Asymptotically cylindrical Calabi–Yau manifolds

Abstract

Let M be a complete Ricci-flat Kähler manifold with one end and assume that this end converges at an exponential rate to [0, ∞) × X for some compact connected Ricci-flat manifold X. We begin by proving general structure theorems for M ; in particular we show that there is no loss of generality in assuming that M is simply-connected and irreducible with Hol(M ) = SU(n), where n is the complex dimension of M . If n > 2 we then show that there exists a projective orbifold M and a divisor D ∈ |−K M | with torsion normal bundle such that M is biholomorphic to M \ D, thereby settling a long-standing question of Yau in the asymptotically cylindrical setting. We give examples where M is not smooth: the existence of such examples appears not to have been noticed previously. Conversely, for any such pair (M , D) we give a short and self-contained proof of the existence and uniqueness of exponentially asymptotically cylindrical Calabi-Yau metrics on M \ D.

Spin(7)-manifolds as generalized connected sums and 3d $$ \mathcal{N}=1 $$ theories

“…Asymptotically Calabi-Yau manifolds were discussed in [38]. Similar to the explicit construction of acyl three-folds in [5,6], we expect to able to construct Z + by excising a fiber X 3 from a compact four-foldZ + with c 1 (Z + ) = [X 3 ], which is fibered by Calabi-Yau three-folds.…”

“…The work [38][39][40] on acyl Calabi-Yau four-folds and acyl G 2 -manifolds and their deformation theory, together with clear criteria when the asymptotic Calabi-Yau three-folds X ± 3 allow a biholomorphic map φ, should clarify under which circumstances a gluing can be found for a given pair of such manifolds. In our examples, we can easily find such diffeomorphisms by realizing X ± 3 as hypersurfaces in toric varieties, so that a diffeomorphic pair can be simply constructed by writing down identical algebraic equations.…”

G2–instantons over twisted connected sums

“…Definition 2.19). Given a choice of hyperkähler structure (ω I , ω J , ω K ) on Σ such that [ω I ] is the restriction of a Kähler class on Z, one can make V := Z \ Σ into an asymptotically cylindrical (ACyl) Calabi-Yau 3-fold, that is, a non-compact Calabi-Yau 3-fold with a tubular end modelled on R + × S 1 × Σ, see Haskins-Hein-Nordström [HHN15]. Then Y := S 1 × V is an ACylG 2 -manifold with a tubular end modelled on…”

Holomorphic bundles for higher dimensional gauge theory

“…A building block (Z, D) [CHNP13, Definition 3.5] is given by a projective morphism f : Z → P 1 such that D := f −1 (∞) is a smooth anticanonical K3 surface, under certain mild topological assumptions; in particular, D has trivial normal bundle. Choosing a convenient Kähler structure on Z, one can make W := Z \ D into an ACyl Calabi-Yau 3-fold, that is, a non-compact CalabiYau with a tubular end modelled on R + × S 1 × D. Then S 1 × W is an ACyl G 2 -manifold with a tubular end modelled on R + × T 2 × D [HHN15,CHNP15].…”