Higher-dimensional log Calabi-Yau pairs

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1. Topology of dual complexes

Definition: A log Calabi–Yau pair is a couple

$(X,B)$ where

$X$ is a projective variety,

$B$ is a boundary divisor,

$(X,B)$ has log canonical singularities, and

$K_X+B \equiv 0$ .

Definition: The dual complex

$\mathcal{D}(X,B)$ of a log Calabi-Yau pair

$(X,B)$ is the dual complex of the divisor

$B_Y^{=1}$ where

$(Y,B_Y)\longrightarrow(X,B)$ is a dlt modification.

A priori, the dual complex of a log Calabi–Yau pair depends on the choice of a dlt modification, i.e., of some partial resolution of the pair. Due to the work of De Fernex–Kollár–Xu, the dual complex

$\mathcal{D}(X,B)$ of a log Calabi–Yau pair

$(X,B)$ is well-defined up to simple homotopy equivalence. The dual complex

$\mathcal{D}(X,B)$ of a log Calabi-Yau pair

$(X,B)$ is known to be a pseudo-manifold which is a

$\mathbb{Q}$ -homology sphere. In dimension 2, the dual complex of a log Calabi–Yau pair is either two points, a line, or a circle. Kollár and Xu proved that the dual complex of a log Calabi-Yau pair of dimension at most 4 is PL-isomorphic to a finite quotient of a sphere.

Definition: The coregularity of a log Calabi-Yau pair

$(X,B)$ is the dimension of a minimal log canonical center of a dlt modification of

$(X,B)$ . Equivalently,

$\text{coreg}(X,B)=\text{dim}(X)-\text{dim}\mathcal{D}(X,B)-1.$

Calabi–Yau pairs of coregularity zero are also called pairs of maximal intersection in the literature. The dual complex of a Calabi–Yau pair of dimension

$n$ and coregularity zero is a CW complex of dimension

$n-1$ .

Recently, Brown has constructed some explicit dual complexes of log Calabi–Yau 4-folds of coregularity zero. For instance, the Lens spaces $L(3,1)$ , $L(5,1)$ and $L(5,2)$ appear as dual complexes of log Calabi–Yau 4-folds. On the other hand, Moraga proved that only finitely many groups appear as the fundamental groups of dual complexes of $n$ -dimensional log Calabi–Yau pairs. It is not clear what are the possible fundamental groups that appear in dimension 4.

Add remark

Problem 1.1.
What are the possible fundamental groups $\pi_1(\mathcal{D}(X,B))$ for a log Calabi–Yau 4-fold $(X,B)$ of coregularity zero? Does $\mathbb{Z}/7\mathbb{Z}$ appear as such a group?
It is a folklore conjecture that the dual complex of log Calabi–Yau pair is the finite quotient of a sphere. However, this conjecture is open for log Calabi–Yau pairs of dimension at least 5. With the current results, we cannot rule out that a Poincare homology $\mathbb{S}^3$ appears as a dual complex of a log Calabi–Yau pair.

Add remark

Problem 1.2.
Does the Poincare homology $\mathbb{S}^3$ arise as the dual complex of a log Calabi–Yau pair?
Analogously to the previous problem, the Wu manifold is another potential example of dual complex of a log Calabi–Yau pair. Providing such an explicit example or introducing tools to rule it out as a possible dual complex would be an step forward in the understanding of dual complexes.

Add remark

Problem 1.3.
Does the Wu manifold appear as the dual complex of a log Calabi-Yau pair?
Figueroa and Moraga proved that any 3-manifold that admits a smooth embedding into $\mathbb{R}^4$ is homotopic to the dual complex of a suitable log smooth simple normal crossing Calabi–Yau 3-fold. This statement is not known for higher-dimensional manifolds. The first step towards this direction is to give explicit examples for 4-manifolds.

Add remark

Problem 1.4.
Does there exist a log smooth simple normal crossing Calabi–Yau variety $X$ for which $\mathcal{D}(X)$ is homotopy equivalent to the Wu manifold?

Cite this as: AimPL: Higher-dimensional log Calabi-Yau pairs, available at http://aimpl.org/higherdimlogcy.

Higher-dimensional log Calabi-Yau pairs

1. Topology of dual complexes

Problem 1.1.

Problem 1.2.

Problem 1.3.

Problem 1.4.