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A variant of the Mukai pairing via deformation quantization. (English) Zbl 1262.19005

The present paper focuses on pairings on the Hochschild cohomology ring of smooth complex projective varieties. There are mainly two such pairings in the literature, namely:
– The Shklyarov pairing, introduced in the DG-framework in the preprint [D. Shklyarov, “Hirzebruch-Riemann-Roch theorems for DG-algebras”, arXiv:0710.1937].
– The Mukai pairing, defined by A. Căldăraru and S. Willerton in [New York J. Math. 16, 61–98 (2010; Zbl 1214.14013)] via Serre duality.
Results of N. Markarian [J. Lond. Math. Soc., II. Ser. 79, No. 1, 129–143 (2009; Zbl 1167.14005)] and the author [New York J. Math. 14, 643–717 (2008; Zbl 1158.19002)] imply that the Mukai pairing on a variety \(X\) is given (up to sign) via the Hochschild-Kostant-Rosenberg isomorphism by the formula \(<a, b>=\int_X a \wedge b \wedge \mathrm{Td} (X)\). Later on, it has been proved by the author [Mosc. Math. J. 10, No. 3, 629–645 (2010; Zbl 1208.14013)] that the two aforementioned pairings were the same up to signs.
The aim of the current paper is to obtain directly the expression of the Shklyarov pairing. The method relies on the theory of deformation quantization as developed in [M. Kashiwara and P. Schapira, Deformation quantization modules. Astérisque 345. Paris: Société Mathématique de France (2012; Zbl 1260.32001)] as well as on the index theorem of P. Bressler, R. Nest and B. Tsygan [Adv. Math. 167, No. 1, 1–25 (2002; Zbl 1021.53064); ibid. 167, No. 1, 26–73 (2002; Zbl 1021.53065)]. Using this, the proof reduces to prove that the Euler class of the structural sheaf \(\mathcal{O}_X\) is the Todd class of \(X\). This fact has been conjectured by Kashiwara in 1991 and proved by the reviewer in [J. Differ. Geom. 90, No. 2, 267–275 (2012; Zbl 1247.32013)]. A completely different proof is presented here.

MSC:

19L10 Riemann-Roch theorems, Chern characters
14C40 Riemann-Roch theorems
53D55 Deformation quantization, star products
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References:

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