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- W4387355864 abstract "The proposals of Joyce [Joy18], and Doan and Walpuski [DW19] on counting closed associative submanifolds depend on various conjectural transitions. This article makes contributions to some of the transitions which arise from the degenerations of associative submanifolds into conically singular (CS) associative submanifolds. First, we study the moduli space of CS associative submanifolds with singularities at a finite number of points locally modeled on associative cones in $mathbf R^7$ and their transversality in a co-closed $G_2$-manifold as well as in a one parameter family of co-closed $G_2$-structures. We prove that the index of the operator controlling the deformation theory depends on the stability-index of the associative cones. As a result, we obtain that in a generic path of co-closed $G_2$-structures there are no CS associative submanifolds having singularities modeled on cones with stability-index greater than one. In particular, this result applies to associative cones whose links are null-torsion holomorphic curves in $S^6$, and to all special Lagrangian cones in $mathbf C^3$ except the Harvey-Lawson $T^2$-cone and a union of two special Lagrangian planes with transverse intersection. Second, we study the desingularizations of CS associative submanifolds in a one parameter family of co-closed $G_2$-structures. As a result, we obtain desingularization results related to the above transitions conjectured by Joyce [Joy18] for CS associative submanifolds with Harvey-Lawson $T^2$-cone singularity and for associative submanifolds with transverse intersection." @default.
- W4387355864 created "2023-10-05" @default.
- W4387355864 creator A5049969908 @default.
- W4387355864 date "2023-10-02" @default.
- W4387355864 modified "2023-10-06" @default.
- W4387355864 title "Deformations and desingularizations of conically singular associative submanifolds" @default.
- W4387355864 doi "https://doi.org/10.48550/arxiv.2310.01741" @default.
- W4387355864 hasPublicationYear "2023" @default.
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