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# Geometry of Calabi-Yau Metrics

Communicated by *Notices* Associate Editor Chikako Mese

## Introduction

Calabi-Yau metrics are named after two mathematicians: E. Calabi and S.-T. Yau. They are fundamental objects in geometry and physics.

Let be a differentiable manifold of dimension A Riemannian metric . defines a smoothly varying family of inner products on the tangent spaces of We fix a point . in and choose an identification between and the Euclidean space Then for any piecewise smooth loop . based at parallel transport along , with respect to the Levi-Civita connection yields an orthogonal transformation The .*holonomy group* of is by definition the subgroup of consisting of all such transformations This is up to conjugation independent of the choices of . and We say . is a *Calabi-Yau metric* if and there is a further identification so that the holonomy group of is contained in In the literature one often uses a weaker notion of local holonomy group which involves only null-homotopic loops . However for our purposes in this article we will take the above more restrictive definition. .

A salient feature of Calabi-Yau metrics is that they have vanishing Ricci curvature, hence provide solutions to the Riemannian vacuum Einstein equation. This can be deduced as a consequence of the Ambrose-Singer holonomy theorem and the Bianchi identity. In terms of the Berger classification of Riemannian holonomy groups, Calabi-Yau metrics are examples of Riemannian metrics with *special holonomy*, which play a pivotal role in string theory.

This article is an expanded version of the notes for some recent colloquia and mini-school lectures given by the author. The main goal here is to explain to the readers some constructions and the geometry of Calabi-Yau metrics; in the meantime we aim to selectively discuss several interesting examples in the field and some recent research progress. Obviously, this article is by no means supposed to be a comprehensive historic survey of the subject. The topics are merely chosen according to the personal taste of the author, and there are many other related papers emphasizing different aspects. Also, for lack of space it is impossible to provide precise references to all the results mentioned below, but the author hopes that interested readers can easily look up further details on their own.

The author would like to thank Xuemiao Chen, John Lott, Holly Mandel, and an anonymous referee for helpful comments that improved the exposition.

## Yau’s Existence Theorem

We first consider the more flexible notion of *Kähler metrics*. A Riemannian metric on a differentiable manifold of dimension is Kähler if its holonomy group is contained in Since . one can define an almost complex structure on , i.e., a tensor field , satisfying that is orthogonal with respect to and is parallel under the Levi-Civita connection. By the Newlander-Nirenberg theorem it follows that is a *complex* manifold, that is to say, near each point one can find complex-valued coordinates such that the transition functions are holomorphic and is represented by for …, , Furthermore, the associated .*Kähler form* defined by satisfies and it can be locally written as , for a real-valued *potential* function In terms of the complex coordinates, the latter means that .

For convenience we also call a Kähler metric. Conversely, any real-valued function such that the matrix is positive definite defines a local Kähler metric via the above formula. If is compact, a Kähler metric defines a non-trivial cohomology class in A standard example is the Fubini-Study metric . on the complex projective space In local affine coordinates . one can take the potential function , Clearly . also restricts to a Kähler metric on any complex submanifold of .

Suppose now is a Kähler metric on a complex manifold Then by definition it is a Calabi-Yau metric if and only if there exists a non-zero complex-valued . -form locally given as , which is parallel with respect to the Levi-Civita connection. This condition is equivalent to saying that , is holomorphic and nowhere vanishing, i.e., is a holomorphic volume form, and moreover the following equation holds:

where

What we have done in the above is to decouple the definition of a Calabi-Yau metric into two ingredients of different flavor. First we need *Calabi-Yau variety*. The existence of

The second ingredient we need is a Kähler metric ^{1}, which is the reason for the name “Calabi-Yau.”

^{1}

We remark that the original Calabi conjecture was stated in a more general form, and has led to a far-reaching program in Kähler geometry for the last few decades, centered around the question of finding canonical Kähler metrics on complex manifolds.

A classical fact is that since

In terms of local complex coordinates this takes the form of a complex Monge-Ampère equation. Yau’s proof of Theorem 0.1 is by solving 2 via a *continuity method*. Write

where again *a priori* estimates, namely, if one can show that for any

Theorem 0.1 immediately produces many examples of non-trivial compact Calabi-Yau metrics from algebraic geometry.

In the above example, when

In light of this, it is interesting to understand more precisely the geometry of the Calabi-Yau metrics resulting from Theorem 0.1. This is the topic that we shall discuss in the rest of this article.

## Calabi-Yau Metrics with Symmetry

By the Bochner technique, having vanishing Ricci curvature implies that a compact Calabi-Yau metric cannot admit any non-trivial continuous symmetry, i.e., any Killing vector field must be parallel. But this does not have to be the case for non-compact manifolds. Indeed, there are explicit constructions of non-compact Calabi-Yau metrics using symmetry which, as we shall see later, often provide models and intuition for understanding the geometry of compact Calabi-Yau metrics near the degeneration limit.

In the simplest setting when the complex dimension is 2, we recall the well-known *Gibbons-Hawking ansatz* GH. Choose a positive harmonic function

where

We claim that *hyperkähler*.

Conversely, any Calabi-Yau metric in 2 complex dimensions with a free

The Calabi-Yau metrics constructed this way have little topology. The situation becomes more appealing if one makes certain variants of the construction. First we can let

At this point we can build various examples. To start with, we can take *Hopf fibration*, explicitly expressed as

If we instead take

Next we make a small change, and take *Taub-NUT* metric. It is a complete Calabi-Yau metric on

Now we choose

Then we obtain the well-known *Eguchi-Hanson space*. It contains an embedded 2-sphere

As a more general example of

In terms of complex geometry, we can choose a complex structure so that the obvious projection map to the Riemannian surface *Calabi ansatz*. The latter is a general way of producing a canonical metric on the total space of a holomorphic vector bundle over a given Kähler manifold. In our setting, we consider an *Chern connection* has curvature form given by *Calabi model space*. When

In TY90, Tian-Yau constructed complete Calabi-Yau metrics asymptotic to the Calabi model space. The underlying complex manifold is the complement of a smooth anti-canonical divisor

Of course one can play with the Gibbons-Hawking ansatz and generate many more examples of non-compact Calabi-Yau metrics. Often they have significance in modeling singularity formations of Calabi-Yau metrics, just like the Eguchi-Hanson spaces.

## Gluing Construction on K3 Surfaces

Now we describe a very different method of constructing compact Calabi-Yau manifolds, via the *gluing* technique. There are many references on this, see for example Don12.

Let

One can easily resolve the singularities of *Kummer construction*. To see this we consider the local model near each singularity. Let *K3 surface*.

We want to construct Calabi-Yau metrics on

This is indeed the Eguchi-Hanson metric in disguise. For

Now given

Now one wants to correct the error and deform *weights* into the Banach spaces. This captures the degenerate geometry and results in uniform weighted elliptic estimates. This is the crucial technical point of the construction, and the upshot is that the correction is possible if

Compared with Yau’s existence theorem, an obvious drawback of the gluing construction is that it can only describe a small open set of the space of all Calabi-Yau metrics. On the other hand, the benefit is that it provides a more precise geometric description of the metrics. More importantly, the gluing technique is also a general construction useful for many other geometric PDEs, where it is impossible to have an analogue of Yau’s existence theorem. For example, it was used by Joyce to construct compact Riemannian manifolds with holonomy group

The gluing construction, viewed in reverse, also yields examples of geometric degenerations of Calabi-Yau metrics, fitting into the general convergence theory of Riemannian metrics with bounded Ricci curvature. The latter, when applied to our setting, states that given a sequence of complete Calabi-Yau metrics *Gromov-Hausdorff limit*, which is a complete metric space *singularities* of

The theory is well-developed when one imposes a further *volume non-collapsing condition*. This means that there exists *regular* set *singular* set *bubble tree structure*. Heuristically, this appears in many other areas of geometric analysis, such as harmonic maps and Yang-Mills connections.

In the above gluing construction, we are in the non-collapsing situation, and as *model* for the singularity formation of Calabi-Yau metrics.

## Moduli Spaces

Let *versal* deformation space of complex structures, which is itself an open set in the complex vector space *Kähler cone*, which is an open cone in the real vector space

It is then natural to study the global structure of the moduli space of Calabi-Yau metrics. In low dimensions, we have the classical results. Namely, when

When

The Torelli theorem guarantees that

The higher-dimensional situation is more challenging and there are very few general results. The following is a folklore question.

When *Reid’s fantasy* hoping that all compact Calabi-Yau varieties can be connected via geometric transitions, through suitable classes of singular Calabi-Yau varieties.

A closely related question is to study the compactification of a fixed connected component of the moduli space of all Calabi-Yau metrics. Namely, we want to know the behavior of a sequence of Calabi-Yau metrics * Kähler degenerations* and *complex structure degenerations*, and we will discuss these in the following sections.

## Kähler Degenerations

Fix a compact Calabi-Yau variety

We divide the discussion in two cases. First we assume *null-locus* of