Superconnections, descent, and monodromy on transversely holomorphic foliations
Abstract
Let X carry a transversely holomorphic foliation, equivalently an elliptic involutive structure V⊂ TℂX, and let 𝒪V be its sheaf of leafwise-constant, transversely holomorphic functions. We construct a finite superconnection model for the derived category of coherent 𝒪V-modules. The key input is a mixed local reduction for finite Maurer--Cartan objects over the mixed de Rham--Dolbeault dga (∧• V∨,dV): a multiplicative homotopy contracts the real directions, after which Block's Dolbeault gauge theorem removes the positive transverse form degrees. For compact X, this gives an exact equivalence between the homotopy category of bounded finite-rank flat V-superconnections and Dbcoh(X,𝒪V), interpolating between the de Rham and Dolbeault realizations. We prove locally finite Čech descent under necessary uniform amplitude and rank bounds, identify the coherent heart with equivariant coherent analytic sheaves on a transverse monodromy groupoid, and characterize descent to ordinary holonomy. For a holomorphic suspension, we identify the full superconnection category, up to Morita equivalence, with the homotopy fixed points of the Dolbeault category of the transversal and derive an equivariant Ext spectral sequence. Examples on S1 and S2 delimit when ordinary monodromy $1$-groupoids can recover the derived category.
Related
- Opers on transversely holomorphic foliations
- Transversely projectively flat foliations
- Stability of closed leaves holomorphic foliations via torsion behavior of groups of germs
- Bounded Leaf Volumes and Analytic Quotients of Singular Holomorphic Foliations
- Holomorphic realizations of pairs of foliations on Riemann surfaces
BibTeX
@misc{zeng2026superconnections,
title = {{Superconnections, descent, and monodromy on transversely holomorphic foliations}},
author = {Qingyun Zeng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04796},
eprint = {2609.04796},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04796},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Superconnections, descent, and monodromy on transversely holomorphic foliations
AU - Qingyun Zeng
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.04796
UR - https://arxiv.org/abs/2609.04796
AB - Let X carry a transversely holomorphic foliation, equivalently an elliptic involutive structure V⊂ T_ℂX, and let 𝒪_V be its sheaf of leafwise-constant, transversely holomorphic functions. We construct a finite superconnection model for the derived category of coherent 𝒪_V-modules. The key input is a mixed local reduction for finite Maurer--Cartan objects over the mixed de Rham--Dolbeault dga (∧^• V^∨,d_V): a multiplicative homotopy contracts the real directions, after which Block's Dolbeault gauge theorem removes the positive transverse form degrees. For compact X, this gives an exact equivalence between the homotopy category of bounded finite-rank flat V-superconnections and D^b_coh(X,𝒪_V), interpolating between the de Rham and Dolbeault realizations. We prove locally finite Čech descent under necessary uniform amplitude and rank bounds, identify the coherent heart with equivariant coherent analytic sheaves on a transverse monodromy groupoid, and characterize descent to ordinary holonomy. For a holomorphic suspension, we identify the full superconnection category, up to Morita equivalence, with the homotopy fixed points of the Dolbeault category of the transversal and derive an equivariant Ext spectral sequence. Examples on S^1 and S^2 delimit when ordinary monodromy $1$-groupoids can recover the derived category.
ER -