The notion of large complex structure limit plays a special role in the theory of mirror symmetry. If X is a Calabi-Yau manifold, a large complex structure limit point is a point in a compactified moduli space of complex structures M X on X which, in some sense, represents the "worst possible degeneration" of the complex structure. This notion was given a precise Hodge-theoretic meaning in [27]. The basic example to keep in mind of this sort of degeneration is the degeneration of a hypersurface of degree n + 1 in P n to a union of the n + 1 coordinate hyperplanes. Mirror symmetry posits the existence of a mirror to X associated to each large complex structure limit point of X. To first approximation, this means that if p ∈ M X is a large complex structure limit point in a compactification of the complex moduli space of X, then there exists a mirrorX and an isomorphism between a neighbourhood of p in M X and the complexified Kähler moduli space ofX which preserves certain additional information, such as the Yukawa couplings (which will not concern us in this paper). This isomorphism is known as the mirror map. Now the Strominger-Yau-Zaslow conjecture [32] suggests that mirror symmetry can be explained by the existence of a special Lagrangian fibration on X when the complex structure on X is near a large complex structure limit point. The mirrorX is then expected to be constructed as the dual of this special Lagrangian fibration. The notion of special Lagrangian is a metric one: it depends on both the complex structure (determined by a holomorphic n-form Ω on X, where n = dim C X), and a Ricci-flat Kähler metric, determined by its Kähler form ω. Thus we expect the existence of special Lagrangian fibrations will depend a great deal on the metric properties of Calabi-Yau manifolds near large complex structure limit points.The simplest example of such a situation occurs for elliptic curves. Consider the family of elliptic curves E α = C/ 1, iα , with α → ∞. We also choose a Ricci-flat, i.e. * Supported in part by NSF grant DMS-9700761, Trinity College, Cambridge, and EPSRCOoguri-Vafa metric in 1998. §1. Identification of large complex structure limits.There are a number of variants of mirror symmetry for K3 surfaces: see especially [10] for mirror symmetry between algebraic families of K3 surfaces and [4] for a more general version. We will use an intermediate version here, following [14], §7, which highlights the role of the special Lagrangian fibration. See also [17], §1. We review this point of view here. This will serve as motivation for Question 1.2 below, which will be addressed in the remainder of the paper. However, the setup of mirror symmetry will not be used again in this paper.Let L be the K3 lattice, L = H 2 (X, Z) for X a K3 surface. Fix a sublattice of L isomorphic to the hyperbolic plane H generated by E and σ 0 , with E 2 = 0, σ 2 0 = −2, and E.σ 0 = 1. We will view mirror symmetry as an involution acting on the moduli space of triples (X, B + iω, Ω) where X is a marked K3 surface, Ω is the class of a...

No abstract

The authors of [16] have proposed a conjectural construction of mirror symmetry for Calabi-Yau threefolds. They argue from the physics that in a neighbourhood of the large complex structure limit (see [11] for the definition of large complex structure limits), any Calabi-Yau threefold X with a mirror Y should admit a family of supersymmetric toroidal 3-cycles. In mathematical terminology, this says that there should be a fibration on X whose general fibre is a special Lagrangian 3-torus T 3 .We recall from [8] the notion of a special Lagrangian submanifold.Definition. Let X be a Kähler manifold of dimension n, with complex structure I and Kähler metric g, whose holonomy is contained in SU (n). Recall that this latter condition is equivalent to the existence of a covariant constant holomorphic n-form Ω on X. We say that Ω is normalized if (−1) n(n−1)/2 (i/2) n Ω ∧Ω = ω n /n! the volume form on X, where ω denotes the Kähler form -thus the normalized holomorphic n-form on X is unique up to a phase factor e iθ . We say that a submanifold M of X is special Lagrangian (we shall take M to be embedded, although more generally one only needs it to be immersed) if a normalized holomorphic n-form Ω can be chosen on X whose real part Re Ω restricts to the volume form on M -the normalization has been chosen here so that Re Ω is a calibration in the sense of [8]. This definition is equivalent to M being an oriented submanifold of real dimension n such that the restriction of ω to M is zero, i.e. M is Lagrangian, together with the existence of a holomorphic n-form Ω whose imaginary part Im Ω also restricts to zero on M [8]. Given a special Lagrangian submanifold M of X, it is shown in [10] that the deformations of M as a special Lagrangian submanifold are unobstructed and that the tangent * Supported in part by NSF grant DMS-9400873 and Trinity College, Cambridge space to the local deformation space may be identified with the harmonic 1-forms on M . The dimension of the local deformation space is therefore dim R H 1 (M, R). For X aCalabi-Yau n-fold and T a special Lagrangian n-torus submanifold, T therefore moves in an n-dimensional family.Definition. A Calabi-Yau n-fold X is said to have a special Lagrangian n-torus fibration if there exists a map of topological manifolds f : X → B whose general fibre is a special Lagrangian n-torus.We have taken B as a topological manifold, but from the results of [10], it will be locally a differentiable manifold with a natural Riemannian metric [10], (3.10), except perhaps at points corresponding to singular fibres. In §3, we shall generalise the notion of special Lagrangian n-torus fibration to the case when the metric on X is allowed to degenerate in a suitably nice way.Let us now take X to be a Calabi-Yau threefold. If X contains a special Lagrangian T 3 , then it moves locally in a three dimensional family. There are hard questions which need to be addressed concerning whether the family obtained foliates the manifold and whether it can be suitably compactified so as to obtain a special...

No abstract

No abstract