K-theoretic virtual localization conjecture

Let XX be an equivariant virtually smooth scheme with fixed-point components XiX_i, inclusion ι:XiX\iota:\bigcup X_i\to X, virtual structure sheaves OXvir{\mathcal O}_X^{\mathop{\text{vir}}} and OXivir{\mathcal O}_{X_i}^{\mathop{\text{vir}}}, and virtual normal bundles NivirN_i^{\mathop{\text{vir}}}. In localized equivariant KK-theory, KK-theoretic virtual localization.

OXvir=i=1sι(OXivir/Λ1(Nivir)).{\mathcal O}_X^{\mathop{\text{vir}}}=\sum_{i=1}^s\iota_*\big({\mathcal O}_{X_i}^{\mathop{\text{vir}}}/\Lambda_{-1}(N_i^{\mathop{\text{vir}}})^\vee\big).

This is the KK-theoretic analogue of virtual localization. In the context of DGDG-schemes, the conjecture has already been proved, so its status is solved.

Sources & referencesView supporting material

Primary source

Barbara Fantechi and Lothar Göttsche, “Riemann-Roch theorems and elliptic genus for virtually smooth Schemes”, arXiv:0706.0988 (2007).

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