ScalingStacks

Consider an algebraic Calabi-Yau degeneration family $X\to S\setminus\{0\}$ as above. We follow [3] to consider the asymptote of $\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}$ as $t\to 0$. Along the way, we will introduce the concept of dual intersection complexes and essential skeletons, which are simplicial complexes encoding the intersection patterns of divisors on the central fibre. An important lesson is that the measure theoretic limit of the Calabi-Yau manifolds is closer to simplicial complexes than algebraic varieties, indicating that the metric limit must be significantly different from Fubini-Study metrics associated to projective embeddings of bounded degree.

A very useful tool is to fill in the central fibre by choosing an snc model (cf. Remark 2) $\mathcal{X}$ over $S$. The central fibre $\mathcal{X}_{0}$ is an snc divisor with components $E_{i}$ for $i\in I$, and we write $\mathcal{X}_{0}=\sum_{i\in I}b_{i}E_{i}$. In the special case of semistable snc models $b_{i}=1$ for $i\in I$; this can always be achieved after finite base change. The canonical divisor $K_{\mathcal{X}}$ is supported on $\mathcal{X}_{0}$ as $K_{X}$ has a trivialising section $\Omega$. We may write $K_{\mathcal{X}}=\sum_{i}(a_{i}+b_{i}-1)E_{i}$, so that the relative log canonical divisor

$$K^{log}_{\mathcal{X}/S}:=K_{\mathcal{X}}-K_{S}+\mathcal{X}_{0,red}-\mathcal{X}_{0}=\sum a_{i}E_{i}.$$

Shifting all $a_{i}$ by a constant $\kappa$ is equivalent to multiplying $\Omega$ by $t^{\kappa}$, which gives an elementary factor $|t|^{2\kappa}$ to $\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}$. Thus we shall always assume $\min a_{i}=0$.

It is useful to introduce a quantitative stratification on $X_{t}$ according to the intersection pattern of $E_{i}$. Let $E_{J}=\cap_{i\in J}E_{i}$ for $J\subset I$, which is irreducible if nonempty. Using the distance function of a fixed smooth background Kähler metric on $\mathcal{X}$, we can write

$$E_{J}^{0}=\{q\in X_{t}|d(q,E_{J})\ll 1\}\setminus\{q\in X_{t}|d(q,E_{J^{\prime}})\ll 1,\quad\text{some }J^{\prime}\supsetneq J\}.$$

Around $\emptyset\neq E_{J}\subset\mathcal{X}$, we denote $p=|J|-1$, and introduce local coordinates $z_{0},\ldots z_{n}$ on $\mathcal{X}$, such that $z_{0},z_{1},\ldots,z_{p}$ are the defining equations of $E_{i}$ for $i\in J$. The conditions on the divisors mean that away from deeper strata we may arrange $t=\prod_{0}^{p}z_{i}^{b_{i}}$, and

$$\Omega=u_{J}\prod_{0}^{p}z_{i}^{a_{i}+b_{i}}d\log z_{i}\wedge\prod_{p+1}^{n}dz_{j}$$

for some local nowhere vanishing holomorphic function $u_{J}$. By definition $\Omega=dt\wedge\Omega_{t}$ along $X_{t}$, so on $E_{J}^{0}$

$$\Omega_{t}=b_{0}^{-1}u_{J}z_{0}^{a_{0}}\ldots z_{p}^{a_{p}}\prod_{1}^{p}d\log z_{i}\wedge\prod_{p+1}^{n}dz_{j},$$

$$\sqrt{-1}^{n^{2}}\Omega_{t}\wedge\overline{\Omega}_{t}=|b_{0}|^{-2}|u_{J}|^{2}|z_{0}|^{2a_{0}}\ldots|z_{p}|^{2a_{p}}\prod_{1}^{p}\sqrt{-1}d\log z_{i}\wedge d\log\bar{z}_{i}\wedge\prod_{p+1}^{n}\sqrt{-1}dz_{j}\wedge d\bar{z}_{j}.$$

Notice also that the local equation $t=\prod_{0}^{p}z_{i}^{b_{i}}$ has $b_{J}=\gcd_{i\in J}b_{i}$ sheets of solutions. Using the polar coordinates by $z_{i}=e^{x_{i}\log|t|+\sqrt{-1}\theta_{i}}$ for $i\in J$, ones sees that the magnitude of $\int_{E_{J}^{0}}\sqrt{-1}^{n^{2}}\Omega_{t}\wedge\overline{\Omega}_{t}$ is $O(|\log|t||^{l})$ for $l=|\{j\in J:a_{j}=0\}|-1$.

The local logarithmic variables $x_{i}=\frac{\log|z_{i}|}{\log|t|}$ lie on the simplex

$$\Delta_{J}=\{\sum_{0}^{p}b_{i}x_{i}=1,\quad 0\leq x_{i}\leq 1\}.$$

These depend on the choice of $z_{i}$, but since the local defining equation of divisors differ by a nowhere vanishing holomorphic function, the ambiguity of $x_{i}$ is only $O(\frac{1}{|\log|t||})$ for $0<|t|\ll 1$. Taking a more global viewpoint, the combinatorial pattern of how these simplices fit together exactly reflects the intersection pattern of the divisors $E_{i}$. Formally, this information is encoded in the dual intersection complex $\Delta_{\mathcal{X}}$ for the snc model $\mathcal{X}$: this is the polyhedral complex whose vertices $v_{i}$ correspond to $E_{i}$, and we assign a simplex $\Delta_{J}$ with vertices $v_{i}$ for $i\in J$ if and only if $E_{J}\neq 0$. The coodinates $x_{j}$ then define a piecewise integral affine structure on $\Delta_{\mathcal{X}}$. Up to the above $O(\frac{1}{|\log|t||})$ ambiguity, we now have a logarithm map $\text{Log}_{\mathcal{X}}:X_{t}\to\Delta_{\mathcal{X}}$, locally described by $x_{i}=\frac{\log|z_{i}|}{\log|t|}$. Consequently, the ‘hybrid’ space $X\sqcup\Delta_{\mathcal{X}}$ is equipped with a natural topology, so that a sequence of points $z_{k}\in X_{t}$ converges to $x\in\Delta_{\mathcal{X}}$ iff $t\to 0$ and $\text{Log}_{\mathcal{X}}(z_{k})\to x$. The name ‘hybrid’ refers to the mixture of algebraic varieties with simplicial objects, which is better suited for measure theoretic limits, than the algebraic family $\mathcal{X}$.

The measure also singles out a distinguished subcomplex $Sk(\mathcal{X})$, called the essential skeleton, consisting of the simplices in $\Delta_{\mathcal{X}}$ whose vertices correspond to $E_{i}$ with $a_{i}=0$. This is where the limit of the normalised CY measure is supported. The dimension of $Sk(\mathcal{X})$ is a measurement of how transcendental the degeneration $X$ is; it is reflected by the growth order of $\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}$. The largest possible value for the dimension is $n$.

In the case of a large complex structure limit, $\dim_{\mathbb{R}}Sk(\mathcal{X})=n$. Let us analyze the CY measure more explicitly, in a semistable snc model. For $E_{J}$ corresponding to an $n$-dimensional simplex in $Sk(\mathcal{X})$, on $E_{J}^{0}$

$$\sqrt{-1}^{n^{2}}\Omega_{t}\wedge\overline{\Omega}_{t}=|u_{J}|^{2}\prod_{1}^{n}\sqrt{-1}d\log z_{i}\wedge d\log\bar{z}_{i}.$$

(7)

Here $u_{J}$ limits to its value $u_{J}(E_{J})$ at the point stratum $E_{J}$, which is called the Poincaré residue of $\Omega$, and is easily seen to be independent of the choice of coordinates $z_{i}$. It is a consequence of the residue theorem on Riemann surfaces that $|u_{J}(E_{J})|^{2}$ is independent of such $J$ [3, Thm. 7.1]. Thus the pushforward to $\Delta_{\mathcal{X}}$ of the normalised CY measure (5) converges smoothly in the interior of $\Delta_{J}$ to a constant multiple of the Lebesgue measure:

$$\text{Log}_{\mathcal{X}*}d\mu_{t}=\text{Log}_{\mathcal{X}*}\frac{\Omega_{t}\wedge\overline{\Omega}_{t}}{\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}}\xrightarrow{t\to 0}d\mu_{0}:=\text{Const}\cdot dx_{1}\ldots dx_{n}.$$

(8)

Notice $dx_{1}\ldots dx_{n}$ is canonically defined due to the presence of an integral affine structure on $\Delta_{J}$. Viewed as a measure on $\Delta_{\mathcal{X}}$, the limit $d\mu_{0}$ has null measure on the complement of the $n$-dimensional faces of $Sk(\mathcal{X})$, as the integral of $d\mu_{t}$ in the corresponding region is $O(\frac{1}{|\log|t||})$. The constant in (8) is independent of $J$ and its sole purpose is to make $d\mu_{0}$ a probability measure.