Special Lagrangian Submanifolds with Isolated Conical Singularities. III. Desingularization, The Unobstructed Case

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 CalabiYau 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 third in a series of five papers [10,11,12,13] 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 [13], which surveys the series, and applies the results to prove some conjectures.The first paper [10] laid the foundations for the series, and studied the regularity of SL m-folds with conical singularities near their singular points. The second paper [11] discussed the deformation theory of compact SL m-folds X with conical singularities in an almost Calabi-Yau m-fold M . This paper and the sequel [12] study desingularizations of compact SL mfolds X with conical singularities. That is, we construct a family of compact, nonsingular SL m-foldsÑ t in M for t ∈ (0, ǫ] such thatÑ t → X as t → 0, in the sense of currents.Having a good understanding of the singularities of special Lagrangian submanifolds will be essential in clarifying the Strominger-Yau-Zaslow conjecture on the Mirror Symmetry of Calabi-Yau 3-folds [20], and also in resolving conjectures made by the author [7] on defining new invariants of Calabi-Yau 3-folds by counting special Lagrangian homology 3-spheres with weights. The series aims to develop such an understanding for simple singularities of SL m-folds.Here is the basic idea of the paper. Let X be a compact SL m-fold with conical singularities x 1 , . . . , x n in an almost Calabi-Yau m-fold (M, J, ω, Ω). 1Choose an isomorphism υ i : C m → T xi M for i = 1, . . . , n. Then there is a unique SL cone C i in C m with X asymptotic to υ i (C i ) at x i . Let L i be an Asymptotically Conical SL m-fold (AC SL m-fold ) in C m , asymptotic to C i at infinity. As C i is a cone it is invariant under dilations, so tC i = C i for all t > 0. Thus tL i = {t x : x ∈ L i } is also an AC SL m-fold asymptotic to C i for t > 0. We explicitly construct a 1-parameter family of compact, nonsingular Lagrangian m-folds N t in (M, ω) for t ∈ (0, δ) by gluing tL i into X at x i , using a partition of unity.When t is small, N t is close to being special Lagrangian (its phase is nearly constant), but also close to being singular (it has large curvature and small injectivity radius). We prove using analysis that for small ǫ ∈ (0, δ) we can deform N t to a special Lagrangian m-foldÑ t in M for all t ∈ (0, ǫ], using a small Hamiltonian deformation. The proof involves a delicate balancin...

Desingularization of Coassociative 4-Folds with Conical Singularities

“…This result is analogous to [10,Proposition 5.8] and the proof is almost identical. Rather than repeating that technical proof with only cosmetic changes, we hope to convince the reader with a heuristic argument.…”

“…Remark The conditions b 1 (Σ i ) = 0 and λ i < −2 mean that this result is most closely analogous to [10,Theorem 7.10]; that is, the main SL desingularization result in the unobstructed case.…”

“…This theorem is an analogue of [10,Theorem 6.12] and the proof presented is essentially an adaptation of the proof given in the reference cited.…”

“…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.…”

Special Lagrangian Submanifolds with Isolated Conical Singularities. v. Survey and Applications

“…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 fifth in a series of five papers [15,16,17,18] 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 , .…”