As discussed in Section 2.1, our motivation was based on studying more general degenerations of Calabi-Yau manifolds. During the preparation of this paper in the Fall of 2018, we also made some preliminary progress towards understanding the case of maximal degenerations (which is related to the SYZ conjecture in mirror symmetry), based on similar ideas to that of Section 2 and 4, and partly motivated by [Mor10]. We hoped in a future paper to work out the details of constructing local models generalizing the Ooguri-Vafa metric to higher dimensions. In January 2019, we received a preprint by Yang Li [Li19] who, partly motivated by [HSVZ18], has essentially achieved most of what we were planning to do (in complex dimension three). For this reason, we decided not to expand in this direction beyond what we have written at the time we learned about [Li19]. On the other hand, we still present the original brief discussions here (so the arguments are rather sketchy and there will be NO theorems). We hope this may still be of some interest to the readers, since it seems to shed a slightly different light from [Li19].

We start with a lemma on Green’s function on certain non-compact spaces, which is relate to Proposition 3.31.

We are interested in studying the Green’s currents in the situation of Section 3.4 with $D$ replaced by the non-compact Calabi-Yau manifold $(\mathbb{C}^{*})^{n}$, and with $H$ replaced by a smooth algebraic hypersurface in $D$ defined by a Laurent polynomial $F$. Here $D$ is endowed with the standard flat Kähler metric

(8.8)

$$\omega_{D}=\sum_{j}\frac{1}{2}\sqrt{-1}\partial\log w_{j}\wedge\bar{\partial}\log w_{j}$$

where $\{w_{1},\cdots,w_{n}\}$ are standard holomorphic coordinates on $(\mathbb{C}^{*})^{n}$.

Denote $\xi_{j}=-\log w_{j}=u_{j}+\sqrt{-1}v_{j}$, which gives an identification $(\mathbb{C}^{*})^{n}$ with $\mathbb{R}^{n}\times(S^{1})^{n}$ equipped with the standard flat product metric. Let $\pi:(\mathbb{C}^{*})^{n}\rightarrow\mathbb{R}^{n}$ be the projection map. The amoeba $\mathcal{A}(F)$ of $F$ is by definition the image $\pi(H)$.

We want to solve

(8.9)

$$\Delta G_{P}=2\pi\cdot\delta_{P}$$

In terms of the coordinates $\{\xi_{j}\}$, we can view $\delta_{P}$ as a matrix of distributions by the decomposition

(8.10)

$$\delta_{P}=\sum_{\alpha,\beta}f_{\alpha\beta}\widehat{\delta}_{P}\frac{\sqrt{-1}}{2}d\xi_{\alpha}\wedge d\bar{\xi}_{\beta}\wedge dz$$

where $\widehat{\delta}_{P}$ is a $2n$-current such that for compactly supported smooth function $\phi$

(8.11)

$$(\widehat{\delta}_{P},\phi)=\int_{P}\phi\dvol_{P}$$

Then by definition it is not difficult to see that at every point on $P$,

(8.12)

$$f_{\alpha\beta}d\xi_{\alpha}\wedge d\xi_{\beta}=\frac{\partial F\wedge\bar{\partial}F}{|\partial F|^{2}}.$$

If we decompose

(8.13)

$$G_{P}=\sum h_{\alpha\beta}\frac{\sqrt{-1}}{2}d\xi_{\alpha}\wedge d\xi_{\beta}\wedge dz$$

Then we need to solve a matrix of distributional equations

(8.14)

$$\Delta h_{\alpha\beta}=f_{\alpha\beta}\widehat{\delta}_{P}.$$

Writing

(8.15)

$$\widehat{\delta}_{P}=\int_{P}\delta_{y}\dvol_{P}(y)$$

then one can write down a solution in the form

(8.16)

$$h_{\alpha\beta}(x)\equiv\int_{P}(G_{y}(x)-\mathcal{G}(y))f_{\alpha\beta}(y)\dvol_{P}(y)$$

where $G_{y}(x)$ is the Green’s function on $Q$ constructed in Lemma 8.1, and $\mathcal{G}(y)$ is a renormalization function to make the integral converge. For example, we can take

(8.17)

$$\mathcal{G}(y)=\frac{c_{n}}{|\pi(y)|^{n-2}+1}$$

Now we consider an illustrating example when $n=2$, and

(8.18)

$$F(w_{1},w_{2})\equiv w_{1}+w_{2}+1.$$

The amoeba $\mathcal{A}(F)$ is a well-known shape on $\mathbb{R}^{2}$ with three branches at infinity. Moreover, it is not difficult to show by direct calculation that $\mathcal{A}(F)$ converges exponentially fast (in the Hausdorff sense) to its tropicalization, $T(F)$ which is given by the union of three half lines $P_{1},P_{2},P_{3}$ emanating from $0$ in $\mathbb{R}^{2}$, along the directions of $e_{1},e_{2},-e_{1}-e_{2}$. In this case, one also expects that the Green’s current $G_{P}$, viewed as a matrix $(h_{\alpha\beta})$, is asymptotic to the matrix of Green’s functions defined using $T(F)$ on $\mathbb{R}^{2}$. This asymptotics should hold in suitable regions away from $T(F)$.

The point is that we should remember more information on $T(F)$ than simply a subspace in $\mathbb{R}^{2}$. Notice each $P_{i}$ is a straight half line and it has a unit normal $n_{i}$ in $\mathbb{R}^{2}$ (well-defined up to sign). Here $n_{i}$ naturally arises if one notices (8.12). Then the following is a well-defined matrix valued distribution on $\mathbb{R}^{3}=\mathbb{R}^{2}\times\mathbb{R}$,

(8.19)

$$\delta_{T(F)}\equiv\sum_{i}\widehat{\delta}_{P_{i}}\cdot n_{i}\otimes n_{i}$$

where we view $P_{i}\subset\mathbb{R}^{3}$ as $P_{i}\times\{0\}$. Then we can solve for a matrix value Green’s function $G_{T(F)}$ for $T(F)$ in $\mathbb{R}^{3}$

(8.20)

$$-\Delta_{\mathbb{R}^{3}}G_{T(F)}=2\pi\delta_{T(F)}.$$

For this purpose we first solve the Green’s function for $P_{1}$ in $\mathbb{R}^{3}$. Again this is easy to write down explicitly as

(8.21)

$$G_{P_{1}}(x)=\int_{0}^{\infty}(\frac{1}{\sqrt{(x_{1}-t)^{2}+x_{2}^{2}+x_{3}^{2}}}-\frac{1}{t+1})dt=-\log(\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}-x_{1})+\log 2$$

This has interesting asymptotics. Writing $r^{2}=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}$ and $u=(x_{2},x_{3})$. If $|x_{1}|\leq C|u|$ then

(8.22)

$$G_{P_{1}}(x)\sim-\log r+\log 2+\frac{x_{1}}{r}-\frac{x_{1}^{2}}{r^{2}}+\cdots$$

If $x_{1}\gg|u|$, then

(8.23)

$$G_{P_{1}}(x)\sim-2\log|u|+\log|x_{1}|+O(|u|^{2}x_{1}^{-2}).$$

Now the Green’s function for $T(F)$ can be written down as a matrix

(8.24)

$$G_{T(F)}=\left[{\begin{array}[]{cc}G_{P_{2}}+\frac{1}{2}G_{P_{3}}&-\frac{1}{2}G_{P_{3}}\\ -\frac{1}{2}G_{P_{3}}&G_{P_{1}}+\frac{1}{2}G_{P_{3}}\\ \end{array}}\right]$$

Away from the three direction, the asymptotics as $r\rightarrow\infty$ is given by

(8.25)

$$-\left[{\begin{array}[]{cc}\frac{3}{2}\log r&-\frac{1}{2}\log r\\ -\frac{1}{2}\log r&\frac{3}{2}\log r\\ \end{array}}\right]$$

Now using the Green’s current $G_{P}$ and its asymptotics at infinity as describe above, one can construct an $S^{1}$ invariant incomplete three dimensional Kähler metrics as in Section 4.1. Notice as in 4.1 there are various parameters. First one can change the flat metric on $\mathbb{R}^{2}$. Also in the equation

(8.26)

$$\partial_{z}^{2}\tilde{\omega}+d_{D}d_{D}^{c}h=0,$$

one is free to add a function of $z$ to $h$, and add a closed $(1,1)$ form on $D=(\mathbb{C}^{*})^{2}$ to $\tilde{\omega}$. For appropriate choices of parameters one can make this Kähler metric approximately Calabi-Yau, and then the goal is to use weighted analysis to perturb to a family of genuine (incomplete) Calabi-Yau metrics. In appropriate scales, these metrics should collapse to a limit which is given as a domain in $\mathbb{R}^{3}$. One unsatisfactory point from our point of view is that comparing with the general expectation in SYZ metric collapsing conjecture, these incomplete metrics live on a too small region, since here the collapsing limit is flat whereas in general we should get a limit which is singular along the union of $P_{i}$’s. In other words, what one constructs here is only an infinitesimal model for the collapsing.

In a different direction. In complex three dimension, one can also consider $T^{2}$ invariant Calabi-Yau metrics. As discussed in Section 2.5, the corresponding dimension reduced equation has slightly different form and the linearized equation in the case when there are stabilizers also motivates us study certain Green’s currents.

Again we consider the model case $Q=\mathbb{R}^{2}\times\mathbb{C}^{*}$ is the quotient space and over $P=P_{1}\cup P_{2}\cup P_{3}\subset\mathbb{R}^{2}\times\{1\}$ we have stabilizers.

In this case we are interested in a matrix valued Dirac current

(8.27)

$$\delta_{P}=\sum\widehat{\delta}_{P_{i}}\cdot n_{i}\otimes n_{i}$$

where $P_{i}$ is naturally viewed as a submanifold in $\mathbb{R}^{2}\times\mathbb{C}^{*}$, and the corresponding matrix valued Green’s function $G_{P}$ satisfying

(8.28)

$$\Delta G_{P}=2\pi\delta_{P}$$

In large scale this is modeled by the corresponding current in $\mathbb{R}^{3}=\mathbb{R}^{2}\times\mathbb{R}$, and this has been discussed in the above. Near the vertex of $P$ one can consider the model $\mathbb{R}^{2}\times\mathbb{C}$, and find the corresponding Green’s function for $P\subset\mathbb{R}^{2}\times 0$. This is similar to the calculation above. For example, one gets

(8.29)

$$G_{P_{1}}(x)=\int_{0}^{\infty}\frac{1}{(x_{1}-t)^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}}dt=\frac{1}{v}(\frac{\pi}{2}+\tan^{-1}\frac{x_{1}}{v})$$

where $u=x_{3}+\sqrt{-1}x_{4}$ is the coordinate on $\mathbb{C}$, and

(8.30)

$$v^{2}=|u|^{2}+x_{2}^{2}.$$

We then define

(8.31)

$$(W_{ij})=\left[{\begin{array}[]{cc}G_{P_{2}}+\frac{1}{2}G_{P_{3}}&-\frac{1}{2}G_{P_{3}}\\ -\frac{1}{2}G_{P_{3}}&G_{P_{1}}+\frac{1}{2}G_{P_{3}}\\ \end{array}}\right]$$

and

(8.32)

$$\tilde{\omega}=\frac{\sqrt{-1}}{2}\Tr(W_{ij})dw\wedge d\bar{w}.$$

Then one can check the equation (2.65) is satisfied, and one obtains away from the singular locus a $\mathbb{T}^{2}$-invariant Kähler metric.

Naively one expects to compactify this metric along singular locus. We compare this with the standard local holomorphic model, which is the standard flat holomorphic structure $(\omega_{\mathbb{C}^{3}},\Omega_{\mathbb{C}^{3}})$ on $\mathbb{C}^{3}$ under the natural $\mathbb{T}^{2}$-action

(8.33)

$$(e^{\sqrt{-1}\theta_{1}},e^{\sqrt{-1}\theta_{2}}).(z_{1},z_{2},z_{3})=(e^{\sqrt{-1}(\theta_{1}+\theta_{2})}z_{1},e^{-\sqrt{-1}\theta_{1}}z_{2},e^{-\sqrt{-1}\theta_{2}}z_{3}).$$

The corresponding quotient map is given by

(8.34)

$$\mathcal{Q}:\mathbb{C}^{3}\rightarrow\mathbb{R}^{2}\oplus\mathbb{C};(z_{1},z_{2},z_{3})\mapsto(\frac{1}{2}(|z_{2}|^{2}-|z_{1}|^{2}),\frac{1}{2}(|z_{3}|^{2}-|z_{1}|^{2}),z_{1}z_{2}z_{3})$$

Also one can compute

(8.35)

$$W_{ij}=\frac{1}{|z_{1}|^{2}|z_{2}|^{2}+|z_{3}|^{2}|z_{1}|^{2}+|z_{2}|^{2}|z_{3}|^{2}}\left[{\begin{array}[]{cc}|z_{1}|^{2}+|z_{3}|^{2}&-|z_{1}|^{2}\\ -|z_{1}|^{2}&|z_{1}|^{2}+|z_{2}|^{2}\\ \end{array}}\right]$$

and

(8.36)

$$\tilde{\omega}=\frac{1}{|z_{1}|^{2}|z_{2}|^{2}+|z_{3}|^{2}|z_{1}|^{2}+|z_{2}|^{2}|z_{3}|^{2}}\cdot\frac{\sqrt{-1}}{2}{dw}\wedge d\bar{w}$$

So comparing with the previous formula they do not naturally match. This suggests that we might need to do something different near the vertex.

Now if we take the above formula of Green’s current, but work instead on $\mathbb{R}^{2}\times\mathbb{C}$, then one can see the above matrix actually has strictly positive lower bound at infinity. This makes us suspect the existence of a complete Calabi-Yau metric on $\mathbb{C}^{3}$ which is approximately the above ansatz at infinity. One approach is by using this ansatz as background metric at infinity and solve the Calabi-Yau equation as in [TY90]. This should be similar to the result of Yang Li constructing a complete Calabi-Yau metric $\mathbb{C}^{3}$ with infinity tangent cone $\mathbb{C}^{2}/\mathbb{Z}_{2}\times\mathbb{C}$. If such a metric can be constructed, then it should have a $\mathbb{T}^{2}$-symmetry and at infinity has $r^{4}$ volume growth and the tangent cone at infinity is $\mathbb{R}^{2}\oplus\mathbb{C}$ with locus of the singular fibration given by the $Y$-vertex. The situation may be analogous to that the Taub-NUT space is fibered over $\mathbb{R}\oplus\mathbb{C}$. The difference is that here we need to have discriminant locus essentially due to topological reasons.

The existence of such a complete Calabi-Yau metric on $\mathbb{C}^{3}$ also resolves the above concern regarding the bad singularity behavior of the ansatz metric near the vertex.

• •

  From the proof of Theorem 1.1, it follows that similar results hold in the following more general situation. We leave it for the readers to check the details.

  • –

    $p:\mathcal{X}\rightarrow\Delta$ is a proper holomorphic map from an $n+1$ dimensional normal complex analytic variety onto a disc $\Delta$ in $\mathbb{C}$.

  • –

    For $t\neq 0$, $X_{t}\equiv p^{-1}(t)$ is a smooth $n$ dimensional compact complex manifold.

  • –

    $X_{0}\equiv p^{-1}(0)$ is a union of two smooth $n$ dimensional Fano manifolds $Y_{1}$ and $Y_{2}$, and $Y_{1}\cap Y_{2}$ is a smooth $n-1$ dimensional Calabi-Yau manifold $D$.

  • –

    $\mathcal{L}$ is a relatively ample holomorphic line bundle on $\mathcal{X}$.

  • –

    Two positive integers $d_{1},d_{2}$, and we denote $k=d_{1}+d_{2}$.

  • –

    Holomorphic sections $f_{,}f_{2},f$ of $\mathcal{L}^{d_{1}},\mathcal{L}^{d_{2}},\mathcal{L}^{d_{1}+d_{2}}$ respectively satisfying

    (8.37)

    $$f_{1}f_{2}+tf=0.$$

  • –

    $Y_{1}\cap Y_{2}$ is a smooth $n-1$ dimensional Calabi-Yau manifold $D$.

  • –

    $\mathcal{X}$ is singular along a smooth divisor $H\subset D$ given by $\{f_{1}=f_{2}=f=t=0\}$. and transverse to $H$ the singularity is modeled on $\{x_{1}x_{2}+tx_{3}=0\}$.

  • –

    There is a holomorphic volume form on the smooth locus of $\mathcal{X}$.

• •

  Theorem 1.1 can be viewed as understanding the first order expansion of the family of Calabi-Yau metrics on $\widehat{\mathcal{X}}$ near $t=0$. One may ask whether it is possible to obtain a refined asymptotic expansion. In spirit, it is similar to the case of family of hyperbolic metrics on nodal degeneration of Riemann surfaces (See the recent work [MZ18] by Melrose-Zhu), and it is very likely similar techniques will be useful here. We thank Dominic Joyce and Xuwen Zhu for conversations on this.

• •

  As is mentioned in the Introduction, it remains an interesting question to directly glue together two Tian-Yau metrics with the same divisor $D$, without a priori assuming the existence of the complex family $\mathcal{X}$. As mentioned in the Introduction, in the case $n=2$ this was done in [HSVZ18] using $SU(2)$-structures, and in the case $n=3$ it is possible to use deformations of $SU(3)$-structures. This would require certain analysis (in particular Liouville theorem) on forms instead of functions. We leave this for future study.

• •

  In Section 5, we proved a Liouville Theorem on the Tian-Yau spaces using elementary analysis on special functions. Although not needed in this paper, it is interesting to see if there is a general Fredholm theory for the analysis of the Laplace operator on such spaces. We asked similar questions in the two dimensional case in [HSVZ18].

• •

  There is a different class of Tian-Yau spaces, constructed on the complement of a smooth anti-canonical divisor in a projective manifold with trivial normal bundle. In particular the ambient manifold can not be Fano. These spaces have different asymptotics at infinity from the ones we considered in this paper. Namely, they are asymptotically cylindrical. Given a smooth Fano manifold $Y$ and a pencil of anti-canonical divisors with smooth base locus $B$, let $Y^{\prime}$ be the blown-up of $Y$ along $B$, and let $D^{\prime}$ be the proper transform of a smooth element $D$ in the pencil. Then there is such an asymptotically cylindrical Calabi-Yau metric on $Y\setminus B$ (in every Kähler class). Asymptotically cylindrical Tian-Yau spaces have been important ingredients in the twisted connected sum construction of examples of compact $G_{2}$ holonomy manifolds. It is interesting to see whether the ideas of this paper can be used to construct new examples of $G_{2}$ holonomy manifolds by gluing together a suitably twisted circle fibration over various pieces.

• •

  As pointed out in Remark 1.1.1, our main result approximately reduces the understanding on the geometry of part of the Calabi-Yau manifolds $(X_{t},\omega_{CY,t})$ (the neck region) for $|t|\ll 1$ to the geometry of the Calabi-Yau metric on the one lower dimensional space $D$. One expects this can possibly lead to an inductive way to study geometry of Calabi-Yau metrics in higher dimensions through iterated degenerations. Correspondingly, it is also interesting to relate the submanifold geometry of the neck region to that of $D$. For example, suppose we have a special Lagrangian fibration on a region in $D$, can we construct special Lagrangian fibrations on the neck which are invariant under the $S^{1}$ action? At the two ends of the neck it is easy to see the pre-image of a special Lagrangian fibration under the projection map is approximately special Lagrangian. Near the singular fibers of the $S^{1}$ fibration the situation is more complicated and one expects certain singular perturbation techniques are needed. There are also similar discussions in [Li19] in the setting of Section 8.1.

• •

  In connection with algebro-geometric study of degenerations of Calabi-Yau manifolds, Theorem 1.1 shows that the normalized Gromov-Hausdorff limit in our setting is topologically the same as the essential skeleton of the degeneration $\mathcal{X}$. In the other extreme case, namely, the case of large complex structure limit of Calabi-Yau manifolds, it is a folklore conjecture (by Gross-Wilson and Kontsevich-Soibelman) that the normalized Gromov-Hausdorff limit is topologically the same as the essential skeleton of the degeneration. It is then natural to expect this conjecture may extend to general degenerations. Also it is also an interesting question to understand the algebro-geometric meaning of the normalized limit measure in Theorem 1.1, see [BJ17] for related algebro-geometric work. In the case $n=2$, there is also a plausible connection with the compactification of moduli space of hyperkähler metrics on K3 manifolds , see [OO18]. We leave all these for future exploration.