# Program

### Schedule

(*) indicates an in-person speaker, and all others will be speaking remotely. All of the talks will be recorded and live-streamed on Zoom. In person talks will be held in room 6214 (André-Aisenstadt Building), and coffee breaks will be held in room 6245.

Time | Monday | Tuesday | Wednesday | Thursday | Friday |
---|---|---|---|---|---|

9:00-10:00 | Anna Maria Fino | Andrew Swann | Ryushi Goto | Éveline Legendre | Thomas Walpuski |

10:00-10:30 | coffee break | coffee break | coffee break | coffee break | coffee break |

10:30-11:30 | Jason Lotay | Charles Boyer | Lorenzo Foscolo* | Andrew Dancer | Christina Tønnesen-Friedman |

11:30-14:00 | Lunch | Lunch | Lunch | Lunch | Lunch |

14:00-15:00 | Yuri Ustinovskiy | Weiyong He | Xiangwen Zhang | Julien Keller | Roschig & Stecker* |

15:00-15:30 | break | break | break | break | break |

15:30-16:30 | Jeffrey Streets* | Uwe Semmelman* | Claude LeBrun* | Vicente Cortés* | Marco Gualtieri* |

# Talks

**Charles Boyer (University of New Mexico)**

**Transverse Kähler holonomy and Sasakian rigidity**

This talk is based on joint work with Hongnian Huang and Christina Tønessen- Friedman. We discuss the rigidity of Sasakian structures under deformations of the transverse Kählerian structure. In particular, we outline a proof showing that if the basic Hodge numbers \(h_B^{0,1}\) and \(h_B^{0,2}\) vanish, then Sasakian structures are rigid under deformations of the transverse Kähler flow. However, it is shown that rigidity fails for transverse hyperkähler structures. These results indicate the algebraic nature of Sasakian geometry.

**Vicente Cortés (Universität Hamburg)**

**Quaternionic-Kähler manifolds with ends of finite volume**

I will explain a construction of complete quaternionic-Kähler manifolds with interesting fundamental groups, which are not locally symmetric. The examples include manifolds which are diffeomorphic to a cylinder over a compact locally homogeneous manifold such that one of the two ends is of finite volume. This is joint work with Danu Thung and Markus Röser, see arXiv:2105.00727.

**Andrew Dancer (University of Oxford)**

**Symplectic duality and implosions**

Symplectic duality is a correspondence between holomorphic symplectic varieties arising as Higgs and Coulomb branches of a class of physical theories. In this talk I report on work identifying candidates for the symplectic duals of hyperkähler implosion spaces. This is joint work with A. Bourget, J. Grimminger, A. Hanany, F. Kirwan, and Z. Zhong.

**Anna Fino (Università di Torino)**

**Balanced metrics and the Hull-Strominger system**

A Hermitian metric on a complex manifold is called balanced if its fundamental form is co-closed. In the talk I will review some general results about balanced metrics and the Hull-Strominger system. In particular, I will show that the Fu-Yau solution to this system on torus bundles over K3 surfaces can be generalized to torus bundles over K3 orbifolds. This is a joint work with G. Grantcharov and L. Vezzoni.

**Lorenzo Foscolo (University College London)**

**Twistor constructions of complete non-compact hyperkähler metrics**

The talk is based on joint work with Roger Bielawski about twistor constructions of higher dimensional non-compact hyperkähler metrics with maximal and submaximal volume growth. In the first part of the talk, based on arXiv:2012.14895, I will discuss the case of hyperkähler metrics with maximal volume growth. Exploiting Namikawa’s theory of Poisson deformations of conic symplectic singularities in algebraic geometry, we produce large families of (generally incomplete) hyperkähler metrics deforming hyperkähler cone metrics. In the second part of the talk, I will report on work in progress about the construction of non-compact hyperkähler metrics generalising the geometry of ALF spaces of dihedral type to higher dimensions. We produce candidate holomorphic symplectic manifolds and twistor spaces from Hilbert schemes of hypertoric manifolds with an action of a Weyl group. The spaces we define are closely related to Coulomb branches of 3-dimensional supersymmetric gauge theories.

**Ryushi Goto (Osaka University)**

**Matsushima-Lichnerowicz type theorems of Lie algebra of automorphisms of generalized Kähler manifolds of symplectic type**

In this talk, we show the reductivity of the Lie algebra of the automorphisms of a generalized complex manifold which admits a compatible generalized Kähler structure of symplectic type with constant scalar curvature, which is an obstruction Theorem of Matsushima and Lichnerowicz type. We discuss deformations starting from an ordinary Kähler manifold with constant scalar curvature and show that nontrivial generalized Kähler structures of symplectic type with constant scalar curvature arise as deformations if the Lie algebra of automorphisms is trivial. If time is permit, we obtain constant scalar curvature generalized Kähler structures on even dimensional semisimple Lie groups.

**Marco Gualtieri (University of Toronto)**

**Brane quantization and generalized Kahler structures**

We will explore the ramifications of the idea that a generalized Kahler metric is a Lagrangian brane in a symplectic Morita equivalence which itself has a space filling brane. This pair of branes, one Lagrangian and one space-filling, allow us to implement a version of the Gukov-Witten quantization procedure, leading to a noncommutative algebra quantizing the Hitchin Poisson structure. This is based on the joint work with Francis Bischoff (https://arxiv.org/abs/2108.01658)

**Weiyong He (University of Oregon)**

**Harmonic and biharmonic almost complex structures**

We study harmonic and biharmonic almost complex structures on compact almost Hermitian manifolds, in particular on dimension four. We prove regularity results for harmonic almost complex structure, parallel to the theory of harmonic maps. We also prove the existence of energy-minimizing smooth biharmonic almost complex structure on dimension four, given a homotopy class. These objects should be interesting for the study of geometry of almost complex manifolds.

**Julien Keller (UQAM)**

**Constant scalar curvature Kähler metrics with cone singularities**

We will review some recent progress on constant scalar curvature Kähler metrics with cone singularities along a divisor. We will discuss the existence of such special metrics, focussing on the particular case of projective bundles. Moreover, we will explain some new results on the relationship between such metrics and uniform log-K stability.

**Claude LeBrun (Stony Brook University)**

**Einstein Manifolds, Kahler Metrics, and Conformal Geometry**

The underlying smooth compact 4-manifold of any Del Pezzo surface admits Einstein metrics, and the known Einstein metrics on each such manifold can be shown to sweep out exactly one connected component of the Einstein moduli space. This talk will highlight some of the remarkable properties of these metrics from the standpoint of conformal geometry, with a pronounced emphasis on the Weyl functional.

**Éveline Legendre (Université Paul Sabatier Toulouse 3)**

**Extremal Sasaki structure and weighted cscK metrics**

Given two commuting Reeb vector fields on a fixed CR manifold of Sasaki type, V. Apostolov and D. Calderbank have given an exact and simple formula between their respective associated transversal scalar curvature. If one of these Reeb vector fields is quasiregular the transversal scalar curvature of the other turns out to be a particular case of weighted cscK metric, as studied by Lahdili, on the orbifold Kähler quotient.

I will discuss parts of a joint work with V.Apostolov and D.Calderbank where we have extended their result to variation within the K\"ahler class of the quotient. This allowed us to translate the Sasaki--Mabuchi functional in terms of weighted Mabuchi functional (and then use Lahdili's result). We also have related the weighted K-stability to the Sasaki K-stability of Collins--Szekelyhidi on some class of test configurations and have shown that, on that class, we can improve their result from K-semistability to K-stability.

**Jason Lotay (University of Oxford)**

**Some remarks on contact Calabi-Yau 7-manifolds**

In geometry and physics it has proved useful to relate \(G_2\) and Calabi-Yau geometry via circle bundles. Contact Calabi-Yau 7-manifolds are, in the simplest cases, such circle bundles over Calabi-Yau 3-orbifolds. These 7-manifolds provide testing grounds for the study of geometric flows which seek to find torsion-free \(G_2\)-structures. They also give useful backgrounds to examine the heterotic \(G_2\) system (also known as the \(G_2\)-Hull-Strominger system), which is a coupled set of equations arising from physics that involves the \(G_2\)-structure and gauge theory on the 7-manifold. I will report on recent progress on both of these directions in the study of contact Calabi-Yau 7-manifolds, which is joint work with H. Sá Earp and J. Saavedra.

**Leon Roschig & Leander Stecker (Philipps-Universität Marburg)**

**Revisiting the Classification of Homogeneous 3-Sasakian and Quaternionic Kähler Manifolds**

This talk is based on joint work with Oliver Goertsches, see arXiv:2110.03603. We provide a new, self-contained proof of the classification of homogeneous 3-Sasakian manifolds, which was originally obtained by Boyer, Galicki and Mann. In doing so, we construct an explicit one-to-one correspondence between simply connected homogeneous 3-Sasakian manifolds and simple complex Lie algebras via the theory of root systems. We also discuss the non-simply connected case and derive the famous classification of homogeneous positive quaternionic Kähler manifolds due to Wolf and Alekseevskii from our results.

**Uwe Semmelmann (Universität Stuttgart)**

**Stability of Einstein metrics**

Einstein metrics can be characterised as critical points of the (normalised) total scalar curvature functional. They are always saddle points. However, there are Einstein metrics which are local maxima of the functional restricted to metrics of fixed volume and constant scalar curvature. These are by definition stable Einstein metrics. Stability can equivalently be characterised by a spectral condition for the Lichnerowicz Laplacian on divergence- and trace-free symmetric 2-tensors, i.e. on so-called tt-tensors: an Einstein metric is stable if twice the Einstein constant is a lower bound for this operator. Stability is also related to Perelman's \(\nu\) entropy and dynamical stability with respect to the Ricci flow.

In my talk I want to discuss the stability condition. I will present a recent result obtained with G. Weingart, which completes the work of Koiso on the classification of stable compact symmetric spaces. Moreover, I will describe an interesting relation between instability and the existence of harmonic forms. This is done in the case of nearly Kähler, Einstein-Sasaki and nearly \(G_2\) manifolds. If time permits I will also explain the instability of the Berger space \(SO(5)/SO(3)\), which is a homology sphere. In this case instability surprisingly is related to the existence of Killing tensors. The latter results are contained in joint work with M. Wang and C. Wang.

**Jeffrey D. Streets (University of California at Irvine)**

**Generalized Ricci flow**

The generalized Ricci flow is a natural extension of the Ricci flow which incorporates torsion. In this talk I will describe fundamental geometric and analytic properties of this equation, leading to global existence and convergence results with applications to complex geometry.

**Andrew Swann (Aarhus University)**

**Nearly Kähler manifolds with torus symmetry**

The currently known compact examples of (strict) nearly Kähler six-manifolds all admit a rank two symmetry group. This talk will describe how multi-moment maps can be used to provide local descriptions of nearly Kähler structures with a two-torus symmetry. We will also discuss behaviour at critical points of the multi-moment map and situations where there is enhanced symmetry. This is joint work with Giovanni Russo.

**Christina Tønnesen-Friedman**

**Sasakian geometry on Sphere bundles over a product of two compact Riemann surfaces**

This talk, which is based on recent work with Charles P. Boyer, will discuss the existence question for extremal and constant scalar curvature Sasaki metrics on three-dimensional sphere bundles over a product of two compact Riemann surfaces. We will apply the so-called fiber join construction for K-contact manifolds, introduced by T. Yamazaki around the turn of the century, to the Sasaki case.

**Yuri Ustinovskiy (New York University)**

**Variational Approach to Generalized Kähler-Ricci Solitons**

This talk concerns (gradient) Generalized Kähler-Ricci solitons (GKRS) - geometric structures independently arising in the context of supersymmetric sigma models, generalized geometry of Hitchin and Gualtieri and uniformization problems in non-Kähler geometry. GKRS are natural generalizations of Einstein Riemannian manifolds and Ricci-solitons, incorporating a torsion term \(H\in \Lambda^3(M).\)

Every Generalized Kähler (GK) structure on a smooth manifold \(M\) admits an infinitesimal variation parametrized by \(C^\infty(M,\mathbb{R})/\mathbb{R}\). Integrating such infinitesimal variations we obtain a notion of GK class, and it makes sense to study the existence and uniqueness questions for GKRS in a given class. Following the fundamental ideas in Kähler geometry, we define a weighted J-functional on the GK class of a log-nondegenerate GK manifold. This functional naturally extends the (weighted) Aubin's functional in Kähler geometry and the (unweighted) J-functional of Apostolov and Streets in nondegenerate GK setting. We prove that a log-nondegenerate GK structure is a critical point for the weighted J-functional if and only if it is a gradient steady GKRS. We use this functional to prove rigidity of solitons in a given GK class, and use the latter to deduce complete classification of compact GKRS for \(\dim M=4\). The key statement in this classification is the uniqueness of the solitons on Hopf surfaces. (This talk is based on the joint work with Vestislav Apostolov and Jeffrey Streets.)

**Thomas Walpuski (Humboldt-Universität zu Berlin)**

**The Gopakumar–Vafa finiteness conjecture**

In 1998, using arguments from M–theory, Gopakumar and Vafa argued that there are integer BPS invariants of symplectic Calabi–Yau 3–folds. Unfortunately, they did not give a direct mathematical definition of their BPS invariants, but they predicted that they are related to the Gromov–Witten invariants by a transformation of the generating series. The Gopakumar–Vafa conjecture asserts that if one defines the BPS invariants indirectly through this procedure, then they satisfy an integrality and a (genus) finiteness condition.

The integrality conjecture has been resolved by Ionel and Parker. A key innovation of their proof is the introduction of the cluster formalism: an ingenious device to side-step questions regarding multiple covers and super-rigidity. Their argument could not resolve the finiteness conjecture, however. The reason for this is that it relies on Gromov’s compactness theorem for pseudo-holomorphic maps which requires an a priori genus bound. It turns out, however, that rather powerful tools from geometric measure theory imply a compactness theorem for pseudo-holomorphic cycles. This can be used to upgrade Ionel and Parker’s cluster formalism and prove both the integrality and finiteness conjecture. This talk is based on joint work with Eleny Ionel and Aleksander Doan.

**Xiangwen Zhang (UCI)**

**A geometric flow for Type IIA superstrings**

The equations of flux compactifications of Type IIA superstrings were written down by Tomasiello and Tseng-Yau. To study these equations, we introduce a natural geometric flow on symplectic Calabi-Yau 6-manifolds. We prove the well-posedness of this flow and establish the Shi-type estimates which provide a criterion for the long time existence. This is based on joint work with Fei, Phong and Picard.