PreprintarXiv licencearXiv
Finiteness and rigidity of optimal destabilizing subvarieties for the J-equation
Abstract
We prove that when the J-equation does not admit a smooth solution, there is always only a finite number of obstructing optimally destabilizing curves and divisors on compact Kähler manifolds of any dimension. For threefolds, this establishes the unconditional finiteness of all optimally destabilizing subvarieties. We moreover characterize the union of these optimal subvarieties, proving that they are contained in the locus where a natural continuity path loses local smooth compactness. Finally, to address subvarieties that are neither curves nor divisors, we prove a rigidity criterion which prevents optimal cycles from moving in compact one-parameter families, with further consequences for their deformation and symmetry properties.
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BibTeX
@misc{sivaram2026finiteness,
title = {{Finiteness and rigidity of optimal destabilizing subvarieties for the J-equation}},
author = {P. Sivaram and Zakarias Sjöström Dyrefelt},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.31122},
eprint = {2608.31122},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.31122},
}
RIS
TY - UNPB
TI - Finiteness and rigidity of optimal destabilizing subvarieties for the J-equation
AU - P. Sivaram
AU - Zakarias Sjöström Dyrefelt
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.31122
UR - https://arxiv.org/abs/2608.31122
AB - We prove that when the J-equation does not admit a smooth solution, there is always only a finite number of obstructing optimally destabilizing curves and divisors on compact Kähler manifolds of any dimension. For threefolds, this establishes the unconditional finiteness of all optimally destabilizing subvarieties. We moreover characterize the union of these optimal subvarieties, proving that they are contained in the locus where a natural continuity path loses local smooth compactness. Finally, to address subvarieties that are neither curves nor divisors, we prove a rigidity criterion which prevents optimal cycles from moving in compact one-parameter families, with further consequences for their deformation and symmetry properties.
ER -