Special Lagrangian Submanifolds with Isolated Conical Singularities. IV. Desingularization, Obstructions and Families
Special Lagrangian Submanifolds with Isolated Conical Singularities. IV. Desingularization, Obstructions and Families
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具有孤立圆锥奇点的特殊拉格朗日子流形。
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发表时间：
2003
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通讯作者：
D. joyce
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作者：
D. joyce
这是第四次在一系列的五个文件研究特殊的拉格朗日子流形（SLV的m-倍）X在（几乎）Calabi-Yau m-foldsM与奇点x1，...，xn局部建模于特殊拉格朗日锥C1，.，Cn在孤立奇点为0的情况下的解。建议读者开始与纸V.纸III和这一个构造desingularizations的X，实现X作为一个limitofa家庭的紧凑，非奇异SL m-倍Ct在M小t > 0。假设L1，...，Ln是λ m中的渐近锥SL m-折叠，其中Li在无穷远处渐近于锥Ci.我们将Li收缩一个小的t > 0，并在xi处将Li粘到X中，其中i= 1，.，n得到了一个单参数的紧致非奇异Lagrange m-折叠族Nt，当t> 0时，我们用分析的方法证明了当t足够小时，我们可以通过一个小的Hamilton变形将Nt变形为一个紧致非奇异的特殊Lagrange m-折叠族Ct.这Ct光滑依赖于t，并作为t→ 0它收敛到奇异SL m-倍X，在这个意义上的currents.Paper III研究了简单的情况下，其中拓扑条件X和李，我们避免障碍的存在Ct。本文考虑了当这些障碍是非平凡的时的更复杂的情形，以及在几乎Calabi-Yaum-folds M（s∈F）族中而不是在单个几乎Calabi-Yaum-folds M中的去奇异化。
This is the fourth in a series of five papers studying special Lagrangian submanifolds(SLV m-folds) X in (almost) Calabi–Yau m-foldsM with singularities x1,..., xnlocally modelled on special Lagrangian conesC1,..., Cn in ℂm with isolated singularities at 0. Readers are advised to begin with Paper V.Paper III and this one construct desingularizations of X, realizing X as a limitof a family of compact, nonsingular SL m-folds Ct in M for small t > 0. Suppose L1,..., Ln are Asymptotically Conical SL m-folds in ℂm, withLi asymptotic to the cone Ciat infinity. We shrink Li by a small t > 0, and gluetLi into X at xi for i= 1,..., n to get a 1-parameter family of compact, nonsingularLagrangianm-folds Nt for small t> 0.Then we show using analysis that when t is sufficiently small we can deform Nt toa compact, nonsingular special Lagrangianm-fold Ct, via a small Hamiltonian deformation. This Ct depends smoothly on t, and as t→ 0 it converges to the singular SL m-fold X, in the sense of currents.Paper III studied simpler cases, where by topological conditions on X and Li we avoid obstructions to the existence of Ct. This paper considers more complex cases when theseobstructions are nontrivial, and also desingularization in families of almost Calabi–Yaum-folds Ms for s∈F, rather than in a single almost Calabi–Yau m-fold M.