## Căldăraru’s conjecture and Tsygan’s formality.(English)Zbl 1252.18035

Summary: We complete the proof of Căldăraru’s conjecture on the compatibility between the module structures on differential forms over poly-vector fields and on Hochschild homology over Hochschild cohomology. In fact we show that twisting with the square root of the Todd class gives an isomorphism of precalculi between these pairs of objects.
Our methods use formal geometry to globalize the local formality quasi-isomorphisms introduced by Kontsevich and Shoikhet. (The existence of the latter was conjectured by Tsygan.) We also rely on the fact – recently proved by the first two authors – that Shoikhet’s quasi-isomorphism is compatible with cap products after twisting with a Maurer-Cartan element.

### MSC:

 18G60 Other (co)homology theories (MSC2010) 18G55 Nonabelian homotopical algebra (MSC2010) 58H05 Pseudogroups and differentiable groupoids

Zbl 0962.18008
Full Text:

### References:

 [1] H. J. Baues, ”The double bar and cobar constructions,” Compositio Math., vol. 43, iss. 3, pp. 331-341, 1981. · Zbl 0478.57027 [2] D. Calaque, ”Formality for Lie algebroids,” Comm. Math. Phys., vol. 257, iss. 3, pp. 563-578, 2005. · Zbl 1079.53138 [3] D. Calaque, V. Dolgushev, and G. Halbout, ”Formality theorems for Hochschild chains in the Lie algebroid setting,” J. Reine Angew. Math., vol. 612, pp. 81-127, 2007. · Zbl 1141.53084 [4] D. Calaque and C. A. Rossi, Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry, European Mathematical Society (EMS), Zürich, 2011. · Zbl 1220.53006 [5] D. Calaque and C. A. Rossi, ”Shoikhet’s conjecture and Duflo isomorphism on (co)invariants,” SIGMA Symmetry Integrability Geom. Methods Appl., vol. 4, p. 060, 2008. · Zbl 1196.53044 [6] D. Calaque and C. A. Rossi, ”Compatibility with cap-products in Tsygan’s formality and homological Duflo isomorphism,” Lett. Math. Phys., vol. 95, iss. 2, pp. 135-209, 2011. · Zbl 1213.81142 [7] D. Calaque and M. Van den Bergh, ”Hochschild cohomology and Atiyah classes,” Adv. Math., vol. 224, iss. 5, pp. 1839-1889, 2010. · Zbl 1197.14017 [8] D. Calaque, C. A. Rossi, and M. van den Bergh, ”Hochschild (co)homology for Lie algebroids,” Int. Math. Res. Not., vol. 2010, iss. 21, pp. 4098-4136, 2010. · Zbl 1215.14006 [9] A. Cualduararu, ”The Mukai pairing. II. The Hochschild-Kostant-Rosenberg isomorphism,” Adv. Math., vol. 194, iss. 1, pp. 34-66, 2005. · Zbl 1098.14011 [10] V. Dolgushev, ”A formality theorem for Hochschild chains,” Adv. Math., vol. 200, iss. 1, pp. 51-101, 2006. · Zbl 1106.53054 [11] V. Dolgushev, D. Tamarkin, and B. Tsygan, ”The homotopy Gerstenhaber algebra of Hochschild cochains of a regular algebra is formal,” J. Noncommut. Geom., vol. 1, iss. 1, pp. 1-25, 2007. · Zbl 1144.18007 [12] V. Dolgushev, D. Tamarkin, and B. Tsygan, Formality of the homotopy calculus algebra of Hochschild (co)chains, 2008. · Zbl 1144.18007 [13] A. A. Voronov and M. Gerstenkhaber, ”Higher-order operations on the Hochschild complex,” Funktsional. Anal. i Prilozhen., vol. 29, iss. 1, pp. 1-6, 96, 1995. · Zbl 0849.16010 [14] E. Getlzer and J. Jones, Operads, homotopy algebra and iterated integrals for double loop spaces. [15] D. Huybrechts and M. Nieper-Wisskirchen, ”Remarks on derived equivalences of Ricci-flat manifolds,” Math. Z., vol. 267, iss. 3-4, pp. 939-963, 2011. · Zbl 1213.32012 [16] M. Kapranov, ”Rozansky-Witten invariants via Atiyah classes,” Compositio Math., vol. 115, iss. 1, pp. 71-113, 1999. · Zbl 0993.53026 [17] M. Kontsevich, ”Deformation quantization of Poisson manifolds,” Lett. Math. Phys., vol. 66, iss. 3, pp. 157-216, 2003. · Zbl 1058.53065 [18] D. Manchon and C. Torossian, ”Cohomologie tangente et cup-produit pour la quantification de Kontsevich,” Ann. Math. Blaise Pascal, vol. 10, iss. 1, pp. 75-106, 2003. · Zbl 1051.53072 [19] N. Markarian, ”The Atiyah class, Hochschild cohomology and the Riemann-Roch theorem,” J. Lond. Math. Soc., vol. 79, iss. 1, pp. 129-143, 2009. · Zbl 1167.14005 [20] A. C. Ramadoss, ”The Mukai pairing and integral transforms in Hochschild homology,” Mosc. Math. J., vol. 10, iss. 3, pp. 629-645, 2010. · Zbl 1208.14013 [21] B. Shoikhet, ”A proof of the Tsygan formality conjecture for chains,” Adv. Math., vol. 179, iss. 1, pp. 7-37, 2003. · Zbl 1163.53350 [22] B. Shoikhet, ”Tsygan formality and Duflo formula,” Math. Res. Lett., vol. 10, iss. 5-6, pp. 763-775, 2003. · Zbl 1042.17016 [23] R. G. Swan, ”Hochschild cohomology of quasiprojective schemes,” J. Pure Appl. Algebra, vol. 110, iss. 1, pp. 57-80, 1996. · Zbl 0865.18010 [24] D. Tamarkin, Another proof of M. Kontsevich formality theorem for $$\mathbbR^n$$, 1998. [25] D. Tamarkin and B. Tsygan, ”Cyclic formality and index theorems,” Lett. Math. Phys., vol. 56, iss. 2, pp. 85-97, 2001. · Zbl 1008.19002 [26] B. Tsygan, ”Formality conjectures for chains,” in Differential Topology, Infinite-Dimensional Lie Algebras, and Applications, Providence, RI: Amer. Math. Soc., 1999, vol. 194, pp. 261-274. · Zbl 0962.18008 [27] M. Van den Bergh, ”On global deformation quantization in the algebraic case,” J. Algebra, vol. 315, iss. 1, pp. 326-395, 2007. · Zbl 1133.14021 [28] P. Xu, ”Quantum groupoids,” Comm. Math. Phys., vol. 216, iss. 3, pp. 539-581, 2001. · Zbl 0986.17003 [29] A. Yekutieli, ”Deformation quantization in algebraic geometry,” Adv. Math., vol. 198, iss. 1, pp. 383-432, 2005. · Zbl 1085.53081 [30] A. Yekutieli, Private communication. [31] A. Yekutieli, ”Continuous and twisted $$L_\infty$$ morphisms,” J. Pure Appl. Algebra, vol. 207, iss. 3, pp. 575-606, 2006. · Zbl 1104.53085
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.