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The space of Fredholm operators of fixed index is stratified by submanifoldsaccording to the dimension of the kernel. Geometric considerations often leadto questions about the intersections of concrete families of elliptic operatorswith these submanifolds: are the intersections non-empty? are they smooth? whatare their codimensions? The purpose of this article is to develop tools toaddress these questions in equivariant situations. An important motivation forthis work are transversality questions for multiple covers of $J$-holomorphicmaps. As an application, we use our framework to give a concise exposition ofWendl's proof of the super-rigidity conjecture.

Equivariant Brill-Noether theory for elliptic operators and super-rigidity of $J$-holomorphic maps