Timetable for Newcastle Complex Geometry Workshop

Thanks to recent work of X.X. Chen and J. Cheng it is now known that a compact Kähler manifold admits a cscK metric if and only if it is geodesically stable. In particular, the Yau-Tian-Donaldson conjecture reduces to the statement that geodesic stability is equivalent to K-stability. Motivated by this, we discuss a notion of "geodesic K-stability" that can be naturally interpreted as a weak version of both geodesic stability and K-stability, thus serving as a possible bridge between the two. As a main result we then deduce various new stability results for non quantizable Kähler manifolds admitting holomorphic vector fields, and explain how to compare geodesic K-stability to the usual notion of K-stability (showing that they are equivalent in many cases).

G2-manifolds are the Riemannian 7-manifolds with G2 holonomy and in many respects can be regarded as analogues of Calabi-Yau 3-folds. In joint work with Mark Haskins and Johannes Nordström we construct infinitely many families of new complete non-compact G2-manifolds (only four such manifolds were previously known). The underlying smooth 7-manifolds are all circle bundles over asymptotically conical Calabi-Yau 3-folds, the metrics are circle-invariant and have an asymptotic geometry that is the 7-dimensional analogue of the geometry of 4-dimensional ALF hyperkähler metrics. After describing the main features of our construction I will concentrate on some illustrative examples, describing how results in algebraic geometry about isolated singularities and their resolutions can be used to produce examples of complete G2-manifolds.

A symplectic Fano manifold is a compact symplectic manifold where the first Chern class is a positive multiple of the cohomology class of the symplectic form. In dimension 4, these manifolds are del Pezzo surfaces, as shown by Ohta and Ono. In dimensions 12 and above, some non-Kähler examples where found by Fine and Panov, using the symplectic twistor space construction of Reznikov. They also conjectured that 6-dimensional symplectic Fano manifolds with a Hamiltonian circle action have a compatible complex projective structure, and hence fall into the famous classification of Iskovskikh, Mori and Mukai.

In this talk, I will discuss a joint work with Dmitri Panov, where we show that 6-dimensional symplectic Fano manifolds with a Hamiltonian circle action are simply connected and satisfy c_1c_2=24. I will also discuss how this implies that these manifolds are (symplectically) birational to CP^3.

Okounkov bodies were introduced by Okounkov in the 90's as a way of generalizing the correspondence between line bundles and polytopes in toric geometry to the setting of ample line bundles on projective manifolds. I will discuss a new way of thinking of Okounkov bodies as arising from certain degenerations of the manifold together with its Kähler structure. This is work in progress together Ya Deng.