Vector bundles induced from jet schemes
Author:
Bailin Song
Journal:
Trans. Amer. Math. Soc. 374 (2021), 2661-2685
MSC (2020):
Primary 17B69, 53C07, 32L10
DOI:
https://doi.org/10.1090/tran/8239
Published electronically:
February 2, 2021
MathSciNet review:
4223029
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: A family of holomorphic vector bundles is constructed on a complex manifold $X$. The spaces of the holomorphic sections of these bundles are calculated in certain cases. As an application, if $X$ is an $N$-dimensional compact Kähler manifold with holonomy group $SU(N)$, the space of holomorphic vector fields on its jet scheme $J_m(X)$ is calculated. We also prove that the space of the global sections of the chiral de Rham complex of a K3 surface is the simple $\mathcal N=4$ superconformal vertex algebra with central charge $6$.
- Joel Ekstrand, Reimundo Heluani, Johan Källén, and Maxim Zabzine, Chiral de Rham complex on Riemannian manifolds and special holonomy, Comm. Math. Phys. 318 (2013), no. 3, 575–613. MR 3027580, DOI https://doi.org/10.1007/s00220-013-1659-4
- Lawrence Ein and Mircea Mustaţă, Jet schemes and singularities, Algebraic geometry—Seattle 2005. Part 2, Proc. Sympos. Pure Math., vol. 80, Amer. Math. Soc., Providence, RI, 2009, pp. 505–546. MR 2483946, DOI https://doi.org/10.1090/pspum/080.2/2483946
- Dominic D. Joyce, Compact manifolds with special holonomy, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2000. MR 1787733
- Victor Kac, Vertex algebras for beginners, 2nd ed., University Lecture Series, vol. 10, American Mathematical Society, Providence, RI, 1998. MR 1651389
- M. Kapranov, Rozansky-Witten invariants via Atiyah classes, Compositio Math. 115 (1999), no. 1, 71–113. MR 1671737, DOI https://doi.org/10.1023/A%3A1000664527238
- A. Kapustin, Chiral de Rham complex and the half-twisted sigma-model. arXiv:hep-th/0504074.
- Andrew R. Linshaw, Gerald W. Schwarz, and Bailin Song, Jet schemes and invariant theory, Ann. Inst. Fourier (Grenoble) 65 (2015), no. 6, 2571–2599 (English, with English and French summaries). MR 3449590
- Andrew R. Linshaw, Gerald W. Schwarz, and Bailin Song, Arc spaces and the vertex algebra commutant problem, Adv. Math. 277 (2015), 338–364. MR 3336089, DOI https://doi.org/10.1016/j.aim.2015.03.007
- Fyodor Malikov, Vadim Schechtman, and Arkady Vaintrob, Chiral de Rham complex, Comm. Math. Phys. 204 (1999), no. 2, 439–473. MR 1704283, DOI https://doi.org/10.1007/s002200050653
- Fyodor Malikov and Vadim Schechtman, Chiral de Rham complex. II, Differential topology, infinite-dimensional Lie algebras, and applications, Amer. Math. Soc. Transl. Ser. 2, vol. 194, Amer. Math. Soc., Providence, RI, 1999, pp. 149–188. MR 1729362, DOI https://doi.org/10.1090/trans2/194/07
- Fyodor Malikov and Vadim Schechtman, Chiral Poincaré duality, Math. Res. Lett. 6 (1999), no. 5-6, 533–546. MR 1739212, DOI https://doi.org/10.4310/MRL.1999.v6.n5.a6
- Bailin Song, The global sections of the chiral de Rham complex on a Kummer surface, Int. Math. Res. Not. IMRN 14 (2016), 4271–4296. MR 3556419, DOI https://doi.org/10.1093/imrn/rnv274
- B. Song, Chiral Hodge cohomology and Mathieu moonshine, arXiv:1705.04060 [math.QA].
- Hermann Weyl, The Classical Groups. Their Invariants and Representations, Princeton University Press, Princeton, N.J., 1939. MR 0000255
- K. Yano and S. Bochner, Curvature and Betti numbers, Annals of Mathematics Studies, No. 32, Princeton University Press, Princeton, N. J., 1953. MR 0062505
Retrieve articles in Transactions of the American Mathematical Society with MSC (2020): 17B69, 53C07, 32L10
Retrieve articles in all journals with MSC (2020): 17B69, 53C07, 32L10
Additional Information
Bailin Song
Affiliation:
School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
Email:
bailinso@ustc.edu.cn
Received by editor(s):
May 3, 2018
Received by editor(s) in revised form:
July 1, 2020, and July 9, 2020
Published electronically:
February 2, 2021
Additional Notes:
The author was supported by National Natrual Science Foundation of China No.11771416
Article copyright:
© Copyright 2021
American Mathematical Society