The diversity of symplectic Calabi-Yau 6-manifolds

Abstract

Given an integer b and a finitely presented group G, we produce a compact symplectic 6‐manifold with c1=0, b2>b, b3>b and π1=G. In the simply connected case, we can also arrange for b3=0; in particular, these examples are not diffeomorphic to Kähler manifolds with c1=0. The construction begins with a certain orientable, four‐dimensional, hyperbolic orbifold assembled from right‐angled 120‐cells. The twistor space of the hyperbolic orbifold is a symplectic Calabi–Yau orbifold; a crepant resolution of this last orbifold produces a smooth symplectic manifold with the required properties.

On the topology of symplectic Calabi-Yau 4-manifolds

“…We want to stress that the results presented above sit at the intersection of symplectic topology and 4-manifold topology. Namely, they provide constraints on the type of M that emerge only in this dimension; it is otherwise known by Fine and Panov [14] that in dimension 6 for any finitely presented group G, there exists a symplectic manifold Z with canonical class K = 0 and π 1 (Z) = G and much latitude on the choice of higher Betti numbers. (In dimension 2, of course, the only admissible G is Z 2 .…”

Non-Kähler SYZ Mirror Symmetry

“…Construction of non-Kähler Calabi-Yau manifolds has been studied in several recent works [Cal58,STY02,GP04,GGP08,Wu06,FP10,FLY12,FP13]. The survey by Fu [Fu10] gave an excellent introduction to this topic.…”

The Conley Conjecture and Beyond

“…In fact, the manifolds meeting this requirement are more numerous than it might seem. As is proved in Fine and Panov (2013), for every finitely presented group G there exists a closed symplectic 6-manifold M with π 1 (M) = G and c 1 (T M) = 0. A basic example of a negative monotone symplectic manifold is a smooth hypersurface of degree d > n + 2 in CP n+1 .…”