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 $n-1$. \[ \]
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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.
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.
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.
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.
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.