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

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

Autoren

  • Wolfgang Ebeling
  • Sabir M. Gusein-Zade

Organisationseinheiten

Externe Organisationen

  • Lomonosov Moscow State University
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Details

OriginalspracheEnglisch
Seiten (von - bis)2543-2556
Seitenumfang14
FachzeitschriftMathematische Nachrichten
Jahrgang291
Ausgabenummer17-18
Frühes Online-Datum24 Juli 2018
PublikationsstatusVeröffentlicht - Dez. 2018

Abstract

Let a finite abelian group G act (linearly) on the space ℝn and thus on its complexification ℂn. Let W be the real part of the quotient ℂn/G (in general W≠ℝn/G). We give an algebraic formula for the radial index of a 1-form 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 ΩGω of Ωω=Ωnℝn,0/ω∧Ωn−1ℝn,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 π−1(W) of W under the natural quotient map.

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An algebraic formula for the index of a 1-form on a real quotient singularity. / Ebeling, Wolfgang; Gusein-Zade, Sabir M.
in: Mathematische Nachrichten, Jahrgang 291, Nr. 17-18, 12.2018, S. 2543-2556.

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

Ebeling W, Gusein-Zade SM. An algebraic formula for the index of a 1-form on a real quotient singularity. Mathematische Nachrichten. 2018 Dez;291(17-18):2543-2556. Epub 2018 Jul 24. doi: 10.48550/arXiv.1708.09219, 10.1002/mana.201700453
Ebeling, Wolfgang ; Gusein-Zade, Sabir M. / An algebraic formula for the index of a 1-form on a real quotient singularity. in: Mathematische Nachrichten. 2018 ; Jahrgang 291, Nr. 17-18. S. 2543-2556.
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N2 - Let a finite abelian group G act (linearly) on the space ℝn and thus on its complexification ℂn. Let W be the real part of the quotient ℂn/G (in general W≠ℝn/G). We give an algebraic formula for the radial index of a 1-form 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 ΩGω of Ωω=Ωnℝn,0/ω∧Ωn−1ℝn,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 π−1(W) of W under the natural quotient map.

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