Asymptotically cylindrical Calabi-Yau manifolds

By M. Haskins, H. Hein, Johannes Nordstrom. 2012

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Let M be a complete Ricci-flat Kahler manifold with one end and assume that this end converges at an exponential rate to [0,oo) x 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 barM and a divisor barD in |-K_barM| with torsion normal bundle such that M is biholomorphic to barM \ barD, thereby settling a long-standing question of Yau in the asymptotically cylindrical setting. We give examples where barM is not smooth: the existence of such examples appears not to have been noticed previously. Conversely, for any such pair (barM, barD) we give a short and self-contained proof of the existence and uniqueness of exponentially asymptotically cylindrical Calabi-Yau metrics on barM \ barD.

Subfields: Applied Mathematics, Geometry and Topology, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/6e72288d6a418dd51e1268a74ce7a5a595b89283

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In Applied Mathematics

In Geometry and Topology

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Citations in the corpus

G2-manifolds and associative submanifolds via semi-Fano 3-folds (2012, background): “See [37] for further discussion of this and other related issues.”
G2-instantons over twisted connected sums (2013, unknown): “with a tubular end modelled on R+ × S1 × Σ, see Haskins–Hein–Nordström [11].”
Asymptotically cylindrical Calabi–Yau 3–folds from weak Fano 3–folds (2012, methodology): “at (2.1) holds (implying that (!; ) is a Calabi-Yau structure). This was done by Tian and Yau [82] in a more general setting. In our special case, a simpler and signicantly shorter proof is given in [31], which in addition proves that !is ACyl with respect to . The asymptotic limit of !has the form ”
Asymptotically conical Calabi-Yau manifolds, III (2012, background): “tructure of the total space of L. Then, modulo diﬀeomorphism, J extends to a smooth complex structure on U. This result is analogous to [14, Theorem 3.1], but Li’s proof is diﬀerent from the proof in [14]. The aim of this appendix is to clarify the relation between the conical and the cylindrical ca”