Geometry Main

The 6th GTSS


Feza Gürsey Institute - Online (Youtube Channel, Slides)
August 2-14, 2021

First Week


TIME              SPEAKER                  TITLE
Aug 2-6
Jesse Madnick
Holomorphic curves in the 6-sphere
Lunch Time
Aug 2-6
Sebastian Heller Constant mean curvature(CMC) surfaces and integrable systems
Aug 2-6
İzzet Coşkun
Topics in Algebraic Curves

Second Week


TIME              SPEAKER                  TITLE
Aug 9-13
Anna Fino Interplays of Complex and Symplectic Geometry
Lunch Time
Aug 9-14
Alexandra Otiman
Topics in locally conformally Kähler (LCK) geometry
Aug 9-13
Michael Albanese Yamabe Invariant of Complex Surfaces

Scientific Commitee


Vicente Cortés University of Hamburg, Germany
İzzet Coşkun University of Illinois at Chicago, USA
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
İlhan İkeda Bosphorus University
Mustafa Kalafat Nesin Mathematical Village

Registration       Poster       Participants       Arrival


The 6th GTSS will be held at the FGE, Feza Gürsey Center for Physics and Mathematics (Online)
which is established in the Kandilli (North) Campus of Boğaziçi University (Bosphorus).
The first photo on top of this webpage demonstrates the view of the venue.
There will be about 6 mini-courses of introductory nature, related to the Geometry-Topology research subjects.
Thursday is holiday.
In the middle of the week (i.e. on Thursday) there is an excursion on the Bosphorus. Do not forget to get your swimsuit with yourself. (For the real events only.)


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.


Holomorphic curves in the 6-sphere

The content of each lecture is as follows.

Lecture 1: G2 Geometry & Associative 3-folds. This lecture is expository, meant as an introduction for non-experts. We discuss Riemannian holonomy in general, and then focus on G2 holonomy and G2-structures. We then discuss the two most important classes of submanifolds in G2 geometry: the associative 3-folds and coassociative 4-folds.

Lecture 2: Nearly-Kahler Geometry & Holomorphic Curves. This lecture is expository, meant as an introduction for non-experts. We introduce the class of nearly-Kahler 2n-manifolds by way of the Gray-Hervella classification. We then specialize to (strict) nearly-Kahler 6-manifolds, explaining their relationship with G2-holonomy cones, and briefly reviewing the known compact examples. Finally, we discuss the two most important classes of submanifolds in nearly-Kahler 6-manifolds: the holomorphic curves and Lagrangian 3-folds.

Lecture 3: The Jacobi Spectrum of Closed Holomorphic Curves in S^6. This lecture is a research talk. Recall that minimal surfaces are area-minimizing to first order, but not necessarily to second-order. The extent to which a minimal surface is (or isn’t) area-minimizing to second-order is encoded by its Jacobi operator. However, for a given minimal surface, computing the spectrum of the Jacobi operator - i.e., the eigenvalues and their multiplicities - is generally non-trivial.
We will discuss a class of minimal surfaces in the round S^6 known as "null-torsion" holomorphic curves. By a remarkable theorem of Bryant, extended by Rowland, every closed Riemann surface may be conformally embedded in S^6 as a null-torsion holomorphic curve. In the case of low genus, we give a simple explicit formula for the multiplicity of the first Jacobi eigenvalue. Moreover, for all genera, we give a simple lower bound for the nullity. We expect that these results will have implications for the deformation theory of asymptotically conical associative 3-folds in R^7.

Lecture 4: Free-Boundary Holomorphic Curves in Nearly-Kahler 6-Manifolds. This lecture is a research talk. We will discuss holomorphic curves with boundary in nearly-Kahler 6-manifolds, analyzing two different boundary conditions. First, we show that a holomorphic curve in a geodesic ball B of S^6 that meets the boundary of B orthogonally must be totally-geodesic. This implies rigidity results for reflection-invariant holomorphic curves in S^6 and associative cones in R^7. Second, we consider holomorphic curves with boundary on a Lagrangian in a nearly-Kahler 6-manifold. By deriving a second variation formula for area, and exploiting a suitable analogue of Riemann-Roch, we obtain a topological lower bound on the Morse index.

  1. R. Bryant - Submanifolds and Special Structures on the Octonians.
    J. Diff. Geom. (1982).

  2. N. Ejiri - The Index of Minimal Immersions of S^2 into S^{2n}.
    Mathematische Zeitschrift. (1983).

  3. A. Fraser and R. Schoen - Uniqueness Theorems for Free Boundary Minimal Disks in Space Forms.
    Int. Math. Res. Not. (2015).

  4. A. Gray and L. M. Hervella - The Sixteen Classes of Almost Hermitian Manifolds and their Linear Invariants.
    Ann. Mat. Pura. Appl. (1980).

  5. F. Reese Harvey and H. Blaine Lawson - Calibrated Geometries.
    Acta Mathematica. (1982).

  6. D. Joyce - Riemannian Holonomy Groups and Calibrated Geometry.
    Oxford University Press. (2007).

  7. J. Madnick - The Second Variation of Null-Torsion Holomorphic Curves in the 6-Sphere.
    ArXiv:2101.09580 (Jan 2021) 35 pages.

  8. J. Madnick - Free-Boundary Problems for Holomorphic Curves in the 6-Sphere.
    Arxiv:2105.10562 (May 2021). 17 pages.

  9. S. Montiel and F. Urbano - Second Variation of Superminimal Surfaces into Self-Dual Einstein 4-Manifolds.
    Trans. AMS. (1997).

Constant mean curvature(CMC) surfaces and integrable systems

These lectures give a detailed introduction to the theory of constant mean curvature surfaces in space forms via gauge theoretic methods and the theory of integrable systems. The main goal is to explain recent results about the construction of new examples and the derivation of geometric quantities in terms of complex analytic data.

Lecture 1: Minimal and CMC surfaces in R3 and S3. We start with a short summary about minimal surfaces in R3. We explain the induced Riemann surface structure, the Hopf differential and the spinorial Weierstrass representation. In the second part, we discuss surfaces of constant mean curvature in R3 and S3. We derive the gauge-theoretic description of these surfaces [7], and explain the Lawson correspondence. We discuss some important examples [8, 6].

Lecture 2: Holomorphic bundles and at connections. We introduce holomorphic bundles over Rie- mann surfaces from various points of view. We study at connections over holomorphic bundles. In particular, we discuss unitary at connections and the famous Narasimhan-Seshadri theorem - this will enable us to deal with intrinsic closing conditions of CMC surfaces in space forms. We introduce the moduli spaces of at connections and moduli spaces of poly-stable holomorphic bundles.

Lecture 3: Loop group factorisation methods. We define the associated family of at connections of a CMC surface in R3 and S3 and study its basic properties. We construct minimal and CMC surfaces from certain maps into the moduli space of at connections satisfying intrinsic and extrinsic closing conditions [3, 4]. The proof uses a loop group factorisation. We discuss the DPW method - the special case of families of holomorphic connections [1, 2].

Lecture 4: Recent results and examples. We discuss recent results about the existence and properties of minimal and CMC surfaces [5, 6]. In particular, we investigate the area of the Lawson surfaces in terms of their genus.

A survey: Gerding, A.; Pedit, F.; Schmitt, N. Constant mean curvature surfaces an integrable systems perspective.
Harmonic maps and differential geometry, 7–39, Contemp. Math., 542, AMS 2011.

  1. J. Dorfmeister, F. Pedit, and H. Wu, Weierstrass type representation of harmonic maps into symmetric spaces.
    Comm. Anal. Geom. 6 (1998), no. 4, 633-668.

  2. S. Heller, Higher genus minimal surfaces in S3 and stable bundles.
    J. Reine Angew. Math., Volume 685 (2013), pp 105-122.

  3. S. Heller, A spectral curve approach to Lawson symmetric CMC surfaces of genus 2.
    Math. Annalen, Volume 360, Issue 3 (2014), pp 607-652.

  4. L. Heller, S. Heller, N. Schmitt, Navigating the Space of Symmetric CMC Surfaces.
    Journal of Differential Geometry, Vol. 110, No. 3 (2018), pp. 413-455.

  5. L. Heller, S. Heller, M. Traizet, Area Estimates for High genus Lawson surfaces via DPW.

  6. A. Bobenko, S. Heller, N. Schmitt, Constant mean curvature surfaces based on fundamental quadrilaterals.
    Math. Physics, Analysis and Geometry, Preprint: arxiv: 2102.03153

  7. N.J. Hitchin, Harmonic maps from a 2-torus to the 3-sphere.
    J. Differential Geom. 31 (1990), 627-710.

  8. H. B. Lawson. Complete minimal surfaces in S3.
    Ann. of Math. (2) 92 (1970), 335-374.

Topics in Algebraic Curves

This will be lectures on selected topics on algebraic curves.
The daily description will be as follows.

Lecture 1: (Mon 2-8-21) Rational curves in projective space

In this lecture, I will describe spaces of rational curves in projective space. I will discuss the stratification of the space according to the splitting type of the restricted tangent bundle and the normal bundle. I will show that the general curve has a balanced normal bundle and following Sacchiero describe the possible splitting types of the normal bundle. Following joint work with Eric Riedl, I will exhibit some pathological behavior of the stratification by the splitting type of the normal bundle.

Lecture 2: (Tue 3-8-21) Rational curves on hypersurfaces

In this lecture, I will discuss the space of rational curves on hypersurfaces in projective space. I will mention several conjectures concerning these spaces and prove some special cases of these conjectures. Following joint work with Eric Riedl and Geoffrey Smith, I will discuss the separable rational connectedness of hypersurfaces and more generally complete intersections in certain homogeneous spaces.

Lecture 3: (Wed 4-8-21) Algebraic hyperbolicity of very general hypersurfaces of large degree

In this lecture, I will discuss bounds on the genus of a curve in a very general hypersurface of degree at least 5 in projective 3-space. Following joint work with Eric Riedl, I will show that a very general surface of degree at least 5 is algebraically hyperbolic.

Lecture 4: (Fri 6-8-21) Normal bundles of Brill-Noether space curves

In this lecture, I will discuss the stability of the normal bundle of general Brill-Noether space curves. Following joint work with Eric Larson and Isabel Vogt, I will show that a general Brill-Noether space curve of degree d and genus g at least 2 has a stable normal bundle if and only if (d,g) is not equal to (5,2) or (6,4).

  1. I. Coskun, E. Larson and I. Vogt, Stability of normal bundles of space curves

  2. I. Coskun and E. Riedl, Normal bundles of rational curves in projective space,
    Mathematische Zeitschrift vol 288 (2018), 803--827.

  3. I. Coskun and E. Riedl, Normal bundles of rational curves on complete intersections
    Communications in Contemporary Mathematics 21 no. 2 (2019).

  4. I. Coskun, Izzet and E. Riedl, Algebraic hyperbolicity of the very general quintic surface in P^3
    Adv. Math. 350 (2019), 1314--1323.

  5. I. Coskun and G. Smith, Very free rational curves in Fano varieties

  6. I. Coskun and J. Starr, Rational curves on smooth cubic hypersurfaces
    International Mathematics Research Notices Article RNP102 (2009)

  7. J. Harris, M. Roth and J. Starr, Rational Curves on hypersurfaces of low degree
    J. Reine Angew. Math., 571 (2004), 73--106

  8. Riedl, Eric and Yang, David. Rational curves on general type hypersurfaces
    Journal of Differential Geometry, 116 no.2 (2020), 393--403.

  9. Riedl, Eric and Yang, David. Kontsevich spaces of rational curves on Fano hypersurfaces
    J. Reine Angew. Math. 748 (2019), 207--225.

Interplays of Complex and Symplectic Geometry

I will present - mostly using examples - how complex and symplectic structures relate on compact manifolds. I will show how symplectic forms taming complex structures are intimately related to a special type of Hermitian metrics, known in the literature also as "pluriclosed" metrics. Then I will present some results on the symplectic Calabi-Yau problem in dimension 4 and on balanced metrics in relation to the Strominger-Hull system. A tentative schedule:

1. Kähler geometry. Examples and non-examples.

2. Pluriclosed metrics

3. Symplectic Calabi-Yau problem

4. Balanced metrics and the Strominger-Hull system

  1. L. Alessandrini, G. Bassanelli, Plurisubharmonic currents and their extension across analytic subsets,
    Forum Math. 5 (1993), 291--316.

  2. L. Bedulli, L. Vezzoni, A parabolic flow of balanced metrics,
    J. Reine Angew. Math. 723 (2017), 79–99.

  3. E. Buzano, A. Fino, L. Vezzoni, The Calabi-Yau equation on the Kodaira-Thurston manifold, viewed as an S1-bundle over a 3-torus,
    J. Differential Geom. 101 (2015), no. 2, 175--195.

  4. E. Buzano, A. Fino, L. Vezzoni, The Calabi-Yau equation for T^2-bundles over the non-Lagrangian case,
    Rend. Semin. Mat. Univ. Politec. Torino 69 (2011), no. 3, 281--298.

  5. S.K. Donaldson, Two-forms on four-manifolds and elliptic equations,
    Inspired by S.S. Chern, 153–172, Nankai Tracts Math. 11, World Scientific, Hackensack N.J., 2006.

  6. A. Fino, Y.Y. Li, S. Salamon, L. Vezzoni, The Calabi–Yau equation on 4-manifolds over 2-tori,
    Trans. Amer. Math. Soc. 365 (2013), no. 3, 1551--1575.

  7. N. Enrietti, A. Fino, L. Vezzoni, Tamed symplectic forms and SKT metrics,
    J. Symplectic Geom. 10 (2012), no. 2, 203–223. Correction: J. Symplectic Geom. 17 (2019), no. 4, 1079–1081.

  8. A. Fino, G. Grantcharov, Properties of manifolds with skew-symmetric torsion and special holonomy,
    Adv. Math. 189 (2004), no. 2, 439--50.

  9. A. Fino, G. Grantcharov, L. Vezzoni, Solutions to the Hull-Strominger system with torus symmetry,

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

  11. A. Fino, F. Paradiso, Balanced Hermitian structures on almost abelian Lie algebras,

  12. A. Fino, N. Tardini, L. Vezzoni, Pluriclosed and Strominger Kähler-like metrics compatible with abelian complex structures,

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

  14. J.-X. Fu, S.-T. Yau, The theory of superstring with flux on non-Kähler manifolds and the complex Monge-Amp`ere equation,
    J. Differential Geom. 78 (2008), no. 3, 369--428.

  15. P. Gauduchon, La 1-forme de torsion d'une variété hermitienne compacte. (French)
    [Torsion 1-forms of compact Hermitian manifolds], Math. Ann. 267 (1984), no. 4, 495--18.

  16. P. Gauduchon, Hermitian connections and Dirac operators.
    Boll. Un. Mat. Ital. B (7) 11 (1997), no. 2, suppl., 257--288.

  17. E. Goldstein, S. Prokushkin, Geometric model for complex non-Kähler manifolds with SU(3) structure,
    Comm. Math. Phys. 251 (2004), no. 1, 65--78.

  18. M. L. Michelsohn, On the existence of special metrics in complex geometry,
    Acta Math. 149 (1982), no. 3-4, 261--295.

  19. D. Phong, S. Picard, X. Zhang, Anomaly flows,
    Comm. Anal. Geom. 26 (2018), no. 4, 955--1008.

  20. J. Streets, G. Tian, A parabolic flow of pluriclosed metrics,
    Int. Math. Res. Not. IMRN 2010, no. 16, 3101--3133.

  21. C. Taubes, Clifford Henry Tamed to compatible: symplectic forms via moduli space integration,
    J. Symplectic Geom. 9 (2011), no. 2, 161--250.

  22. V. Tosatti, B. Weinkove, S.T. Yau, Taming symplectic forms and the Calabi–Yau equation,
    Proc. London Math. Soc. 97 (2008), no. 2, 401--424.

  23. V. Tosatti, B. Weinkove, The Calabi–Yau equation on the Kodaira–Thurston manifold,
    J. Inst. Math. Jussieu 10 (2011), no. 2, 437--447.

  24. M. Verbitsky, Rational curves and special metrics on twistor spaces.
    Geom. Topol. 18 (2014), no. 2, 897--909.

  25. B. Weinkove, The Calabi-Yau equation on almost-Kähler four-manifolds,
    J. Differential Geom. 76 (2007), no. 2, 317--349.

  26. S.T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge–Ampère equation, I,
    Comm. Pure Appl. Math. 31 (1978), no. 3, 339–411.

Topics in locally conformally Kähler (LCK) geometry

In this series of lectures I will introduce locally conformally Kähler (lcK) metrics, give explicit examples of compact manifolds admitting such structures and study their topological and cohomological properties. We shall focus in the latter part of the course on Vaisman manifolds.

  1. F. A. Belgun, On the metric structure of non-Kähler complex surfaces,
    Math. Ann. 317 (2000), 1-40.

  2. M. Brunella, Locally conformally Kähler metrics on Kato surfaces,
    Nagoya Math. J., 202 (2011), p. 77-81.

  3. P. Gauduchon, Le theoreme de l'excentricite nulle,
    C. R. Acad. Sci. Paris Ser. A-B 285 (1977), no. 5, A387-A390.

  4. P. Gauduchon, L. Ornea, LCK metrics on Hopf surfaces,
    Ann. Inst. Fourier, 48 (1998), 1107-1127.

  5. N. Istrati, A. Otiman, M. Pontecorvo, On a class of Kato manifolds,
    Int. Math. Res. Not. (IMRN), Vol. 2021, No. 7, pp. 5366-5412.

  6. M. de Leon, B, Lopez, J.C. Marrero, E. Padron, On the computation of the Lichnerowicz-Jacobi cohomology,
    J. Geom. Phys. 44 (2003), 507-522.

  7. K. Oeljeklaus, M. Toma, Non-Kähler compact complex manifolds associated to number fields,
    Ann. Inst. Fourier, Grenoble, 55 (2005), no. 1, 161-171.

  8. L. Ornea, M. Verbitsky, A report on locally conformally Kähler manifolds,
    Contemporary Mathematics 542, 135-150, 2011.

  9. A. Otiman, M. Toma, Hodge decomposition for Cousin groups and Oeljeklaus-Toma manifolds,
    Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) Vol. XXII (2021), 485-503.

  10. K. Tsukada, Holomorphic forms and holomorphic vector fields on compact generalized Hopf manifolds,
    Compositio Mathematica, vol.93, no.1(1994), p.1-22.

  11. I. Vaisman, On locally conformal almost Kähler manifolds,
    Israel J. Math. 24 (1976), no. 3-4, 338-351.

  12. I. Vaisman, On locally and globally conformal Kähler manifolds,
    Trans. AMS 262(1980), 533-542.

Yamabe Invariant of Complex Surfaces

The Yamabe invariant is a real-valued diffeomorphism invariant which arises from Riemannian geometry. Somewhat surprisingly, the Yamabe invariant of a complex surface is intimately related to complex geometric properties of the surface. LeBrun used Seiberg-Witten theory to show that the sign of the Yamabe invariant of a compact Kähler surface is determined by its Kodaira dimension. Recent work has investigated the extent to which this relationship persists in the non-Kähler setting.

The relevant papers that will be discussed are as follows.

  1. LeBrun - Kodaira Dimension and the Yamabe Problem.

  2. Albanese - The Yamabe invariants of Inoue surfaces, Kodaira surfaces, and their blowups.

  3. Albanese & LeBrun - Kodaira Dimension and the Yamabe Problem, II.

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