We shall describe an approximate ansatz for producing Calabi-Yau metrics, which simultaneously generalizes the Calabi ansatz on the total space of a positive line bundle over a compact Calabi-Yau manifold, and the semiflat metrics coming from torus invariant dimensional reductions of Calabi-Yau metrics. A very closely related ansatz in the context of polarized algebraic degenerations, was discovered in [14] in an attempt to give a conjectural differential geometric interpretation to the non-archimedean Monge-Ampère equation. Some of our terminologies will therefore reflect the non-archimedean origin of this ansatz.

Let $L_{1},\ldots L_{m}$ be positive line bundles over an $(n-m)$-dimensional compact Calabi-Yau manifold $Y$ with positively curved smooth Hermitian metrics $h_{L_{1}},\ldots h_{L_{m}}$, and let $Z$ be the total space of $L_{1}\oplus\ldots\oplus L_{m}\to Y$. Let $r_{i}$ be the radius distance function on $L_{i}$. In local holomorphic trivializations of $L_{i}$, the Hermitian metric on $L_{i}$ can be written in terms of local potentials as $h_{L_{i}}=e^{-2\phi_{i}}$, and the radius distance $r_{i}=|\xi_{i}|e^{-\phi_{i}}$, where $\xi_{i}$ are the local fibre coordinates on $L_{i}$. Our task is to construct approximate Calabi-Yau metrics on the region $\{0<r_{i}\ll 1,\forall i\}$ inside $Z$, on the line bundle $L\to Z$ obtained by pulling back a (semi-positive) line bundle $L\to X$. For our main intended applications, $L$ is in fact trivial, and $m=2$. An important conceptual point is that these ansatz metrics will be incomplete in the $r_{i}\sim 1$ region. Their aim is to provide local metric models on the generic regions of complete Calabi-Yau manifolds, and the global problem would also involve the nontrivial step of finding partial completions of these ansatz metrics. Concretely, this incompleteness means in the $r_{i}\sim 1$ region this particular ansatz breaks down, and one must find some alternative ansatz.

It would be helpful to keep in mind that the metric will look like an iterated fibration. On the smallest scale, we have the $T^{m}$ tori, coming from the circle directions of the line bundles $L_{1},\ldots L_{m}$. These are fibred over compact manifolds diffeomorphic to $Y$, which are close to being Calabi-Yau with length scale much bigger than the tori, and this torus fibration structure is in turn fibred over $m$ noncompact real directions, corresponding roughly to the $\log r_{i}$ variables. The length scale of the base is much larger than the intermediate length scale of the $Y$-fibres.

Let $h_{L}$ be a smooth Hermitian metric on $L\to Y$, which we pull back to $L\to Z$. For our main applications, $L$ is trivial and $h_{L}=1$. To zeroth approximation, we try the ansatz (which shall be improved later)

$$h_{1}=h_{L}\exp\left(-2u(-\log r_{1},\ldots,-\log r_{m})\right)$$

where $u$ is some smooth convex function, which should be thought of as the leading order Kähler potential in a particular normalisation convention. Write $x_{i}=-\log r_{i}$, where the sign is chosen so that $x_{i}>0$ in the region of interest. We calculate the Kähler metric (the positivity is not automatic, and amounts to an extra assumption)

$$\begin{split}&-dd^{c}\log h_{1}^{1/2}=-dd^{c}\log h_{L}^{1/2}+dd^{c}u\\ =&-dd^{c}\log h_{L}^{1/2}+\sum\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}d\log r_{i}\wedge d^{c}\log r_{j}-\sum\frac{\partial u}{\partial x_{i}}dd^{c}\log r_{i}.\end{split}$$

Recall in local coordinates $r_{i}=|\xi_{i}|e^{-\phi_{i}}$. The Hessian term contains a term

$$\frac{\sqrt{-1}}{4\pi}\sum\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}d\log\xi_{i}\wedge d\overline{\log\xi_{j}},$$

which for $|\xi_{i}|\ll 1$ exponentially dominates the $T^{m}$-tori and the $m$ base directions, since $\sum d\log\xi_{i}\wedge d\overline{\log\xi_{i}}$ is exponentially larger than $\sum d\xi_{i}\wedge d\overline{\xi_{i}}$ in the logarithmic coordinates. The term

$$-\sum\frac{\partial u}{\partial x_{i}}dd^{c}\log r_{i}=\sum\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i},$$

since $dd^{c}\log|\xi_{i}|=0$ holds for $\xi_{i}\neq 0$, which is valid on the region $0<r_{i}\ll 1$ under consideration. The ansatz will only be used in the region with

$$|D^{2}u|\ll 1\ll|Du|.$$

As such, we can essentially ignore the other contributions produced by the Hessian term.

There are compact directions diffeomorphic to $Y$. For this, notice for each fixed value of $x=(x_{1},\ldots x_{m})$, we get a $T^{m}$-invariant subset $Z_{x}\subset Z$, which projects down to $Y$ via the natural map $Z\to Y$. This projection map is topologically the quotient map by the $T^{m}$-action, so $Z_{x}/T^{m}$ is naturally diffeomorphic to $Y$. Since $Z_{x}/T^{m}$ has no a priori complex structure, one cannot say that this is a biholomorphism. Nevertheless, the natural Riemannian metric on $Z_{x}$ induces a metric on $Z_{x}/T^{m}$ by looking at the transverse directions to the $T^{m}$-action, and via the diffeomorphism this is close to the Kähler metric on $Y$

$$-dd^{c}\log h_{L}^{1/2}+\sum\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i}.$$

(1)

This expression requires some explanation: by definition $-dd^{c}\log h_{L}^{1/2}$ and $dd^{c}\phi_{i}$ make sense on $Y$. The term $\frac{\partial u}{\partial x_{i}}$ is a function of $x$, and for fixed $x$ it is merely a constant coefficient on $Y$. Notice the metric (1) on $Y$ lies in the Kähler class $\mathcal{D}u=c_{1}(L)+\sum\frac{\partial u}{\partial x_{i}}c_{1}(L_{i})$. The reason we are able to acquire a cohomology class which is not obvious in the topological setup, comes from the distributional terms of $dd^{c}\log|\xi_{i}|$ being discarded in the above calculations. We will refer to this $\mathcal{D}u$ as an ‘effective Kähler class’, since it is only present in the effective approximate description of the metric. Its size affects the length scale of the metric on the $Y$-fibres.

The next goal is to improve the metric so that the $Y$-fibres approximately have the Calabi-Yau metrics in the same class $c_{1}(L)+\sum\frac{\partial u}{\partial x_{i}}c_{1}(L_{i})$. This is conceptually similar to the semi-Ricci-flat metrics in the context of holomorphic fibred Calabi-Yau manifolds [18]. We take

$$h=h_{1}\exp(-2\phi_{x}),$$

where for each $x$, we associate some potential $\phi_{x}$ on $Y$, which pulls back to $Z_{x}$, so varying over all $x$ we get a function on an open subset of $Z$. We shall assume that the dependence on $x$ is sufficiently weak. Then the dominant effect is to change the metric on $Z_{x}/T^{m}\simeq Y$ to

$$-dd^{c}\log h_{L}^{1/2}+\sum\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i}+dd^{c}\phi_{x},$$

(2)

and the other effects in the $\log\xi_{i}$ directions are suppressed. We choose $\phi_{x}$ so that the effective fibre metrics (2) are the unique Calabi-Yau metrics in the cohomology class $\mathcal{D}u(x)$. This determines $\phi_{x}$ up to fibrewise constants depending on $x$. For the purpose of constructing approximate Calabi-Yau metrics, the choice is inessential as long as the $x$-derivatives are small enough, just like what happens for semi-Ricci-flat metrics.

To leading order,

$$\begin{split}&(-dd^{c}\log h^{1/2})^{n}\approx\frac{n!}{(n-m)!}\det(D^{2}u)\prod_{1}^{m}\frac{\sqrt{-1}}{4\pi}d\log\xi_{i}\wedge d\overline{\log\xi_{i}}\\ &\wedge(-dd^{c}\log h_{L}^{1/2}+\sum\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i}+dd^{c}\phi_{x})^{n-m}\\ =&\frac{n!\int_{Y}(\mathcal{D}u)^{n-m}}{(n-m)!}\det(D^{2}u)(\prod_{1}^{m}\frac{\sqrt{-1}}{4\pi}d\log\xi_{i}\wedge d\overline{\log\xi_{i}})\wedge\sqrt{-1}^{(n-m)^{2}}\Omega_{Y}\wedge\overline{\Omega}_{Y},\end{split}$$

where $\Omega_{Y}$ is the holomorphic volume form on $Y$, normalized to

$$\sqrt{-1}^{(n-m)^{2}}\int_{Y}\Omega_{Y}\wedge\overline{\Omega}_{Y}=1.$$

The term $\int_{Y}(\mathcal{D}u)^{n-m}$ is intersection theoretic, and is polynomial in the derivative of $u$. The natural holomorphic form on $Z$ is up to a global constant

$$\Omega=\prod_{1}^{m}d\log\xi_{i}\wedge\Omega_{Y},$$

(3)

which is well defined independent of trivializations. We see that

$$(-dd^{c}\log h^{1/2})^{n}\approx\text{const }\Omega\wedge\overline{\Omega}\det(D^{2}u)\int_{Y}(\mathcal{D}u)^{n-m}.$$

Thus provided the various assumptions involved in the approximation are satisfied, then the condition for the metric $-dd^{c}\log h^{1/2}$ to define an approximate Calabi-Yau metric is

$$\det(D^{2}u)\int_{Y}(\mathcal{D}u)^{n-m}=\text{const}.$$

(4)

This is a Monge-Ampère type PDE on the real $m$-dimensional base, which we call the non-archimedean Monge-Ampère equation (NA MA for short) on account of a very similar construction in [14]. The associated almost Calabi-Yau metric is called the generalized Calabi ansatz.

The generalized Calabi ansatz has two familiar special cases:

For $m=n$, then $Y$ is just a point, and $0<r_{i}\ll 1$ amounts to the subset $|\xi_{i}|\ll 1$ inside $(\mathbb{C}^{*})^{n}$. The NA MA equation is just the real MA equation

$$\det(D^{2}u)=\text{const}.$$

The ansatz then produces a Calabi-Yau metric, known as the semiflat metric, familiar in the SYZ conjecture [15].

For $m=1$, and $L$ trivial, then $Z$ is the total space of a positive line bundle $L_{1}$ over an $(n-1)$-dimensional Calabi-Yau manifold $Y$. We then take $h_{L_{1}}$ to be the Hermitian metric corresponding to the Calabi-Yau metric in $c_{1}(L_{1})$. We can also view $h_{L_{1}}$ as a function on $Z$, equal to $r_{L_{1}}^{2}$. Up to a normalising constant, the Calabi ansatz is

$$\omega_{Cal}=\frac{n}{n+1}dd^{c}(-\log h_{L_{1}}^{1/2})^{(n+1)/n}.$$

We compare this to the NA MA equation (4) in this case: $u$ is a function of a single real variable $x$, satisfying

$$u^{\prime\prime}u^{\prime n-1}=\text{const}.$$

Up to constant $u=x^{(n+1)/n}$. Thus the NA MA equation reproduces the Calabi ansatz.

A special case of the generalized Calabi ansatz is when $L$ is trivial, and $L_{i}=d_{i}L_{0}$ for some positive line bundle $L_{0}$ and positive integers $d_{i}>0$. In this case, we can take $h_{L}=1$, and $h_{L_{i}}$ is the suitable tensor power of $h_{L_{0}}$, where $h_{L_{0}}$ can be chosen to correspond to the Calabi-Yau metric in the class $c_{1}(L_{0})$. The ansatz metric is simply

$$\begin{split}dd^{c}u=\sum\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}d\log r_{i}\wedge d^{c}\log r_{j}+\sum\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i}\\ =\sum\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}d\log r_{i}\wedge d^{c}\log r_{j}+(\sum\frac{\partial u}{\partial x_{i}}d_{i})dd^{c}\phi_{0}.\end{split}$$

Observe that for rank reasons

$$(\sum\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}d\log r_{i}\wedge d^{c}\log r_{j})^{m+1}=0,\quad(dd^{c}\phi_{0})^{n-m+1}=0.$$

We compute the volume form using binomial expansion

$$(dd^{c}u)^{n}=\frac{n!}{(n-m)!}(\sum\frac{\partial u}{\partial x_{i}}d_{i})^{n-m}\det(D^{2}u)(\prod_{1}^{m}d\log r_{i}\wedge d^{c}\log r_{i})\wedge(dd^{c}\phi_{0})^{n-m}.$$

Since $(dd^{c}\phi_{0})^{n-m}$ already exhaust all base terms, we can replace $\log r_{i}$ by $\log|\xi_{i}|$, and obtain

$$(dd^{c}u)^{n}=\frac{n!}{(n-m)!}(\sum\frac{\partial u}{\partial x_{i}}d_{i})^{n-m}\det(D^{2}u)(\prod_{1}^{m}\frac{\sqrt{-1}}{4\pi}d\log\xi_{i}\wedge d\overline{\log\xi_{i}})\wedge(dd^{c}\phi_{0})^{n-m}.$$

The Calabi-Yau condition on $dd^{c}\phi_{0}$ and the normalization on $\Omega_{Y}$ imply

$$(dd^{c}\phi_{0})^{n-m}=(\int_{Y}c_{1}(L_{0})^{n-m})\sqrt{-1}^{(n-m)^{2}}\Omega_{Y}\wedge\overline{\Omega}_{Y}.$$

The conclusion is that

The relevance of the generalized Calabi ansatz to the Tian-Yau problem (cf. Question 2.4) is as follows. Let $\bar{X}$ be a smooth Fano manifold, and $D=\sum D_{i}$ be an anticanonical divisor, which we assume to be simple normal crossing, with only multiplicity one components. Let $X=\bar{X}\setminus D$, then $X$ has a nowhere vanishing holomorphic volume form $\Omega$, and one can ask when this admits a complete Calabi-Yau metric.

An important intuition to keep in mind, is that most of the volume growth near the infinity of $X$ in fact concentrates near the deeper intersection strata $D_{J}=\cap_{i\in J}D_{i}$ of the component divisors $D_{i}$. Let $m$ denote the maximal $|J|$ for which $D_{J}$ is non-empty, then from the volume growth perspective, the neighbourhood of the $m$-fold intersection loci are the generic regions in the Calabi-Yau $X$. For any such subset $J$ with $|J|=m$, the neighbourhood of $D_{J}$ is essentially the total space of $\oplus_{j\in J}\mathcal{O}(D_{j})|_{D_{J}}$, which corresponds to $Z$, and an application of the adjunction formula shows $D_{J}$ is a compact Calabi-Yau, which corresponds to $Y$. Up to inessential normalization constants, the holomorphic volume form is (3) up to negligible errors. In many examples all the $\mathcal{O}(D_{i})$ are ample. The possible relevance of $L$ comes from the prescribed global Kähler class on $X$. The picture we would like to advocate, for which our present paper is a very special case, is that one can find new Tian-Yau type metrics on $X$, whose behaviour in the generic region is modelled on the generalized Calabi ansatz. Morever, further details from this paper suggests the whole problem is inductive on $m$, in the sense that what happens in non-generic regions is related to the generalized Calabi ansatz with smaller $m$.

While we believe the generalized Calabi ansatz has wide applicability, both in the Tian-Yau problem, and in the collapsing polarized degeneration problem as described in [14], solving the NA MA equation (4) is practically quite nontrivial for $m\geq 2$. We shall now specialize to the proportional line bundle case of section 2.3, and further assume $m=2$. The NA MA equation becomes a PDE with two independent variables

$$\det(D^{2}u)(d_{1}\frac{\partial u}{\partial x_{1}}+d_{2}\frac{\partial u}{\partial x_{2}})^{n-2}=\text{const}.$$

(5)

Motivated by the Calabi ansatz, we wish to look for homogeneous solutions. A preliminary dimensional analysis is useful:

$$u\sim O(|x|^{\alpha}),\quad Du\sim O(|x|^{\alpha-1}),\quad D^{2}u\sim O(|x|^{\alpha-2}).$$

Thus we want

$$2(\alpha-2)+(n-2)(\alpha-1)=0,\quad\alpha=\frac{n+2}{n}.$$

We try the ansatz

$$x_{2}=\frac{d_{2}}{d_{1}}tx_{1},\quad u(x_{1},x_{2})=x_{1}^{\frac{n+2}{n}}v(t).$$

(6)

Routine computation then reduces the NA MA equation to an ODE:

We mentioned in the beginning that the generalized Calabi ansatz geometrically describes an iterated fibration, which is supposedly the model for the generic region near infinity on the noncompact Calabi-Yau $X=\bar{X}\setminus D$. Figuring out the order of magnitude of various length scales is essentially a matter of dimensional analysis. In the generic region $t$ is of order $\sim 1$, $v$ is smooth and of order $\sim 1$. Our homogeneous ansatz prescribes

$$u\sim O(|x|^{(n+2)/n}),\quad|du|\sim O(|x|^{2/n}),\quad|D^{2}u|\sim O(|x|^{(2-n)/n}).$$

We have $|D^{2}u|\ll 1\ll|du|$. From the descriptions in section 2.1, the Hessian term is responsible for the base and torus direction of the metric, while $du$ is responsible for the effective Kähler class, and therefore the size of the fibres diffeomorphic to $Y\simeq D_{1}\cap D_{2}$. Thus

$$\text{diam}(T^{2})=O(|D^{2}u|^{1/2})=O(|x|^{(2-n)/2n}),\quad\text{diam}(Y)=O(|du|^{1/2})=O(|x|^{1/n}),$$

the distance to the origin is of order

$$O(\int|D^{2}u|^{1/2})=O(|x|^{(2+n)/2n}),$$

and the volume within $|x|\leq r$ is $O(r^{2})$ from the two log directions of the base. In terms of geodesic distance to the origin, the volume grows with power $4n/(n+2)$.

For $|x|\gg 1$, this reaffirms the intuition that the $T^{2}$ length scale is far smaller than $Y\simeq D_{1}\cap D_{2}$, which is far smaller than the real 2-dimensional base.

The ODE (7) is posed on $0<t<\infty$, and we now discuss the boundary conditions to put at $t=0$ and $t=\infty$. The infinity boundary can be reduced to the $t=0$ case by the symmetry of the ODE, which geometrically comes from the symmetry between $x_{1},x_{2}$ (up to elementary scaling). Geometrically one would like to find a partial completion of the generalized Calabi ansatz, near the infinity of $\bar{X}\setminus D_{1}\cup D_{2}$. The generalized Calabi ansatz works in the neighbourhood of $D_{1}\cap D_{2}$, and one would like to understand what happens near $D_{1}\setminus D_{2}$ and $D_{2}\setminus D_{1}$.

The simplest boundary conditions to imagine is for the ODE to remain analytic at $t=0$. In this case, one would specify $v(0)=v_{0}>0$ and $v^{\prime}(0)>-\frac{n+2}{n}v_{0}$, and the ODE is non-singular in a neighbourhood of $t=0$, so that $v$ has a Taylor expansion at $t=0$. Geometrically, the size of $T^{2}$ remains of order $O(x_{1}^{\frac{2-n}{2n}})$ and the effective Kähler class of $Y$ remains of order $O(x_{1}^{2/n})$ as $t\to 0$. There seems to be no known metric near $D_{1}\setminus D_{2}$ that one may attempt to glue to the generalized Calabi ansatz. Intuitively, one cannot suddenly ‘switch off’ the effective Kähler class at $t=0$.

We are thus led to look for boundary conditions such that $\frac{n+2}{n}v+(1-t)v^{\prime}\to 0$ as $t\to 0$, which intuitively means the effective Kähler class ‘switches off gradually’. The following type of boundary conditions is perhaps best motivated by fixing $v_{0}>0$, and trying to decrease $v^{\prime}(0)$ until $v^{\prime}(0)+\frac{n+2}{n}v_{0}$ tends to zero. As $t\to 0$, we require $v^{\prime}$ has some subleading power law behaviour

$$\begin{cases}&v=v_{0}+O(t),\\ &v^{\prime}=-\frac{n+2}{n}v_{0}+at^{\beta}+O(t),\\ &v^{\prime\prime}=a\beta t^{\beta-1}+O(1).\end{cases}$$

(9)

Here $0<\beta<1$ and $a>0$ are some constants to be determined. Thus

$$vv^{\prime\prime}-\frac{2}{n+2}v^{\prime 2}=av_{0}\beta t^{\beta-1}+O(1),$$

$$\frac{n+2}{n}v+(1-t)v^{\prime}=at^{\beta}+O(t),$$

and the ODE predicts

$$\beta-1+(n-2)\beta=0,\quad\beta=\frac{1}{n-1},$$

and

$$\frac{av_{0}}{n-1}a^{n-2}=\text{const},\quad a\propto v_{0}^{-1/(n-1)}.$$

The main previously known source of complete Calabi-Yau metrics on the complement of a singular anticanonical divisor are due to Hein [10] on certain complex surfaces. These examples are constructed on rational elliptic surfaces, which in particular admit elliptic fibrations onto $\mathbb{P}^{1}$ with fibers lying in the anticanonical linear system. Hein constructs complete Calabi-Yau metrics asymptotic to the semi-flat Calabi-Yau metrics discovered by Greene-Shapere-Vafa-Yau [7]. Interestingly, for Kodaira type $I_{b}$ singular fibers, these metrics turn out to be the same as the metrics constructed by Tian-Yau [16] on the complement of an ample anticanonical divisors in a del Pezzo surface [4, 5, 11]. In what follows we collect some sporadic comparisons to other works in the literature which are not directly connected to the Tian-Yau problem.