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

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

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

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

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

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 $E_{J}^{0}$

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:

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.

A major caveat is that snc models are highly non-unique, because one can always blow up a given snc model along some locus inside the central fibre. The intrinsic information concerning polarized degenerations, such as the limiting behaviour of metrics and volume measures, are independent of particular snc models.

The effect of blow up on the dual intersection complex and essential skeleton is well understood (cf. [49, A.4]). The intuitive picture is as follows:

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  If we blow up an snc model $\mathcal{X}$ along a smooth irreducible subvariety properly contained inside $E_{J}=\cap_{j\in J}E_{j}$, but not contained in any smaller intersection stratum, then the new dual intersection complex contains $\Delta_{\mathcal{X}}$, while introducing a new vertex, and some new wings over certain faces of $\Delta_{\mathcal{X}}$. The essential skeleton remains intact.

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  If we blow up an snc model $\mathcal{X}$ along some $E_{J}=\cap_{j\in J}E_{j}$, then the new dual intersection complex is a subdivision of $\Delta_{\mathcal{X}}$. If the simplex $\Delta_{J}$ corresponding to $E_{J}$ is a face of the essential skeleton $Sk(\mathcal{X})$, then there is an induced subdivision on $Sk(\mathcal{X})$, and otherwise the essential skeleton remains intact.

While $\Delta_{\mathcal{X}}$ generally becomes larger under blow ups, the essential skeleton $Sk(\mathcal{X})$ is only subdivided, and its piecewise affine structure (in particular its homeomorphism type) is a birational invariant, denoted as $Sk(X)$. The birational invariance is not surprising: the essential skeleton is the measure theoretic limit of $X_{t}$, a property independent of the choice of models.

In an attempt to extract limiting information from the SYZ picture, Kontsevich and Soibelman [48][49] proposed the following picture. Let $X_{t}$ be a polarized degeneration of Calabi-Yau manifolds near the large complex structure limit, then

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  (Kähler class normalization) The rescaled metrics $\omega_{CY,t}\in\frac{1}{|\log|t||}c_{1}(\mathcal{O}(1))$ have nontrivial finite diameter Gromov Hausdorff subsequential limits.

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  There is an affine structure on the essential skeleton $Sk(X)$ away from a Hausdorff codimension two singular subset. On the smooth locus, there is a Riemannian metric obtained as the Hessian of local solutions to the real Monge-Ampère equation. This metric agrees with the Gromov-Hausdorff limit, which is conjecturally independent of the subsequence.

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  (Topology) Under the strict Calabi-Yau condition $h^{p,0}(X_{t})=0$ for $p<n$, the essential skeleton is homeomorphic to $S^{n}$.

The heuristic idea is that the essential skeleton should be the base of the hypothetical SYZ fibration, and the SYZ fibration is approximated by logarithm maps, at least in the generic region. We briefly comment on the status of the Kontsevich-Soibelman conjecture (cf. also [53, section 4.5]):

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  The uniform diameter estimate independent of small $t$

  $$C^{-1}\leq\text{diam}(X,\omega_{CY,t})\leq C,$$

  is recently established in joint work with Tosatti [56], using primarily Riemannian geometric methods. This together with Gromov compactness proves the Kähler class normalization prediction.

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  The prediction about the existence of a metrically preferred affine structure away from codimension two on the base, is part of the general lore of the SYZ conjecture (cf. section 2.4), even though there is insufficient evidence. Generally speaking, the simplicial complex structure on $Sk(X)$ induces a piecewise affine structure, and improving it to an affine structure away from codimension two, would require highly nontrivial choices. The author is not aware of a completely satisfactory answer to the following elementary question:

  002M

  Question 5. Given a one-parameter family of quartic K3 surfaces near the large complex structure limit, without special symmetry, how can we determine the location of singular points on $Sk(X)$? How can we write down the affine structure on the regular locus explicitly?

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  The topology of the essential skeleton is an active research topic in birational geometry. The homeomorphism between $Sk(X)$ with $S^{n}$ is verified for many examples. In general, it is known [62][63] that $Sk(X)$ is a ‘pseudomanifold’, its rational homology groups agree with $S^{n}$, and its fundamental group has trivial profinite completion, but the actual homeomorphism type is still elusive, and supposedly requires appealing to the Poincaré conjecture.