Special Lagrangian Submanifolds with Isolated Conical Singularities. IV. Desingularization, Obstructions and Families

Abstract

Special Lagrangian m-folds (SL m-folds) are a distinguished class of real mdimensional minimal submanifolds which may be defined in C m , or in Calabi-Yau m-folds, or more generally in almost Calabi-Yau m-folds (compact Kähler m-folds with trivial canonical bundle). We write an almost Calabi-Yau m-fold as M or (M, J, ω, Ω), where the manifold M has complex structure J, Kähler form ω and holomorphic volume form Ω. This is the fourth in a series of five papers [7,8,9,10] studying SL m-folds with isolated conical singularities. That is, we consider an SL m-fold X in an almost Calabi-Yau m-fold M for m > 2 with singularities at x 1 , . . . , x n in M , such that for some special Lagrangian cones C i in T xi M ∼ = C m with C i \ {0} nonsingular, X approaches C i near x i in an asymptotic C 1 sense. Readers are advised to begin with the final paper [10], which surveys the series, and applies the results to prove some conjectures.The first paper [7] laid the foundations for the series, and studied the regularity of SL m-folds with conical singularities near their singular points. The second paper [8] discussed the deformation theory of compact SL m-folds X with conical singularities in an almost Calabi-Yau m-fold M .The third paper [9] and this one study desingularizations of compact SL m-folds X with conical singularities. That is, we construct a family of compact, nonsingular SL m-foldsÑ t in M for t ∈ (0, ǫ] withÑ t → X as t → 0, in the sense of currents. In [9] we did this for simple situations, working in a single almost Calabi-Yau m-fold (M, J, ω, Ω), and making topological assumptions to avoid problems with obstructions to the existence ofÑ t . This paper extends the results of [9] to more complicated situations, in which there are topological obstructions to the existence of desingularizations N t of X, and to desingularizations in families of almost Calabi-Yau m-foldsHaving a good understanding of the singularities of special Lagrangian submanifolds will be essential in clarifying the Strominger-Yau-Zaslow conjecture 2 Special Lagrangian geometryWe introduce special Lagrangian submanifolds (SL m-folds) in two different geometric contexts. First, in §2.1, we define SL m-folds in C m . Then §2.2 discusses SL m-folds in almost Calabi-Yau m-folds, compact Kähler manifolds equipped with a holomorphic volume form, which generalize Calabi-Yau manifolds. Some references for this section are Harvey and Lawson [3] and the author [6]. Special Lagrangian submanifolds in C mWe begin by defining calibrations and calibrated submanifolds, following [3].Definition 2.1 Let (M, g) be a Riemannian manifold. An oriented tangent k-plane V on M is a vector subspace V of some tangent space T x M to M with dim V = k, equipped with an orientation. If V is an oriented tangent k-plane on M then g| V is a Euclidean metric on V , so combining g| V with the orientation on V gives a natural volume form vol V on V , which is a k-form on V . Now let ϕ be a closed k-form on M . We say that ϕ is a calibration on M if for every oriented k-plane V ...

Desingularization of Coassociative 4-Folds with Conical Singularities

“…The primary source of inspiration for the ideas in this paper, other than the author's earlier research, is the work of Joyce [8], [9], [10], [11] and [12] on special Lagrangian submanifolds with conical singularities, in particular [10] on desingularization. There are a number of analogies between Joyce's work and the material in this article: in particular, SL and coassociative submanifolds are both defined by the vanishing of a closed differential form and nearby deformations of these submanifolds can be identified with graphs of small forms.…”

Singularities of Special Lagrangian Submanifolds

“…This equation arises by requiring the projection of an error term to the eigenspaces of ∆ t with small eigenvalues to be zero. In [21,Th. 6.13] we remove assumption (b), extending Theorem 7.1 to the case λ i 0, and allowing Y (L i ) = 0.…”

“…However, all the main results of §2.4, §5 and §7 have extensions to families, which can be found in [18,19,20,21,22]. The discussion of index of singularities in §8.1, and its applications in §8.3 and §8.4, would also be improved by extending it to families.…”

“…We begin in §2 with a brief introduction to special Lagrangian geometry and (almost) Calabi-Yau m-folds. Sections 3-7 survey the author's series of papers [18,19,20,21,22] on SL m-folds with isolated conical singularities, a large class of singularities which are simple enough to study in detail. The last and longest section, §8, suggests directions for future research and gives some open problems.…”

“…The bulk of the paper summarizes the author's work [18,19,20,21,22] on SL m-folds X with isolated conical singularities. That is, near each singular point x, X is modelled on an SL cone C in C m with isolated singularity at 0.…”