An algebraic formula for the index of a 1-form on a real quotient singularity

Mathematische Nachrichten. 2018. Vol. 291. No. 17-18. P. 2543–2556.

Gusein-Zade S.

Let a finite abelian group G act (linearly) on the space R^n and thus on its complexification C^n. Let W be the real part of the quotient C^n/G (in general W \neq R^n/G). We give an algebraic formula for the radial index of a 1-form \omega on the real quotient W. It is shown that this index is equal to the signature of the restriction of the residue pairing to the G-invariant part \Omega^G_\omega of \Omega_\omega=\Omega^n_{R^n,0}/\omega \wedge \Omega^{n-1}_{R^n,0}. For a G-invariant function f, one has the so-called quantum cohomology group defined in the quantum singularity theory (FJRW-theory). We show that, for a real function f, the signature of the residue pairing on the real part of the quantum cohomology group is equal to the orbifold index of the 1-form df on the preimage \pi^{-1}(W) of W under the natural quotient map.

Keywords: index, индекс, group actions, действия групп, 1-form, real quotient singularity, signature formula, 1-форма, вещественная фактор-особенность, сигнатурная формула