Narrow Results

These are the notes for lectures given at the Sanya winter school in complex analysis and geometry in January 2016. In Sec. 1, we review the meaning of Ricci curvature of Kähler metrics and introduce the problem of finding Kähler–Einstein metrics. In Sec. 2, we describe the formal picture that leads to the notion of K-stability of Fano manifolds, which is an algebro-geometric criterion for the existence of a Kähler–Einstein metric, by the recent result of Chen–Donaldson–Sun. In Sec. 3, we discuss algebraic structure on Gromov–Hausdorff limits, which is a key ingredient in the proof of the Kähler–Einstein result. In Sec. 4, we give a brief survey of the more recent work on tangent cones of singular Kähler–Einstein metrics arising from Gromov–Hausdorff limits, and the connections with algebraic geometry.

In this paper, we show that K-semistable Fano manifolds with the smallest alpha invariant are projective spaces. Singular cases are also investigated.

We generalize the definition of alpha invariant to arbitrary codimension. We also give a lower bound of these alpha invariants for K-semistable $\mathbb{Q}$-Fano varieties and show that we can characterize projective spaces among all K-semistable Fano manifolds in terms of higher codimensional alpha invariants. Our results demonstrate the relation between alpha invariants of any codimension and volumes of Fano manifolds in the characterization of projective spaces.

For a variety $X$, a big $\mathbb{Q}$-divisor $L$ and a closed connected subgroup $G\subset \mathrm{\text{Aut}}(X,L)$ we define a $G$-invariant version of the $\delta$-threshold. We prove that for a Fano variety $(X,-K_X)$ and a connected subgroup $G\subset \mathrm{\text{Aut}}(X)$ this invariant characterizes $G$-equivariant uniform $K$-stability. We also use this invariant to investigate $G$-equivariant $K$-stability of some Fano varieties with large groups of symmetries, including spherical Fano varieties. We also consider the case of $G$ being a finite group.

We use the equivariant localization formula to prove that the Donaldson–Futaki invariant of a compact smooth (Kähler) test configuration coincides with the Futaki invariant of the induced action on the central fiber when this fiber is smooth or have orbifold singularities. We also localize the Donaldson–Futaki invariant of the deformation to the normal cone.

We completely classify K-stability of log del Pezzo hypersurfaces with index 2.

Using the Abban–Zhuang theory and the classification of 3-dimensional log smooth log Fano pairs due to Maeda, we prove that almost all threefold log Fano pairs $(X,D)$ of Maeda type with reducible boundary $D$ are K-unstable for any values of the boundary coefficients. We also correct several inaccuracies in Maeda’s classification.

We compute K-semistable domains for various examples of log pairs.

The wonderful compactification $X_m$ of a symmetric homogeneous space of type AIII$(2,m)$ for each $m\ge 4$ is Fano, and its blowup $Y_m$ along the unique closed orbit is Fano if $m\ge 5$ and Calabi–Yau if $m=4$. Using a combinatorial criterion for K-polystability of smooth Fano spherical varieties obtained by Delcroix, we prove that $X_m$ admits a Kähler–Einstein metric for each $m\ge 4$ and $Y_m$ admits a Kähler–Einstein metric if and only if $m=4,5$.

This article is a broad-brush survey of two areas in differential geometry. While these two areas are not usually put side-by-side in this way, there are several reasons for discussing them together. First, they both fit into a very general pattern, where one asks about the existence of various differential-geometric structures on a manifold. In one case we consider a complex Kähler manifold and seek a distinguished metric, for example a Kählera–Einsten metric. In the other we seek a metric of exceptional holonomy on a manifold of dimension 7 or 8. Second, as we shall see in more detail below, there are numerous points of contact between these areas at a technical level. Third, there is a pleasant contrast between the state of development in the fields. These questions in Kähler geometry have been studied for more than half a century: there is a huge literature with many deep and far-ranging results. By contrast, the theory of manifolds of exceptional holonomy is a wide-open field: very little is known in the way of general results and the developments so far have focused on examples. In many cases these examples depend on advances in Kähler geometry.