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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 $B_Y^{=1}$ where $(Y,B_Y)\longrightarrow(X,B)$ is a dlt modification.

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

Recently, Brown has constructed some explicit dual complexes of log Calabi–Yau 4-folds. 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.


    1. 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?


        • Add remark

          Problem 1.2.

          Does the Poincare homology $\mathbb{S}^3$ arise as the dual complex of a log Calabi–Yau pair?


            • 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.


                • 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.