Abstract. We provide a significant extension of the twisted connected sum construction of G2-manifolds, ie Riemannian 7-manifolds with holonomy group G2, first developed by Kovalev; along the way we address some foundational questions at the heart of the twisted connected sum construction. Our extension allows us to prove many new results about compact G2-manifolds and leads to new perspectives for future research in the area. Some of the main contributions of the paper are:(i) We correct, clarify and extend several aspects of the K3 "matching problem" that occurs as a key step in the twisted connected sum construction. (ii) We show that the large class of asymptotically cylindrical Calabi-Yau 3-folds built from semi-Fano 3-folds (a subclass of weak Fano 3-folds) can be used as components in the twisted connected sum construction. (iii) We construct many new topological types of compact G2-manifolds by applying the twisted connected sum to asymptotically Calabi-Yau 3-folds of semi-Fano type studied in [18]. (iv) We obtain much more precise topological information about twisted connected sum G2-manifolds; one application is the determination for the first time of the diffeomorphism type of many compact G2-manifolds. (v) We describe "geometric transitions" between G2-metrics on different 7-manifolds mimicking "flopping" behaviour among semi-Fano 3-folds and "conifold transitions" between Fano and semi-Fano 3-folds. (vi) We construct many G2-manifolds that contain rigid compact associative 3-folds. (vii) We prove that many smooth 2-connected 7-manifolds can be realised as twisted connected sums in numerous ways; by varying the building blocks matched we can vary the number of rigid associative 3-folds constructed therein. The latter result leads to speculation that the moduli space of G2-metrics on a given 7-manifold may consist of many different connected components, and opens up many further questions for future study. For instance, the higher-dimensional gauge theory invariants proposed by Donaldson may provide ways to detect G2-metrics on a given 7-manifold that are not deformation equivalent.

We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau 3-folds starting with (almost) any deformation family of smooth weak Fano 3-folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau 3-folds; previously only a few hundred ACyl Calabi-Yau 3-folds were known. We pay particular attention to a subclass of weak Fano 3-folds that we call semi-Fano 3-folds. Semi-Fano 3-folds satisfy stronger cohomology vanishing theorems and enjoy certain topological properties not satisfied by general weak Fano 3-folds, but are far more numerous than genuine Fano 3-folds. Also, unlike Fanos they often contain P 1 s with normal bundle O(−1) ⊕ O(−1), giving rise to compact rigid holomorphic curves in the associated ACyl Calabi-Yau 3-folds.We introduce some general methods to compute the basic topological invariants of ACyl Calabi-Yau 3-folds constructed from semi-Fano 3-folds, and study a small number of representative examples in detail. Similar methods allow the computation of the topology in many other examples.All the features of the ACyl Calabi-Yau 3-folds studied here find application in [17] where we construct many new compact G 2 -manifolds using Kovalev's twisted connected sum construction. ACyl Calabi-Yau 3-folds constructed from semi-Fano 3-folds are particularly well-adapted for this purpose.

Abstract. We study homogeneous special Lagrangian cones in C n with isolated singularities. Our main result constructs an infinite family of special Lagrangian cones in C 3 each of which has a toroidal link. We obtain a detailed geometric description of these tori. We prove a regularity result for special Lagrangian cones in C 3 with a spherical link -any such cone must be a plane. We also construct a one-parameter family of asymptotically conical special Lagrangian submanifolds from any special Lagrangian cone.

Abstract. There is a rich theory of so-called (strict) nearly Kähler manifolds, almostHermitian manifolds generalising the famous almost complex structure on the 6-sphere induced by octonionic multiplication. Nearly Kähler 6-manifolds play a distinguished role both in the general structure theory and also because of their connection with singular spaces with holonomy group the compact exceptional Lie group G2: the metric cone over a Riemannian 6-manifold M has holonomy contained in G2 if and only if M is a nearly Kähler 6-manifold.A central problem in the field has been the absence of any complete inhomogeneous examples. We prove the existence of the first complete inhomogeneous nearly Kähler 6-manifolds by proving the existence of at least one cohomogeneity one nearly Kähler structure on the 6-sphere and on the product of a pair of 3-spheres. We conjecture that these are the only simply connected (inhomogeneous) cohomogeneity one nearly Kähler structures in six dimensions. IntroductionAt least since the early 1950s (see Steenrod's 1951 book [45, 41.22]) it has been well known that viewing S 6 as the unit sphere in Im O endows it with a natural nonintegrable almost complex structure J defined via octonionic multiplication. Since J is compatible with the round metric g rd , the triple (g rd , J, ω), where ω(·, ·) = g rd (J·, ·), defines an almost-Hermitian structure on S 6 . Its torsion has very special properties: in particular, dω is the real part of a complex volume form Ω. Appropriately normalised, the pair (ω, Ω) defines an SU(3)-structure on S 6 which by construction is invariant under the exceptional compact Lie group G 2 Aut(O).Octonionic multiplication also defines a G 2 -invariant 3-form ϕ on Im O byWe call this the standard G 2 -structure on R 7 . Regarding R 7 as the Riemannian cone over (S 6 , g rd ), ϕ and its Hodge dual are given in terms of (ω, Ω):Conversely, viewing S 6 as the level set r = 1 in R 7 , the SU(3)-structure (ω, Ω) is recovered from ϕ and * ϕ by restriction and contraction by the scaling vector field ∂ ∂r . More generally, consider a 7-dimensional Riemannian cone C = C(M ) over a smooth compact manifold (M 6 , g). Suppose that the holonomy of the cone is contained in G 2 . Then there exists a pair of closed (in fact parallel) differential forms ϕ and * ϕ, pointwise equivalent to the model forms on R 7 and homogeneous with respect to scalings on C. Just as above, viewing M as the level set r = 1 in C, the restriction and contraction by ∂ ∂r of ϕ and * ϕ define an SU(3)-structure (ω, Ω) on M satisfying (1.1). In particular, the closedness of ϕ and * ϕ is equivalent to There are other possible equivalent definitions of a nearly Kähler 6-manifold. By relating the holonomy reduction of the cone C(M ) to the existence of a parallel spinor instead of a pair of distinguished parallel forms, nearly Kähler 6-manifolds can be characterised as those 6-manifolds admitting a real Killing spinor [34]. Alternatively, one could give a definition in terms of Gray-Hervella torsion classes of almost Her...

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.