The Journal of Geometric Analysis

, Volume 29, Issue 1, pp 83–104 | Cite as

Virtual Residue and an Integral Formalism

  • Huai-Liang Chang
  • Mu-Lin Li


We generalize Grothendieck’s residues \(Res\frac{\psi }{s}\) to virtual cases, namely cases when the zero loci of the section s has dimension larger than the expected dimension (zero). We also provide an exponential-type integral formalism for the virtual residue, which can be viewed as an analogue of the Mathai–Quillen formalism for localized Euler classes.


Grothendieck residue Landau Ginzburg model Mirror symmetry 

Mathematics Subject Classification

32J27 32J81 



The authors thank Ugo Bruzzo, Jun Li, Eric Sharpe, Si Li, Qile Chen, Zheng Hua, Huijun Fan, Yongbin Ruan, Edward Witten for helpful discussions. Special thanks to Si Li for informing us the operators \(T_\rho ,R_\rho \) in section three. Finally, we would like to express our appreciation to the referee for pointing out how to improve the paper and providing many valuable suggestions in rewriting the manuscript to make the paper more readable. Partially supported by Hong Kong GRF Grant 16301515 and 16301717.


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Authors and Affiliations

  1. 1.Department of MathematicsHong Kong University of Science and TechnologyHong KongHong Kong
  2. 2.College of Mathematics and EconometricsHunan UniversityChangshaChina

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