Generalized Complex Geometry

Calabi-Yau 3-folds, i.e. complex 3-dimensional Kähler manifolds with holonomy group contained in SU(3), play a central role in complex, symplectic and algebraic geometry, mathematical physics, and more.  It is natural to seek extensions beyond the setting of Kähler geometry.

In this talk, I will discuss compact non-Kähler Hermitian 3-folds $M$ with $\partial \bar \partial$-closed fundamental $2$-form, whose Bismut-Ricci form is identically zero. These geometries first appeared in the 1980’s in mathematical physics as string backgrounds extending the Kähler Calabi-Yau condition, also providing motivating examples of generalized Kähler solitons, but they have also been considered in the mathematics literature as a Hermite-Einstein geometry associated the bundle $T^{1,0} \oplus T^*_{1,0}$ with its Bismut holomorphic structure.

It turns out that compact non-Kähler Hermitian manifolds with $\partial \bar \partial$-closed fundamental $2$-form and zero Bismut-Ricci form have a canonical symmetry reduction to complex co-dimension one. In complex dimension 3, the transversal geometry is actually Kähler and governed by a single 6th-order scalar PDE for the relative Kähler potential of the underlying Kähler metric. In the quasi-regular case, i.e. when the $3$-fold fibers over a compact complex orbifold surface, the PDE has an infinite dimensional momentum map interpretation, similar to the much studied Kähler metrics of constant scalar curvature. We use this to obtain obstructions for the existence of solutions in terms of the automorphism group, paralleling results by Futaki and Calabi-Lichnerowicz-Matsushima in the cscK case. As an application, we characterize the standard Samelson’s examples, i.e. the Hermitian 3-folds which are covered by either $SU(2) \times {\mathbb R }^3$ or $SU(2) \times SU(2)$ equipped with a bi-invariant metric and a left invariant complex structure, as the only regular examples having Bott-Chern number $h_{BC}^{1,1} =2$. We also obtain explicit solutions of the PDE on orthotoric Kähler orbifold surfaces, which yield infinitely many non-Kähler $3$-folds with $\partial \bar \partial$-closed fundamental $2$-form and zero Bismut-Ricci form on $S^3 × S^3$ and $S^1 × S^2 × S^3$, which are not locally isometric to a Samelson geometry. These appear to be the first such examples. 

This talk is based on joint works with Barbaro, Lee and Streets, and Lahdili and Lee, available on arXiv:2408.09648 and arXiv:2601.04937.

In this talk I will explain how generalized geometry can also be used to reframe SKT geometry. I will use this framing to develop the Hodge theory of generalized Kahler and SKT manifolds. I will illustrate the theory with a collection of examples.

Generalized geometry arose as a robust and interesting mathematical framework for understanding the geometry of the target manifold M of a 2-dimensional Sigma model with N=(2,2) worldsheet supersymmetry (or N=2 space-time susy). This leads to generalized Kähler structures (GKS) on M satisfying natural conditions on the curvature, equivalent to a pair of solutions of the physical Killing spinor equations. Mathematically, a GKS defines a pair of commuting generalized complex structures \cJ_1, \cJ_2, and has an interesting interplay with Poisson geometry and the theory of higher differentiable stacks. From a metric viewpoint, nonetheless, the physical supersymmetry equations on a compact manifold imply that the only solutions are Kähler, and one is back to classical complex geometry.

One way to avoid this "Maldacena-Nuñez no-go theorem" for GKS, accommodating compact non-Kähler examples with canonical geometries, is to drop 1/2 of each generalized complex structure (CGS) of the GKS, in such a way that

1/2 * GCS + 1/2 * GCS \neq 1 * GCS,

contradicting basic arithmetics. The corresponding physical theory has then N=(0,2) worldsheet supersymmetry (or N=1 space-time susy), and relates to more classy geometries, such as conformally balanced hermitian metrics, pluriclosed solitons, and Hermitian-Yang-Mills connections. Interestingly, the relation to physics is still present, and one can speculate on mirror symmetry and holography for non-Kahler manifolds. In this talk I will overview different aspects of these 1/2 + 1/2 geometries (which do not involve any CGS, sorry about that).

I will discuss the geometric structures arising from two very natural variations of the setup of generalized complex geometry. On the one hand, I will review Bn-generalized complex geometry, where cosymplectic and normal almost contact structures join symplectic and complex ones. On the other hand, I will introduce Cn-generalized complex geometry, which is joint work with Filip Moučka and can combine complex structures with pseudo-Riemannian metrics.

The theory of constant scalar curvature K\”ahler (cscK) metrics emerged from Calabi’s original visionary work in the 1950’s.  This subject acquires deep structure through the Fujiki-Donaldson moment map interpretation, and is guided by the central YTD conjecture which is expected to yield deep consequences for complex and algebraic geometry.  In recent years the Fujiki-Donaldson framework has been extended to geometries first arising in (2,2)-supersymmetric quantum field theory, now known as generalized K\”ahler (GK) structures.  In this talk I will develop the notion of generalized Ricci soliton/cscGK structure, then formulate a YTD conjecture in this setting which interacts delicately with the holomorphic Poisson geometry of K\”ahler manifolds.  We solve this conjecture over Fano K\”ahler-Einstein manifolds.  Based on joint works with V. Apostolov, B. Pym, and Y. Ustinovskiy.