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Localizing a Dirac operator via J-holomorphic curves
Abstract
Let (M,J) be a compact almost hermitian $4$-manifold with a smooth embedded J-holomorphic curve C representing the canonical class. Motivated by the symplectic Bogomolov--Miyaoka--Yau conjecture, we choose a twisted spinc Dirac operator 𝒟 on M satisfying ind𝒟=3c2(M)−c12(M). We then use the curve C to construct a complex-linear perturbation 𝒜 of 𝒟 whose singular set is Zαsqcup C, where Zα=α−1(0) for a transverse section α of Λ1,0M nonvanishing on C. Applying Maridakis' index localization theorem, we express ind𝒟 as the sum of localized contributions: 3c2(M) from Zα and −c12(M) from C.
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BibTeX
@misc{lee2026localizing,
title = {{Localizing a Dirac operator via $J$-holomorphic curves}},
author = {Junho Lee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.07177},
eprint = {2609.07177},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.07177},
}
RIS
TY - UNPB
TI - Localizing a Dirac operator via J-holomorphic curves
AU - Junho Lee
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.07177
UR - https://arxiv.org/abs/2609.07177
AB - Let (M,J) be a compact almost hermitian $4$-manifold with a smooth embedded J-holomorphic curve C representing the canonical class. Motivated by the symplectic Bogomolov--Miyaoka--Yau conjecture, we choose a twisted spin^c Dirac operator 𝒟 on M satisfying ind𝒟=3c_2(M)−c_1^2(M). We then use the curve C to construct a complex-linear perturbation 𝒜 of 𝒟 whose singular set is Z_αsqcup C, where Z_α=α^−1(0) for a transverse section α of Λ^1,0M nonvanishing on C. Applying Maridakis' index localization theorem, we express ind𝒟 as the sum of localized contributions: 3c_2(M) from Z_α and −c_1^2(M) from C.
ER -