Complex Hantzsche-Wendt manifolds

Abstract

Complex Hantzsche-Wendt manifolds are flat K\"ahler manifolds with holonomy group $\mathbb{Z}_2^{n-1}\subset SU(n)$. They are important example of Calabi-Yau manifolds of abelian type. In this paper we describe them as quotients of a product of elliptic curves by a finite group $\tilde{G}$. This will allow us to classify all possible integral holonomy representations and give an algorithm classifying their diffeomorphism types.Comment: 19 pages, 1 tabl

Symmetries of complex flat manifolds

“…On the other hand, if n ∈ N Aut 0 (T ) (G) fixes all curves E i , then it may act on any of them by an element of the group −ξ ≃ Z 6 . Thus N Aut 0 (T ) (G) ∼ = Z 4 6 ⋊ S 3 . Now it is easy to observe, that under action of N Aut 0 (T ) (G) on the H 1 (G, T ) all cohomology classes corresponding to fixed-point free actions are in the same orbit.…”

“…In order to calculate N α we need to fix a cohomology class of H 1 (G, T ), or equivalently fix a choice of 3-torsion points a, b, c, d, for example all equal to 1 3 (1 − ξ). For this class α we have that N α is exactly Z 4 3 ⋊ S 3 . Thus N α /G ∼ = Z 2 3 ⋊ S 3 , and Aut(M) is an extension:…”

“…As our next example, we will examine complex Hantzsche-Wendt manifolds (or shortly -CHW manifolds) of complex dimension 3. By definition, these are complex flat manifolds with holonomy group Z 2 2 ⊂ SU(3) (see [4]). There are four possible integral holonomy representations.…”

“…All CHW threefolds with this property are diffeomorphic, and their fundamental group has CARAT symbol min.185.1.1.21. Still, there are infinitely many such manifolds up to biholomorphism, which follows from the structure theorem of [4]. In our case this theorem simplifies to the following statement: Theorem 5.1 Manifold M is a CHW threefold with diagonal integral holonomy representation if and only if M is a orbit space T / G, where T = E 1 × E 2 × E 3 is a product of some elliptic curves and G is a group generated by mappings g1 , g2 : T → T such that: Let M = T / G be as above, G = π( G), g 1 = π(g 1 ) and g 2 = π(g 2 ).…”

Spin-Structures on Real Bott Manifolds

“…, n ([2], Proposition 1.3). Moreover, from the definition of matrix P ∈ S d×n we can write equations ( 6) and (7) as follows ( 8)…”