Abstract
A twisted \(\bar{\partial }_f\)-Neumann problem associated to a singularity \((\mathscr {O}_n, f)\) is established. By relating it to the Koszul complex for Toeplitz n-tuples \((f_1,\ldots ,f_n)\), where \(f_i=\frac{\partial f}{\partial z_i}\), on Bergman space \(B^0(D)\), this \(\bar{\partial }_f\)-Neumann problem is solved. Moreover, the cohomology of the \(L^2\)-holomorphic Koszul complex \((B^*(D),{\partial }f\wedge )\) can be computed explicitly.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Taylor, J.L.: A joint spectrum for several commuting operators. J. Funct. Anal. 6, 172 (1970)
Eschmeier, J., Putinar, M.: Spectral Decompositions and Analytic Sheaves. The Clarendon Press Ox-ford University Press, Oxford Science Publications, New York (1996)
Putinar, M.: Private Communication (2015)
Cecotti, S.: \(N=2\) Landau–Ginzburg vs. Calabi–Yau models: non-perturbative aspects. Int. J. Mod. Phys. A 6, 1749 (1991)
Cecotti, S., Vafa, C.: Topological anti-topological fusion. Nucl. Phys. B 367, 359–461 (1991)
Chiodo, A., Ruan, Y.: Landau–Ginzburg/Calabi–Yau correspondence of quintic three-fold via symplectic transformations. Invent. Math. 182(1), 117–165 (2010)
Jarvis, T., Francis, A.: A Brief Survey of FJRW Theory. arXiv:1503.01223 [math.AG]
Fan, H., Jarvis, T., Ruan, Y.: The Witten equation, mirror symmetry and quantum singularity theory. Ann. Math. 108(3), 1–106 (2013)
Gaiotto, G., Moore, G., Witten, E.: Algebra of the infrared: string field theoretic structures in massive \(\mathbb{N}=(2,2)\) field theory in two dimension, in preparation
Kapranov, M., Kontsevich, M., Soibelman, Y.: Algebra of the infrared and secondary polytopes. Adv. Math. 300(4), 616–671 (2016)
Fan, H.: Schrödinger Equations, Deformation Theory and \(tt^*\)-Geometry. arXiv:1107.1290 [math-ph]
Saito, K., Takahashi, A.: From primitive forms to frobenius manifolds, preprint (2008)
Li, C.-C., Li, S., Saito, K.: Primitive Forms via Polyvector Fields. arXiv:1311.1659 [math.AG]
Folland, G.B., Kohn, J.J.: The Neumann Problem for the Cauchy–Riemann Complex. Princeton University Press and University of Tokyo Press, New Jersey (1972)
Chen, S.-C., Shaw, M.-C.: Partial Differential Equations in Several Complex Variables. In: Yau , S.-T(ed.) AMS/IP Studies in Advanced Mathematics. American Mathematical Society, Providence, RI; 19, International Press, Boston (2001)
Goldberg, S.: Unbounded Linear Operators: Theory and Applications. Dover, New York (1985)
Author information
Authors and Affiliations
Corresponding author
Additional information
Huijun Fan: Supported by NSFC (11271028), NSFC (11325101), and Doctoral Fund of Ministry of Education of China (20120001110060).
Rights and permissions
About this article
Cite this article
Wen, H., Fan, H. A twisted \({\overline{\partial }}_{f}\)-Neumann problem and Toeplitz n-tuples from singularity theory. manuscripta math. 156, 63–80 (2018). https://doi.org/10.1007/s00229-017-0953-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-017-0953-4