Abstract
We introduce a notion of Hochschild Lefschetz class for a good coherent \(\mathcal{D }\)-module on a compact complex manifold, and prove that this class is compatible with the direct image functor. We prove an orbifold Riemann–Roch formula for a \(\mathcal{D }\)-module on a compact complex orbifold.
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Notes
This is actually a slight abuse of terminology. In fact, \(\mathcal{H }\mathcal{H }(\widehat{\mathcal{E }}_X, \widehat{\mathcal{E }}_X)\) is an object in the derived category of sheaves of \(\mathbb{C }\)-vector spaces on \(X\).
See Eq. (1) for the definition of \(\mathcal{H }\mathcal{H }(\widehat{\mathcal{E }}_X, \widehat{\mathcal{E }}_X^\gamma )\).
This is a minor abuse of terminology.
\(X^\gamma \) is a disjoint union of embedded submanifolds possibly of different dimensions.
As did in [9].
By a \(\gamma \)-equivariant element in \(D_{\text{ coh}}^{b}(X)\), we mean an element \(\mathcal{K }\) in \(D_{\text{ coh}}^{b}(X)\) together with a morphism from \(\mathcal{K }\) to \(\gamma _{*}\mathcal{K }\), which is denoted by \(\hat{\gamma }\).
\(\gamma _{X_{1}}\) (resp., \(\gamma _{X_{2}}\)) denotes the holomorphic diffeomorphism \(\gamma \) acting on \(X\) (resp., \(X_2\)).
As in Sect. 2, we abuse terminology here: \(\mathcal{H }\mathcal{C }^-(\mathcal{A })\) and \(\mathcal{H }\mathcal{H }(\mathcal{A })\) are objects in the derived category of sheaves of \(\mathbb{C }\)-vector spaces on \(Q\). Also, when \(Q=X:=T^{*}M\) as in Sect. 2 and when \(\mathcal{A }=\widehat{\mathcal{E }}_X, \,\mathcal{H }\mathcal{H }(\mathcal{A })\) as defined in [1] is isomorphic to \(\mathcal{H }\mathcal{H }(\mathcal{A })\) as defined in Sect. 2.
References
Bressler, P., Nest, R., Tsygan, B.: Riemann–Roch theorems via deformation quantization, I, II. Adv. Math. 167(1), 1–25, 26–73 (2002)
Dolgushev, V., Etingof, P.: Hochschild cohomology of quantized symplectic orbifolds and the Chen–Ruan cohomology. Int. Math. Res. Notices 27, 1657–1688 (2005)
Engeli, M., Felder, G.: A Riemann–Roch–Hirzebruch formula for traces of differential operators. Ann. Sci. Éc. Norm. Supér. (4) 41(4), 621–653 (2008)
Fedosov, B.V.: On \(G\)-trace and \(G\)-index in deformation quantization. Lett. Math. Phys. 52, 29–49 (2000)
Feigin, B., Felder, G., Shoikhet, B.: Hochschild cohomology of the Weyl algebra and traces in deformation quantization. Duke Math. J. 127(3), 487–517 (2005)
Felder, G., Tang, X.: Equivariant Lefschetz number of differential operators. Math. Z. 266(2), 451–470 (2010)
Guillermou, S.: Lefschetz class of elliptic pairs. Duke Math. J. 85(2), 273–314 (1996)
Kashiwara, M.: D-modules and Microlocal Calculus. Translations of Mathematical Monographs, vol. 217. American Mathematical Society, New York (2003)
Kashiwara, M., Schapira, P.: Deformation quantization modules. Astésrique, arXiv:1003.3304. (to appear, 2012)
Neumaier, N., Pflaum, M., Posthuma, H., Tang, X.: Homology of formal deformations of proper étale groupoids. J. Reine Angew. Math. 593, 117–168 (2006)
Pflaum, M., Posthuma, H., Tang, X.: An algebraic index theorem for orbifolds. Adv. Math. 210(1), 83–121 (2007)
Pflaum, M., Posthuma, H., Tang, X.: Cyclic cocycles in deformation quantization and higher index theorems. Adv. Math. 223(6), 1958–2021 (2010)
Polesello, P., Schapira, P.: Stacks of quantization-deformation modules over complex symplectic manifolds. Int. Math. Res. Notices 49, 2637–2664 (2004)
Ramadoss, A.: Some notes on the Feigin–Losev–Shoikhet integral conjecture. J. Noncommut. Geom. 2, 405–448 (2008)
Schapira, P., Schneider, J.: Elliptic pairs. I. Relative finiteness and duality. Index theorem for elliptic pairs. Astérisque 224, 5–60 (1994)
Schapira, P., Schneiders, J.: Elliptic pairs. II. Euler class and relative index theorem. Index theorem for elliptic pairs. Astérisque 224, 61–98 (1994)
Acknowledgments
We would like to thank Pierre Schapira and Giovanni Felder for interesting discussions. A.R. is supported by the Swiss National Science Foundation for the project “Topological quantum mechanics and index theorems” (Ambizione Beitrag Nr. PZ00P2_127427/1). X.T. is partially supported by NSF Grant DMS-0900985.
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Ramadoss, A., Tang, X. & Tseng, HH. Hochschild Lefschetz class for \(\mathcal{D }\)-modules. Math. Z. 275, 367–388 (2013). https://doi.org/10.1007/s00209-012-1139-0
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DOI: https://doi.org/10.1007/s00209-012-1139-0