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Quotient spaces and critical points of invariant functions for \(C^ \ast\) actions. (English) Zbl 0767.32008
Consider a linear actions of \(\mathbb{C}^*\) on \(\mathbb{C}^{n+1}\) whose weights are not all of the same sign. We study the fundamental algebraic properties of the sheaves of invariant and basic differential forms for such an action, and use these to define an algebraic multiplicity for critical points of functions invariant under \(\mathbb{C}^*\) actions. We also relate this multiplicity to the topology of the Milnor fibre of the function on the quotient space.
32C38 Sheaves of differential operators and their modules, \(D\)-modules
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
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