Rosa. Einstein metrics from the Calabi ansatz via Derdziński duality.
    Available on the ArXiv at

  5. Oliveira, Gonçalo; Sena-Dias, Rosa. Minimal Lagrangian tori and action-angle coordinates.
    Trans. Am. Math. Soc. 374, No. 11, 7715-7742 (2021).
    Available on the ArXiv at

Topics in SKT (Strong Kähler with torsion) geometry

In this series of lectures I will introduce strong Kähler with torsion (SKT) metrics on complex manifolds, give explicit examples of compact complex manifolds admitting such structures and study their existence on special classes of manifolds. We will then discuss their relation with curvature properties.

Partial Bibliography

1. N. Egidi. Special metrics on compact complex manifolds,
Differential Geom. Appl. 14. (2001), no.3, 217-234.

2. A. Fino, M. Parton, S. Salamon. Families of strong KT structures in six dimensions,
Comment. Math. Helv. 79, no. 2, 317-340 (2004)

3. A. Fino, N. Tardini. Some remarks on Hermitian manifolds satisfying Kähler-like conditions,
Math. Z., 298 (1), (2021) 49-68.

4. A. Fino, N. Tardini, L. Vezzoni. Pluriclosed and Strominger Kähler-like metrics compatible with abelian complex structures.
Bull. London Math. Soc. 2022, 54(5), pp. 1862-72.

5. A. Fino, A. Tomassini. Blow-ups and resolutions of strong Kähler with torsion metrics.
Adv. Math. 221 (2009), no.~3, 914-935.

6. A. Fino, A. Tomassini. On blow-ups and cohomology of almost complex manifolds,
Diff. Geom. Appl. 30 (2012), 520-529.

7. S. Ivanov, G. Papadopoulos. Vanishing theorems and string backgrounds,
Classical and Quantum Gravity, 18 (2001), no.6, 1089-1110.

8. D. Popovici. Non-Kähler Hodge Theory and Deformations of Complex Structures.
preprint book:

9. N. Tardini, A. Tomassini. On geometric Bott-Chern formality and deformations,
Ann. Mat. Pura Appl. (4) 196 (2017), no.~1, 349-362.

10. S. T. Yau, Q. Zhao, F. Zheng. On Strominger Kähler-like manifolds with degenerate torsion,
Trans. Am. Math. Soc. 376(5), (2023) pp. 3063-3085.

11. Q. Zhao, F. Zheng. Strominger connection and pluriclosed metrics,
J. für die Reine und Angew. Math. (2023) 796, 245-267

12. Q. Zhao, F. Zheng. Complex nilmanifolds and Kähler-like connections,
arXiv:1904.09707 [math.DG].

Locally conformal Kähler geometry on Lie groups

During this talks I will introduce some basic definitions involved in the no- tion of locally conformal Kähler (LCK) manifolds (complex structure, symplec- tic form, Kahler metric, etc.). I’ll focus on LCK structures on solvmanifolds, compact quotinents Γ\G, where G is a solvable Lie group, and Γ a co-compact discrete subgroup. We will see some examples, recent result and open problem in this topic.

  1. A. Andrada, M. Origlia. Locally conformally Kähler solvmanifolds: a survey.
    Complex Manifolds (6) (2019), 65–87.

  2. S. Dragomir, L. Ornea. Locally conformal Kähler geometry.
    Progress in Mathematics 155, Birkhäuser Boston, 1998.

  3. L. Ornea. Locally conformal Kähler manifolds. A selection of results.
    Lect. Notes Semin. Interdiscip. Mat. IV (2005), 121–152.

  4. L. Ornea, M. Verbitsky. A report on locally conformally Kähler manifolds.
    in Harmonic maps and differential geometry, Contemp. Math. 542 (2011), 135–149.

Elements of complex differential geometry via applications to drug design

The goal is to give a self-contained survey of some topics in complex differential geometry in the simplest setting imaginable. This actually turns out to have very useful applications in chemistry to the field of drug discovery, which I will then outline.

1: Differential forms, complex analysis, line bundles, sections, and curvature.

2: Quantisation, Theorems of Kodaira and Tian/Bouche. Applications to Chemistry and drug design.


1. R Berman, J Keller: Bergman Geodesics in “Complex Monge-Ampere Equations and geodesics in the space of Kaehler metrics,
Lecture Notes in Math 2038, V, Guedj, Ed. Springer-Verlag 2012

2. Daniel J. Cole, Stuart J. Hall, Thomas Murphy, Rachael Pirie. Riemannian Geometry and Molecular Similarity II: Kähler Quantization
Available on the ArXiv at

3. G. Szekelyhidi. An introduction to Extremal Kaehler metrics, GTM, 152, AMS.

Special Hermitian Metrics on Compact Complex Manifolds

We will present several classes of possibly non-Kähler special Hermitian metrics (SKT, Hermitian-symplectic, balanced, Gauduchon, strongly Gauduchon) on complex manifolds and the cohomologies (Bott-Chern, Aeppli, Dolbeault, De Rham) that are relevant in their study. We will then study these metrics in the context of deformations of complex structures and the Frölicher spectral sequence.

  1. J.-P. Demailly. Complex Analytic and Algebraic Geometry. self published e-book, 1997.

  2. D. Popovici. Non-Kähler Hodge Theory and Deformations of Complex Structures. preprint book.

Topics on ad-invariant metrics

We will introduce ad-invariant metrics on Lie algebras from an algebraic/Lie theoretical point of view, reviewing properties and constructions. Later, we will consider the geometry of the bi-invariant metric that they induce and address their (pseudo-)Riemannian geometrical properties, among which, their isometry group will be of relevance. If time permits, the existence and uniqueness of ad-invariant metrics on Lie algebras will be treated.

  1. D. Conti, V. del Barco and F. A. Rossi. Uniqueness of ad-invariant metrics.
    To appear in Tohoku Math. J. Available on the ArXiv at

  2. B. O'Neil. Semi-Riemannian Geometry with applications to relativity.
    Academic Press, 1983.

  3. G. Ovando. Lie algebras with ad-invariant metrics, a survey-guide.
    Rend. Sem. Mat. Univ. Pol. Torino, 74 (1), 243 - 268, 2016.

  4. G. Ovando and V. del Barco. Isometric actions on pseudo-Riemannian nilmanifolds.
    Ann. Glob. Anal. Geom. 45 (2) (2014) 95-110.


Activities are supported by University of Bonn and Boğaziçi University