ScalingStacks

While this paper focuses on the algebraic case, one may wonder what happens for ‘transcendental families’. A prototypical examples is the family of degree $n+2$ hypersurfaces in $\mathbb{CP}^{n+1}$:

$$X_{t}=\{\sum_{I}a_{I}t^{\lambda_{I}}x^{I}=0\}\subset\mathbb{CP}^{n+1},$$

where $x^{I}$ denote the degree $n+2$ monomials, $a_{I}$ are coefficients chosen suitably generically, and $\lambda_{I}$ are exponents chosen suitably. When $\lambda_{I}$ are sufficiently irrational, this does not fit into our framework, yet the metric SYZ conjecture makes sense.

There seem to be two natural strategies. One is to make the NA pluripotential theory work over NA fields without a discrete valuation (cf. [7] for the latest progress), and the other is to develop the framework of real MA equation on polyhedral sets such as $Sk(X)$ without explicit reference to NA geometry.

Let $(X,L)$ be an algebraic degeneration family with an ample polarization, and $(\mathcal{X},\mathcal{L})$ be a semistable snc model. This is not required to be CY, nor do we impose any maximal degeneration condition. It induces a formal model by base change. Consider the metric $\left\lVert\cdot\right\rVert=\left\lVert\cdot\right\rVert_{\mathcal{L}}e^{-\phi}$ on $(X_{K},L)$, where we assume $\phi=\phi\circ r_{\mathcal{X}}$ and $\phi$ is ‘sufficiently smooth’ on $\Delta_{\mathcal{X}}$. Our plan is to use the heuristic logic at the end of section 3.4 to guess a formula for the NA measure over the interior of any face $\Delta_{J}\subset\Delta_{\mathcal{X}}$, in terms of differential operators. The answer will involve an interesting correction factor to the real MA measure.

Let $\Delta_{J}$ correspond to $E_{J}=\cap_{i\in J}E_{i}$ as usual. We first explain how to associate a class in $H^{1,1}(E_{J})$ to each $x\in\text{Int}(\Delta_{J})$. Let $x_{i}$ be the local affine coordinate corresponding to $E_{i}$ for any $i\in I$; we will only need $E_{i}\cap E_{J}\neq\emptyset$. We introduce an overparametrisation: let $u$ be a function of all $\{x_{i}\}_{i\in I}$, such that $u$ agrees with $\phi$ at least on a neighbourhood of $\text{Int}(\Delta_{J})\subset\Delta_{\mathcal{X}}$, and we assume $u$ is smooth. Here $x_{i}$ are treated as independent variables for $u$, even though on $\Delta_{\mathcal{X}}$ they satisfy various linear constraints. Consider the class in $H^{1,1}(E_{J})$ defined by the affine linear combination of derivatives:

$$\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert)=c_{1}(\mathcal{L})-\sum_{I}\frac{\partial u}{\partial x_{i}}c_{1}(\mathcal{O}(E_{i}))$$

(12)

which depends only on $\phi$ because in $H^{1,1}(E_{J})$

$$c_{1}(\mathcal{O}(\sum_{I}E_{i}))=c_{1}(\text{div}(dt))=0,\quad c_{1}(\mathcal{O}(E_{i}))=0\quad\forall E_{i}\cap E_{J}=\emptyset.$$

Notice $\left\lVert\cdot\right\rVert=\left\lVert\cdot\right\rVert_{\mathcal{L}}e^{-\phi}$ and $\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert)$ are invariant under the change

$$\mathcal{L}\to\mathcal{L}+\sum_{I}d_{i}E_{i},\quad\phi\to\phi+\sum_{I}d_{i}\phi_{E_{i}}$$

where $d_{i}\in\mathbb{R}$ and $\phi_{E_{i}}$ denote the model functions associated to $E_{i}$. Thus the function $\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert)$ on $\text{Int}(\Delta_{J})$ is intrinsically associated to $\left\lVert\cdot\right\rVert$.

Our strategy is to consider a family of Hermitian metrics $h_{t}$ on $L\to X_{t}$ such that $h_{t}^{1/|\log|t||}\to\left\lVert\cdot\right\rVert^{2}$ in the hybrid topology on $X\sqcup\Delta_{\mathcal{X}}$, and take the limit of the complex MA measures associated to the curvature forms of $h_{t}$. Introduce smooth Hermitian metrics $h_{\mathcal{L}}$ and $h_{E_{i}}$ on $\mathcal{L}$ and $\mathcal{O}(E_{i})$ for $i\in I$. Use these to produce smooth functions $r_{i}$ on $\mathcal{X}$ for $i\in I$, such that near $E_{i}\subset\mathcal{X}$ the singularity is governed by $r_{i}\sim|z_{i}|e^{-\phi_{i}}$ for smooth local $\phi_{i}$, where $z_{i}$ are local defining equations for $E_{i}$, and away from $E_{i}$ the function $r_{i}$ is smooth and bounded positively from below. In order for $h_{t}$ to have the appropriate convergence behaviour as $t\to 0$ around $\text{Int}(\Delta_{J})\subset X\sqcup\Delta_{\mathcal{X}}$, we use the ansatz

$$h_{t}\sim h_{\mathcal{L}}\exp\left(2\log|t|u(\frac{\log r_{i}}{\log|t|},i\in I)\right)$$

where $r_{i}$ are regarded as smooth functions on $X_{t}$. The curvature form in $(X_{t},c_{1}(L))$ is

$$-dd^{c}\log h_{t}^{1/2}\sim-dd^{c}\log h_{\mathcal{L}}^{1/2}-\log|t|dd^{c}u,$$

$$dd^{c}u=\sum_{i,j\in I}\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}\frac{1}{|\log|t||^{2}}d\log r_{i}\wedge d^{c}\log r_{j}+\frac{1}{\log|t|}\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\log r_{i}.$$

Our goal is to extract the limiting contribution of $(-dd^{c}\log h_{t}^{1/2})^{n}$ to $\text{Int}(\Delta_{J})$ as $t\to 0$. We need to separate this contribution from $\Delta_{J^{\prime}}$ with $J^{\prime}\supsetneq J$; algebraically this means taking the measure contribution near $E_{J}\subset\mathcal{X}$, but away from deeper strata $E_{J^{\prime}}$. To formalize this, we consider the quantitative statum on $X_{t}$ (cf. section 3.1)

$$E_{J,t}^{\epsilon}=\{q\in X_{t}|d(q,E_{J})<\epsilon\}\setminus\bigcup_{J^{\prime}\supsetneq J}\{q\in X_{t}|d(q,E_{J^{\prime}})<\epsilon\},\quad\epsilon\ll 1.$$

We need to compute

$$\lim_{\epsilon\to 0}\lim_{t\to 0}\text{Log}_{\mathcal{X}*}\left((-dd^{c}\log h_{t}^{1/2})^{n}\mathbin{\vrule height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule height=0.55974pt,depth=0.0pt,width=5.59721pt}E_{J,t}^{\epsilon}\right)$$

where the order of limit is important.

For fixed $\epsilon$, after deleting terms suppressed by order $O(\frac{1}{|\log|t||})$, we have the asymptote on $E_{J,t}^{\epsilon}\subset X_{t}$ as $t\to 0$:

$$dd^{c}u\sim\sum_{0}^{p}\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}\frac{1}{|\log|t||^{2}}\frac{\sqrt{-1}}{4\pi}d\log z_{i}\wedge d\log\bar{z}_{j}+\frac{1}{\log|t|}\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\log r_{i},$$

where $z_{0},\ldots z_{p}$ are the local defining functions of the divisors $E_{i}\subset\mathcal{X}$, and $E_{J}=\cap_{0}^{p}E_{i}$. Using that $t=z_{0}\ldots z_{p}$ locally around $E_{J}$, we can eliminate the $z_{0}$ variable to write

$$dd^{c}u\sim\sum_{1}^{p}\frac{\partial^{2}\phi}{\partial x_{i}\partial x_{j}}\frac{1}{|\log|t||^{2}}\frac{\sqrt{-1}}{4\pi}d\log z_{i}\wedge d\log\bar{z}_{j}+\frac{1}{\log|t|}\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\log r_{i},$$

hence

$$\begin{split}-dd^{c}\log h_{t}^{1/2}\sim&\sum_{1}^{p}\frac{\partial^{2}\phi}{\partial x_{i}\partial x_{j}}\frac{1}{|\log|t||}\frac{\sqrt{-1}}{4\pi}d\log z_{i}\wedge d\log\bar{z}_{j}\\ &-dd^{c}\log h_{\mathcal{L}}^{1/2}-\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\log r_{i}.\end{split}$$

(13)

Notice that $dd^{c}\log r_{i}\sim-dd^{c}\phi_{i}$ has a smooth extension to the central fibre as $t\to 0$; the singular effect is eliminated by $dd^{c}\log|z_{i}|=0$ on $X_{t}$. The term $\sum_{1}^{p}\frac{\partial^{2}\phi}{\partial x_{i}\partial x_{j}}\frac{1}{|\log|t||}\frac{\sqrt{-1}}{4\pi}d\log z_{i}\wedge d\log\bar{z}_{j}$ dominates in the directions transverse to $E_{J}$, and the term $-dd^{c}\log h_{\mathcal{L}}^{1/2}+\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i}$ dominates in the directions tangential to $E_{J}$.

The measure asymptote is now

$$\begin{split}(-dd^{c}\log h_{t}^{1/2})^{n}\sim&p!\det(D^{2}\phi)\left(-dd^{c}\log h_{\mathcal{L}}^{1/2}+\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i}\right)^{n-p}\wedge\\ &\prod_{i=1}^{p}\frac{1}{2\pi|\log|t||}d\log|z_{i}|\wedge d\arg{z}_{i},\end{split}$$

(14)

where $D^{2}\phi=(\frac{\partial^{2}\phi}{\partial x_{i}\partial x_{j}})_{p\times p}$ is the Hessian matrix of $\phi$. Thus the pushforward measure has the limit as $t\to 0$:

$$\begin{split}\lim_{t\to 0}\text{Log}_{\mathcal{X}*}\left((-dd^{c}\log h_{t}^{1/2})^{n}\mathbin{\vrule height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule height=0.55974pt,depth=0.0pt,width=5.59721pt}E_{J,t}^{\epsilon}\right)=p!\det(D^{2}\phi)|dx_{1}\ldots dx_{p}|\\ \times\int_{E_{J,0}^{\epsilon}}\left(-dd^{c}\log h_{\mathcal{L}}^{1/2}+\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i}\right)^{n-p}.\end{split}$$

Taking $\epsilon\to 0$,

$$\begin{split}\lim_{\epsilon\to 0}\lim_{t\to 0}\text{Log}_{\mathcal{X}*}\left((-dd^{c}\log h_{t}^{1/2})^{n}\mathbin{\vrule height=6.88889pt,depth=0.0pt,width=0.55974pt\vrule height=0.55974pt,depth=0.0pt,width=5.59721pt}E_{J,t}^{\epsilon}\right)=p!\det(D^{2}\phi)|dx_{1}\ldots dx_{p}|\\ \times\int_{E_{J}}\left(-dd^{c}\log h_{\mathcal{L}}^{1/2}+\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i}\right)^{n-p}.\end{split}$$

Here an interesting topological effect takes place. Even though $dd^{c}\log r_{i}$ starts life as an exact form on $X_{t}$, it acquires a first Chern class in the process of smooth extension to the central fibre $\mathcal{X}_{0}$, because we are removing the distributional contribution $dd^{c}\log|z_{i}|=[E_{i}]$. Thus on $E_{J}$, the smooth closed (1,1)-form

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

lies in the $H^{1,1}$ class

$$c_{1}(\mathcal{L})-\sum_{I}\frac{\partial u}{\partial x_{i}}c_{1}(\mathcal{O}(E_{i})),$$

which is exactly the class $\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert)$ we introduced earlier (cf. (12)). We have thus obtained a formula for the double limit:

$$p!\det(D^{2}\phi)|dx_{1}\ldots dx_{p}|(\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert)^{n-p}\cdot E_{J}).$$

Notice all auxiliary choices are eliminated at this stage. According to our heuristic logic that the NA MA measure should be the limit of the corresponding complex MA measures on $X_{t}$, we conclude the heuristic formula for the NA MA measure over $\text{Int}(\Delta_{J})$

$$r_{\mathcal{X}*}MA(\left\lVert\cdot\right\rVert)=p!\det(D^{2}\phi)|dx_{1}\ldots dx_{p}|(\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert)^{n-p}\cdot E_{J}).$$

(15)

Notice the RHS is a differential operator in the potential $\phi$, because the intersection theoretic term $\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert)$ is affine linear in the first order derivatives of $\phi$. The formula exhibits a curious mixture of intersection theory with real MA operator.

In this section we will speculate on the concrete meaning of the Boucksom-Favre-Jonsson solution to the NA Calabi conjecture (cf. section 3.5). Let $(X,L)$ be a polarized algebraic degeneration family of Calabi-Yau manifolds, which needs not be a maximal degeneration. Let $(\mathcal{X},\mathcal{L})$ be a semistable snc model. An analogue of the Lebesgue measure $d\mu_{0}=\lim_{t\to 0}\text{Log}_{\mathcal{X}*}d\mu_{t}$ exists in this setting, which is supported on the essential skeleton $Sk(X)\subset\Delta_{\mathcal{X}}$. It can be found by the ideas outlined in section 3.1. To describe it, denote $m=\dim Sk(X)$, and $\Delta_{J}$ be any $m$-dimensional face of $Sk(X)$, corresponding to $E_{J}=\cap_{0}^{m}E_{j}$ with local defining functions $z_{0},\ldots z_{m}$. The holomorphic volume form $\Omega$ on $\mathcal{X}$ induces a holomorphic volume form on $E_{J}$, called its Poincaré residue $\text{Res}_{E_{J}}(\Omega)$, determined by the equality in $K_{\mathcal{X}}|_{E_{J}}$:

$$\Omega=\text{Res}_{E_{J}}(\Omega)\wedge td\log z_{0}\wedge\ldots d\log z_{m}.$$

This is independent of the choice of local coordinates $z_{i}$. Then up to normalising $\Omega$ by a global multiplicative constant, on the interior of $\Delta_{J}$,

$$d\mu_{0}=|dx_{1}\ldots dx_{m}|\cdot\int_{E_{J}}\sqrt{-1}^{(n-m)^{2}}\text{Res}_{E_{J}}(\Omega)\wedge\overline{\text{Res}_{E_{J}}(\Omega)},$$

(16)

Notice it is proportional to $|dx_{1}\ldots dx_{m}|$, which explains the name ‘Lebesgue measure’. The union of the $m$-dimensional open faces of $Sk(X)$ has the full $d\mu_{0}$-measure, which is one by our normalisation.

We are interested in the distinguished solution $\left\lVert\cdot\right\rVert_{CY}=\left\lVert\cdot\right\rVert_{\mathcal{L}}e^{-\phi_{0}}$ to the NA MA equation on $(X_{K}^{an},L)$ defined by

$$MA(\left\lVert\cdot\right\rVert_{CY})=(L^{n})d\mu_{0}.$$

(17)

To give an interpretation, we boldly assume that $\phi_{0}=\phi_{0}\circ r_{\mathcal{X}}$, which can be regarded as a strong version of the comparison property. Then we are in the setting of section 5.2.

We begin with the interior of $m$-dimensional faces of $Sk(X)$, which can be viewed as the generic region. Comparing with the heuristic formula (15), and making regularity assumptions on $\phi_{0}$, we deduce a second order PDE

$$\det(D^{2}\phi_{0})(\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert_{CY})^{n-m}\cdot E_{J})=\frac{(L^{n})}{m!}\int_{E_{J}}\sqrt{-1}^{(n-m)^{2}}\text{Res}_{E_{J}}(\Omega)\wedge\overline{\text{Res}_{E_{J}}(\Omega)},$$

(18)

where $(\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert_{CY})^{n-m}\cdot E_{J})$ defines a polynomial in the gradient of $\phi_{0}$.

On any other face $\text{Int}(\Delta_{J})\subset\Delta_{\mathcal{X}}$ of dimension $p$, not necessarily on $Sk(X)$,

$$\det(D^{2}\phi_{0})(\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert_{CY})^{n-p}\cdot E_{J})=0.$$

There are two obvious mechanisms for this to happen:

• •

  We may have the homogeneous real MA equation $\det(D^{2}\phi_{0})=0$. There is a basic mechanism for this: notice that the model choice $\mathcal{X}$ is not canonical, and one can blow up to obtain higher models. For simple blowups, either one subdivides an existent face, or one creates new faces. On any new face $\det(D^{2}\phi_{0})=0$ is automatic, because $\phi_{0}=\phi_{0}\circ r_{\mathcal{X}}$ implies that $\phi_{0}$ is independent of the new coordinate variable.

• •

  We may have $\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert_{CY})^{n-p}\cdot E_{J}=0$, which is an algebraic condition on the gradient of $\phi_{0}$. The author speculates that this option happens for $(m-1)$-dimensional faces on $Sk(X)$, and its role is to match the gradients when we cross from one $m$-dimensional open face of $Sk(X)$ to another. In Remark 5.1 we suggest $\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert_{CY})$ may be a nef class on $E_{J}$, and our equation precisely says this class has zero volume. The birational geometric significance seems well worth investigating. We now discuss some of the simplest ways this mechanism could work:

It is natural to ask what happens to the CY metrics for non-maximal polarized algebraic degenerations. Recall maximal degenerations require the essential skeleton to have dimension $n$. The other extreme case, where $\dim Sk(X)=0$, corresponds to degenerations with a uniform volume noncollapsing condition, and is by now quite well understood [50]: the metric limit is a Calabi-Yau variety with klt singularities. The case $0<\dim Sk(X)<n$ is expected to exhibit a mixture of both the algebraic and NA/tropical behaviours. There is no systematic theory, but the CY metrics are described in some sporadic examples [25][42]. In such examples, a key role is played by a generalized Gibbons-Hawking ansatz, which expresses CY metrics with some torus symmetry in terms of both symplectic moment coordinates and complex coordinates on the Kähler quotient. In the generic region, the asymptotic behaviour of these metrics is captured by the Calabi ansatz, which describes the metric solely in complex coordinates using the Kähler potential [42, chapter 2]. The procedure to pass from the Calabi ansatz to the generalized Gibbons-Hawking ansatz is akin to the Legendre transform.

The Calabi ansatz is as follows. Let $(Y,\Omega_{Y})$ be a compact $(n-1)$-dimensional CY manifold with an ample line bundle $E$, and let $h$ denote the Hermitian metric on $E\to Y$ whose curvature form is the Calabi-Yau metric on $Y$ in the class $c_{1}(E)$, so the length $r=h^{1/2}$ defines a function on the total space $E$. Let $\xi$ denote a local holomorphic fibre coordinate. The subset $\{0<r\lesssim 1\}\subset E$ has a nowhere vanishing holomorphic volume form $\Omega_{E}=\Omega_{Y}\wedge d\log\xi$, defined indpendent of $\xi$. Then the metric ansatz

$$\omega_{E}=dd^{c}(-\log r)^{(n+1)/n}$$

defines a CY metric on $\{0<r\lesssim 1\}\subset E$ compatible with $\Omega_{E}$.

Very little is known about the relationship between the NA MA solution and the metric degenerations, even in the aforementioned cases where the metric is well understood. Relating these is likely to require first proving some version of the comparison property, which is a major open problem. Notwithstanding all technical difficulties, we speculate the following heuristic principle: in the small $t$ limit, inside the generic region on $X_{t}$, the NA solution should approximately describe the behaviour of the Kähler potential at large scales, and obliterate the information on small compact directions. Based on this principle, we will arrive at a generalised Calabi ansatz, which is a candidate limiting description of the CY metrics in the generic region, at least in some idealized situations.

We assume the setting of section 5.3, and denote the NA MA solution as $\left\lVert\cdot\right\rVert_{CY}=\left\lVert\cdot\right\rVert_{\mathcal{L}}e^{-\phi_{0}}$. Let $\Delta_{J}$ be an $m$-dimensional face of $Sk(X)$. We impose the simplifying assumptions:

• •

  There is no $E_{i}$ among $i\in I$ intersecting $E_{J}$ transversely in $\mathcal{X}$. Consequently, the Poincaré residue $\text{Res}_{E_{J}}(\Omega)$ is nowhere vanishing on $E_{J}$. In this situation, the complex geometric picture around $E_{J}\subset\mathcal{X}$ is modelled on the total space of the rank $m$ vector bundle $\tilde{\pi}:\oplus_{0}^{m}\mathcal{O}(E_{i})\to E_{J}$, with a trivialisation $\mathcal{O}(\sum E_{i})\simeq\mathcal{O}$ provided by the coordinate $t$. The holomorphic volume form for $|t|\ll 1$ is approximately

  $$\Omega\sim\tilde{\pi}^{*}\text{Res}_{E_{J}}(\Omega)\wedge td\log z_{0}\wedge\ldots d\log z_{m},$$

  so the holomorphic volume form $\Omega_{t}$ on $X_{t}$ is approximately

  $$\Omega_{t}\sim(-1)^{n-m}\tilde{\pi}^{*}\text{Res}_{E_{J}}\wedge d\log z_{1}\wedge\ldots d\log z_{m}.$$

• •

  For every $x\in\text{Int}(\Delta_{J})$, the class $\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert_{CY})\in H^{1,1}(E_{J})$ is Kähler (cf. Remark 5.1).

• •

  The NA solution $\phi_{0}$ is smooth, and in particular strictly convex on $\text{Int}(\Delta_{J})$.

We recycle the computation in section 5.2, to construct the overparametrisation $u$ of $\phi_{0}$, and produce the Hermitian metric $h_{t}$ on $(X,c_{1}(L))$, so that $h_{t}^{1/|\log|t||}$ naturally converges to $\left\lVert\cdot\right\rVert_{CY}^{2}$ in the hybrid topology. Up to $O(\frac{1}{|\log|t||})$ relative error, we have the metric asymptote (13) and the measure asymptote (14). This procedure is essentially dictated by the NA information. In particular, for each $x\in\text{Int}(\Delta_{J})$, the class

$$[-dd^{c}\log h_{\mathcal{L}}^{1/2}+\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i}]=\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert_{CY})\in H^{1,1}(E_{J})$$

is fixed by the NA MA solution.

Notice at this stage we still have the freedom to adjust $h_{\mathcal{L}}$ and $r_{i}$ which enter the construction in section 5.2. This information controls the small scale metric, and is not directly visible to the NA MA solution. For each $x\in\text{Int}(\Delta_{J})$, let $\phi_{x}$ be the solution to the complex MA equation on $E_{J}$:

$$\frac{\left(-dd^{c}\log h_{\mathcal{L}}^{1/2}+\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i}+dd^{c}\phi_{x}\right)^{n-m}}{\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert_{CY})^{n-m}}=\frac{\text{Res}_{E_{J}}(\Omega)\wedge\overline{\text{Res}_{E_{J}}(\Omega)}}{\int_{E_{J}}\text{Res}_{E_{J}}(\Omega)\wedge\overline{\text{Res}_{E_{J}}(\Omega)}}.$$

(19)

If we wish to completely determine $\phi_{x}$ we need to fix a normalisation, but this choice is not essential. Under our hypotheses the dependence of $\phi_{x}$ on $x$ can be made smooth.

We then modify $h_{t}$ into $\tilde{h}_{t}=h_{t}e^{-2\phi_{x}}$, where $x=\text{Log}_{\mathcal{X}}(z)$ on $X_{t}$. The metric asymptote (13) then implies that on $\text{Log}_{\mathcal{X}}^{-1}(\text{Int}(\Delta_{J}))\subset X_{t}$ for $|t|\ll 1$, up to $O(\frac{1}{|\log|t||})$ relative error,

$$\begin{split}-dd^{c}\log\tilde{h}_{t}^{1/2}\sim&\sum_{1}^{p}\frac{\partial^{2}\phi_{0}}{\partial x_{i}\partial x_{j}}\frac{1}{|\log|t||}\frac{\sqrt{-1}}{4\pi}d\log z_{i}\wedge d\log\bar{z}_{j}\\ &-dd^{c}\log h_{\mathcal{L}}^{1/2}+\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i}+dd^{c}\phi_{x}.\end{split}$$

(20)

The normalisation ambiguity on $\phi_{x}$ is suppressed by $O(\frac{1}{|\log|t||})$ relative to the term $\sum_{1}^{p}\frac{\partial^{2}\phi_{0}}{\partial x_{i}\partial x_{j}}\frac{1}{|\log|t||}\frac{\sqrt{-1}}{4\pi}d\log z_{i}\wedge d\log\bar{z}_{j}$. In particular, we see $-dd^{c}\log\tilde{h}_{t}^{1/2}$ is positive definite, so defines a Kähler metric.

We then compute its volume form up to $O(\frac{1}{|\log|t||})$ relative error:

$$\begin{split}(-dd^{c}\log\tilde{h}_{t}^{1/2})^{n}\sim&m!\det(D^{2}\phi_{0})\left(-dd^{c}\log h_{\mathcal{L}}^{1/2}+\sum_{I}\frac{\partial u}{\partial x_{i}}dd^{c}\phi_{i}+dd^{c}\phi_{x}\right)^{n-m}\wedge\\ &\prod_{i=1}^{m}\frac{1}{2\pi|\log|t||}d\log|z_{i}|\wedge d\arg{z}_{i},\\ \sim m!\det(D^{2}\phi_{0})&\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert_{CY})^{n-m}\frac{\text{Res}_{E_{J}}(\Omega)\wedge\overline{\text{Res}_{E_{J}}(\Omega)}}{\int_{E_{J}}\text{Res}_{E_{J}}(\Omega)\wedge\overline{\text{Res}_{E_{J}}(\Omega)}}\wedge\\ &\prod_{i=1}^{m}\frac{\sqrt{-1}}{4\pi|\log|t||}d\log z_{i}\wedge d\log\bar{z}_{i}\\ \sim\frac{m!}{(4\pi|\log|t||)^{m}}&\det(D^{2}\phi_{0})\mathcal{D}_{J}(x,\left\lVert\cdot\right\rVert_{CY})^{n-m}\frac{\sqrt{-1}^{n^{2}}\Omega_{t}\wedge\overline{\Omega_{t}}}{\int_{E_{J}}\sqrt{-1}^{(n-m)^{2}}\text{Res}_{E_{J}}(\Omega)\wedge\overline{\text{Res}_{E_{J}}(\Omega)}}.\end{split}$$

where we use the metric asymptote (20), the construction of $\phi_{x}$ (19), and the asymptote of $\Omega_{t}$ in terms of the Poincaré residue. Now applying the PDE interpretation of NA MA equation (18), we see

$$(-dd^{c}\log\tilde{h}_{t}^{1/2})^{n}\sim\frac{(L^{n})}{(4\pi|\log|t||)^{m}}\sqrt{-1}^{n^{2}}\Omega_{t}\wedge\overline{\Omega_{t}}.$$

(21)

This means the metric $-dd^{c}\log\tilde{h}_{t}^{1/2}$ is approximately Calabi-Yau up to $O(\frac{1}{|\log|t||})$ relative error in the region on $X_{t}$ corresponding to the (slightly shrinked) interior of $\Delta_{J}$. We call $-dd^{c}\log\tilde{h}_{t}^{1/2}$ the generalised Calabi ansatz, and we expect this to model the CY metric on the generic region of $X_{t}$ in the class $c_{1}(L)$ up to small error. This provides a very tight relation between the NA MA equation and the degenerating CY metrics on $X_{t}$.

We now explain how to see the Calabi ansatz as a special case. The analogue of $\mathcal{X}$ is the total space of the rank 2 vector bundle $E\oplus E^{-1}\to Y$, so the det bundle has a canonical trivialisation coordinate $t$, which defines $X_{t}\subset E\oplus E^{-1}$. Equivalently $X_{t}$ can be viewed as a submanifold of $E$. There is a holomorphic form on the total space $E\oplus E^{-1}$, given by $\Omega=\Omega_{Y}\wedge td\log z_{0}\wedge d\log z_{1}$, where $z_{1}$ is a local fibre coordinate on $E\to Y$, and $z_{0}z_{1}=t$. There is an induced holomorphic volume form $\Omega_{t}$ on $X_{t}$ defined by $\Omega=dt\wedge\Omega_{t}$, calculated to be $\Omega_{t}=(-1)^{n-1}\Omega_{Y}\wedge d\log z_{1}$, compatible up to sign with the Calabi ansatz setup. Now assume $E$ is ample, and is endowed with a Hermitian metric $h$ whose curvature form is the CY metric in $(Y,c_{1}(E))$. Denote $r=h^{1/2}$ as the length function on $E$. The analogue of $\mathcal{L}$ is $\mathcal{O}$. By the generalized Calabi ansatz prescription, we should find a function $\phi:\mathbb{R}_{+}\to\mathbb{R}$ satisfying the special case of the NA MA equation (18)

$$\phi^{\prime\prime}(\phi^{\prime})^{n-1}=\text{const},$$

and produce the metric $-\log|t|dd^{c}\phi(\frac{\log r}{\log|t|})$ on $X_{t}$. The solution $\phi(x)=x^{(n+1)/n}$ reproduces the Calabi ansatz up to scaling.