Spectrum of equivariant cohomology as a fixed point scheme

Hausel, Tamás; Rychlewicz, Kamil

doi:10.48550/arxiv.2212.11836

Cited by 1 publication

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“…We recall and generalise this theorem in the next section. The generalisation is the main result of [66], a joint work of the author and Tamás Hausel.…”

Section: Singular Varietiesmentioning

confidence: 86%

“…In turn, it was noticed that this fixed point scheme is isomorphic to the spectrum of equivariant cohomology of Gr(k, n), and thus the Hitchin system on these minuscule upward flows can be modelled as the spectrum of equivariant cohomology of Grassmannians. Motivated by that, we have shown in [66] that the appearance of the spectrum of equivariant cohomology as a fixed point scheme is not a coincidence, and holds in more general situations. This provides a generalisation of the result of Brion and Carrell.…”

Section: Introductionmentioning

confidence: 99%

“…Chapter 3 is based on Section 2 of [66]. We introduce the notion of a principally paired group and develop the theory of Kostant sections in this generality.…”

Section: Introductionmentioning

confidence: 99%

“…In Chapter 6, we then describe additional results of the author that extend the results of [66]. First, we tackle the case of wilder singularities than the results of [66] allow. Second, we relax the assumptions on regularity of the action, as in Theorem 1.…”

Section: Introductionmentioning

confidence: 99%

“…a parabolic subgroup of a reductive group. The fundamental theorem 5.2.11 from co-authored article [66] says that if the principal nilpotent has a unique zero, then the zero scheme over the Kostant section is isomorphic to the spectrum of the equivariant cohomology ring, remembering the grading in terms of a C × action. A similar statement is proved also for the G-invariant functions on the total zero scheme over the whole Lie algebra.…”

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confidence: 99%

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The positivity of local equivariant Hirzebruch class for toric varieties

Rychlewicz

2020

Bull. London Math. Soc.

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The central object of investigation of this paper is the Hirzebruch class, a deformation of the Todd class, given by Hirzebruch (for smooth varieties). The generalization for singular varieties is due to Brasselet-Schürmann-Yokura. Following the work of Weber, we investigate its equivariant version for (possibly singular) toric varieties. The local decomposition of the Hirzebruch class to the fixed points of the torus action and a formula for the local class in terms of the defining fan are recalled. After this review part, we prove the positivity of local Hirzebruch classes for all toric varieties, thus proving false the alleged counterexample given by Weber.

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“…We recall and generalise this theorem in the next section. The generalisation is the main result of [66], a joint work of the author and Tamás Hausel.…”

Section: Singular Varietiesmentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

“…Chapter 3 is based on Section 2 of [66]. We introduce the notion of a principally paired group and develop the theory of Kostant sections in this generality.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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