The Kuranishi space of joint deformations of Calabi--Yau manifolds and holomorphic vector bundles
Abstract
In this paper, we study joint deformations of pairs (X,E), where X is a compact complex manifold and E→ X is a holomorphic vector bundle. We prove that (X,E) has unobstructed deformations if X is a Fujiki manifold with torsion canonical bundle and H2(X,End0E)=0, where End0 E denotes the trace-free endomorphism bundle. When the canonical bundle is trivial, the Fujiki assumption can be replaced by three weak ∂∂̅-conditions. We construct examples satisfying these conditions whose Frölicher spectral sequences do not degenerate at E1. These results provide non-Kähler extensions of the theorems of Li--Pan and Iacono--Manetti. Without this vanishing assumption, we obtain both unobstructed and obstructed pairs. In Thomas's example, we prove unobstructedness of the pair, although the bundle has obstructed deformations with the manifold fixed. On the other hand, we prove that every strict projective Calabi--Yau manifold of dimension at least three admits a simple bundle with obstructed joint deformations. We also construct obstructed pairs with Hermitian flat bundles on complex tori. These results answer two questions raised by Felten.
Related
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BibTeX
@misc{zhang2026kuranishi,
title = {{The Kuranishi space of joint deformations of Calabi--Yau manifolds and holomorphic vector bundles}},
author = {Runze Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08776},
eprint = {2610.08776},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08776},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Kuranishi space of joint deformations of Calabi--Yau manifolds and holomorphic vector bundles
AU - Runze Zhang
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08776
UR - https://arxiv.org/abs/2610.08776
AB - In this paper, we study joint deformations of pairs (X,E), where X is a compact complex manifold and E→ X is a holomorphic vector bundle. We prove that (X,E) has unobstructed deformations if X is a Fujiki manifold with torsion canonical bundle and H^2(X,End^0E)=0, where End^0 E denotes the trace-free endomorphism bundle. When the canonical bundle is trivial, the Fujiki assumption can be replaced by three weak ∂∂̅-conditions. We construct examples satisfying these conditions whose Frölicher spectral sequences do not degenerate at E_1. These results provide non-Kähler extensions of the theorems of Li--Pan and Iacono--Manetti. Without this vanishing assumption, we obtain both unobstructed and obstructed pairs. In Thomas's example, we prove unobstructedness of the pair, although the bundle has obstructed deformations with the manifold fixed. On the other hand, we prove that every strict projective Calabi--Yau manifold of dimension at least three admits a simple bundle with obstructed joint deformations. We also construct obstructed pairs with Hermitian flat bundles on complex tori. These results answer two questions raised by Felten.
ER -