The main purpose of this paper is to prove a general result about the geometry of holomorphic line bundles over Kähler manifolds. This result is essentially a verification of a conjecture of Tian [22] and Tian has, over many years, highlighted the importance of the question for the existence theory of Kähler-Einstein metrics. We will begin by stating this main result.

We consider data $(X,g,J,L,A)$ where $(X,g)$ is a compact Riemannian manifold of real dimension $2n$, $J$ is a complex structure which respect to which the metric is Kähler, $L$ is a Hermitian line bundle over $X$ and $A$ is a connection on $L$ with curvature $-i\omega$, where $\omega$ is the Kähler form. We will often just write $X$ as an abbreviation for this data. We suppose the metric satisfies fixed upper and lower bounds on the Ricci tensor

(1.1)

$$-\frac{g}{2}\leq{\rm Ric}\leq g.$$

(The particular bounds we have chosen are just convenient normalisations; any other fixed bounds would do.) For $V,c>0$ let ${\mathcal{K}}(n,c,V)$ denote the class of all such data such that the volume of $X$ is $V$ and the “non-collapsing” condition

(1.2)

$${\rm Vol}\ B_{r}\geq c\frac{\pi^{n}}{n!}r^{2n}$$

holds. Here $B_{r}$ is any metric $r$-ball in $X$, $r$ is any number less than the diameter of $X$ and the normalising factor $\pi^{n}/n!$ is the volume of the unit ball in $\mbox{${\mathbb{C}}$}^{n}$.

The connection induces a holomorphic structure on $L$ and for each positive integer $k$ there is a natural $L^{2}$ hermitian metric on the space $H^{0}(X,L^{k})$. Recall that the “density of states” (or Bergman) function $\rho_{k,X}$ is defined by

$$\rho_{k,X}=\sum|s_{\alpha}|^{2},$$

where $(s_{\alpha})$ is any orthonormal basis of $H^{0}(X,L^{k})$. An equivalent definition is that $\rho_{k,X}(x)$ is the maximum of $|s(x)|^{2}$ as $s$ runs over the holomorphic sections with $L^{2}$ norm $1$. Thus, to establish a lower bound on $\rho_{k,X}(x)$ we have to produce a holomorphic section $s$ with $L^{2}$ norm not too large and with $|s(x)|$ not too small. Write

$$\underline{\rho}(k,X)=\min_{x\in X}\rho_{k,X}(x).$$

Standard theory, a part of the Kodaira Embedding Theorem, asserts that for each fixed $X$ we have $\underline{\rho}(k,X)>0$ for large enough $k$. Our main result can be thought of as an extension of this statement which is both uniform over ${\mathcal{K}}(n,c,V)$ and gives a definite lower bound.

The proof involves a combination of the Gromov-Hausdorff convergence theory— developed by Anderson, Cheeger, Colding, Gromoll, Gromov, Tian and others over the past 30 years or so—and the “Hörmander technique” for constructing holomorphic sections. When $n=2$ the theorem was essentially proved by Tian in [21] and the overall scheme of our proof is similar. The theorem provides the foundations for a bridge between the differential geometric convergence theory and algebraic geometry, leading to the following result (as indicated by Tian).

(More precise statements, and more detailed information, are given in Section 4 below.)

Many of the ideas and arguments required to derive this are similar to those of Ding and Tian in [9] who considered Fano manifolds with Kähler-Einstein metrics. Then the limit is a “$\mathbb{Q}$-Fano” variety, as Ding and Tian conjectured.

In Section 2 we review relevant background in convergence theory and complex differential geometry. Given this background, the rest of the proof is essentially self-contained. This proof is given in Section 3. We begin by reducing Theorem 1.1 to a “local” statement (Theorem 3.2)involving a point in a Gromov-Hausdorff limit space and attention is then focused on a tangent cone at this point. In Section 4 we give the proof of Theorem 1.2. We also establish some further relations between the differential geometric and algebro-geometric theories. In Section 5 we include a more detailed analysis of tangent cones in the 3-dimensional case, showing that these are cones over Sasaki-Einstein orbifolds and discuss the likely picture in the higher dimensional situation.

Our main interest throughout this paper is in the case when $X$ is a Fano manifold, the metric is Kähler-Einstein with positive Ricci curvature and $L=K_{X}^{-1}$. Then the Ricci bound (1.1) holds trivially and the non-collapsing condition (1.2) is automatic for a suitable $c$ (in fact with $c=V(2n-1)!!/(2^{n+1}(2n-1)^{n}\pi^{n})$). But the general hypotheses we have made above seem to give the natural context for the discussion here, although the applications outside the Fano case may be limited. In the case of Kähler-Einstein metrics of negative or zero Ricci curvature there is of course a complete existence theory due to Aubin and Yau. It would be interesting to characterise the non-collapsing condition in this situation algebro-geometrically.

We would like to emphasise that this is a “theoretical” paper in the following sense. Our purpose is to establish that some high powers $k_{0},k_{1}$ of a positive line bundle have certain good properties uniformly over manifolds in ${\mathcal{K}}(n,c,V)$ and Gromov-Hausdorff limits thereof. For many reasons one would like to know values of $k_{0},k_{1}$ which are, first, explicitly computable and, second, realistic. (That is, not too different from the optimal values which, in reality, yield these good properties.) This paper is theoretical in that we will not attempt to do anything of this kind. Of our two foundations—the Hörmander technique and convergence theory—-the first is quite amenable to explicit estimates but the second is not. So it is unclear whether even in principle one could extract any computable numbers. Of course this is an important question for future research. Given this situation we have not attempted to make the arguments in Section 3 and 4 efficient, in the sense that (even if one somehow had effective constructions for the building blocks) our arguments from those building blocks who lead to huge, completely unrealistic, numbers. This is connected to certain definite and tractable mathematical questions which we take up briefly again at the very end of the paper, where we formulate a conjectural sharper version of Theorem 1.1 (Conjecture 5.15).

We finish this introduction with some words about the origins of this paper. While the question that we answer in Theorem 1.1 is a central one in the field of Kähler-Einstein geometry, it is not something that the authors have focused on until recently. The main construction in this paper emerged as an off-shoot of a joint project by the first-named author and Xiuxiong Chen, studying the slightly different problem of Kähler-Einstein metrics with cone singularities along a divisor. A companion article by the first named author and Chen, developing this related theory, will appear shortly. Both authors are very grateful to Chen for discussions of these matters, extending over many years.