ScalingStacks

Consider an algebraic Calabi-Yau degeneration family $X\to S\setminus\{0\}$ as in the Introduction. We are given the holomorphic volume forms $\Omega_{t}$ on the fibres $X_{t}$, and let us follow [3] to consider the question of calculating the asymptote of $\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}$ as $t\to 0$. Since we only care about small $t$, we are free to shrink $S$. For instance, we may assume $dt$ is nowhere vanishing on $S$.

A very useful tool is to fill in the central fibre by choosing an snc model (cf. Remark 1.2) $\mathcal{X}$ 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. The measure calculation above explains why $\Delta_{\mathcal{X}}$ (or rather the essential skeleton inside, see below) is a better candidate notion as a limit of $X_{t}$ than the algebraic limit $\mathcal{X}_{0}$. Intuitively, the Calabi-Yau measure in the limit becomes mutually orthogonal with the measure of any fixed Fubini-Study metric, and the region carrying most of CY measure looks ‘small’ from the algebraic perspective.

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 $\int_{X_{t}}\Omega_{t}\wedge\overline{\Omega}_{t}$. In the case of a maximal degeneration, $\dim_{\mathbb{R}}Sk(\mathcal{X})=n$. Let us analyze the CY measure more explicitly for maximal degenerations, in a semistable snc model. For $E_{J}$ corresponding to an $n$-dimensional simplex in $Sk(\mathcal{X})$, on $E_{J}^{0}$

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

(4)

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 (1) converges smoothly in the interior of $\Delta_{J}$ to a constant multiple of the Lebesgue measure:

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

(5)

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 (5) is independent of $J$ and its sole purpose is to make $d\mu_{0}$ a probability measure.