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.