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.
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.