Geometric index theorems for holomorphic foliations and curves
Abstract
Let M be a complex surface endowed with a holomorphic projective connection and let C β M be a compact smooth holomorphic curve. Given two singular holomorphic foliations β± and π’ near C, with C not π’-invariant, we attach to every point p β C an index Ind(β±,π’,C,p) β β of β± relative to the reference π’, and we prove that these indices add up to Tπ’ Β· C. When C is β±-invariant the index is the Camacho--Sad index corrected by the order of tangency of π’ with C, and the index formula reduces to the Camacho--Sad formula. The main applications are two contact formulas. At a tangency point p of π’ with C, let k(π’,C,p) be the ratio of the curvatures at p of the leaf of π’ and of the curve; it is a projective invariant, although each curvature separately depends on a choice of metric. If π’ has no singular points on C and only simple tangencies with C, then βp 1/(1 β k(π’,C,p)) = 2/3( C Β· C + g β 1 ), where g is the genus of C: the number of tangencies depends on π’, but this weighted count does not. For a pencil of lines in the projective plane it is the PlΓΌcker formula for the class of a plane curve. The second formula asserts that, for a curve in general position with respect to β± and π’ which is not a geodesic, the total index of the tangencies of β± with π’ along C equals the number of tangencies of π’ with C minus one third of the number of inflection points of C; in particular, it does not depend on β±.
Related
- Opers on transversely holomorphic 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
- Singular Levi-flats and holomorphic foliations on projective spaces
BibTeX
@misc{camacho2026geometric,
title = {{Geometric index theorems for holomorphic foliations and curves}},
author = {CΓ©sar Camacho and Rudy Rosas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22279},
eprint = {2609.22279},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22279},
}
RIS
TY - UNPB
TI - Geometric index theorems for holomorphic foliations and curves
AU - CΓ©sar Camacho
AU - Rudy Rosas
PY - 2026
DA - 2026/09/12
PB - arXiv
DO - 10.48550/arxiv.2609.22279
UR - https://arxiv.org/abs/2609.22279
AB - Let M be a complex surface endowed with a holomorphic projective connection and let C β M be a compact smooth holomorphic curve. Given two singular holomorphic foliations β± and π’ near C, with C not π’-invariant, we attach to every point p β C an index Ind(β±,π’,C,p) β β of β± relative to the reference π’, and we prove that these indices add up to Tπ’ Β· C. When C is β±-invariant the index is the Camacho--Sad index corrected by the order of tangency of π’ with C, and the index formula reduces to the Camacho--Sad formula. The main applications are two contact formulas. At a tangency point p of π’ with C, let k(π’,C,p) be the ratio of the curvatures at p of the leaf of π’ and of the curve; it is a projective invariant, although each curvature separately depends on a choice of metric. If π’ has no singular points on C and only simple tangencies with C, then β_p 1/(1 β k(π’,C,p)) = 2/3( C Β· C + g β 1 ), where g is the genus of C: the number of tangencies depends on π’, but this weighted count does not. For a pencil of lines in the projective plane it is the PlΓΌcker formula for the class of a plane curve. The second formula asserts that, for a curve in general position with respect to β± and π’ which is not a geodesic, the total index of the tangencies of β± with π’ along C equals the number of tangencies of π’ with C minus one third of the number of inflection points of C; in particular, it does not depend on β±.
ER -