Equivariant Brill-Noether theory for elliptic operators and superrigidity of \(J\)-holomorphic maps. (English) Zbl 07643021
Summary: The space of Fredholm operators of fixed index is stratified by submanifolds according to the dimension of the kernel. Geometric considerations often lead to questions about the intersections of concrete families of elliptic operators with these submanifolds: Are the intersections nonempty? Are they smooth? What are their codimensions? The purpose of this article is to develop tools to address these questions in equivariant situations. An important motivation for this work are transversality questions for multiple covers of \(J\)-holomorphic maps. As an application, we use our framework to give a concise exposition of Wendl’s proof of the superrigidity conjecture.
MSC:
| 47-XX | Operator theory |
| 32-XX | Several complex variables and analytic spaces |
| 55N34 | Elliptic cohomology |
| 58J70 | Invariance and symmetry properties for PDEs on manifolds |
| 58C99 | Calculus on manifolds; nonlinear operators |
| 57M60 | Group actions on manifolds and cell complexes in low dimensions |
| 53D45 | Gromov-Witten invariants, quantum cohomology, Frobenius manifolds |
| 53D35 | Global theory of symplectic and contact manifolds |
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