BALANCING THE 2-SYSTOLES OF SOME KÄHLER MANIFOLDS WITH POSITIVE SCALAR CURVATURE
Abstract
Theorems by Bray-Brendle-Neves and Zhu about positive scalar curvature metrics on products of a 2-sphere and an n-torus suggests that positive scalar curvature suggests and appropriate topological assumptions should lead to the existence of a topologically non trivial 2-spheres of small area, which can be stated as upper bound on the 2-systole of such manifolds. Recent progress have been made in this direction by Sha and Tsiamis under the additional Kähler assumption while Checcini-Hirsh-Ziedler, Stryker and Tsiamis showed similar upper bounds on the stable 2-systole using index theoretic methods. We prove here similar inequalities for some Kähler manifold which control the relative sizes of the representatives of a well chosen set of homology classes. For instance on (ℂℙ1×ℂℙ1 , ω) with a positive scalar curvature Kähler metric we quantitavely show that largeness of one factor imposes smallness of the other one.
Related
- Fundamental groups of compact Kähler manifolds with semi-positive holomorphic sectional curvature
- Convergence of cohomogeneity-one Lagrangian mean curvature flow in positive Kähler-Einstein manifolds
- Complete Kähler Manifolds with pinched Ricci curvature
- An L1 bound for Ricci curvature on Kähler manifolds
- Constant kth-mixed curvature on locally conformal Kähler manifolds
BibTeX
@misc{auvray2026balancing,
title = {{BALANCING THE 2-SYSTOLES OF SOME KÄHLER MANIFOLDS WITH POSITIVE SCALAR CURVATURE}},
author = {Hugues Auvray and Thomas Richard},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.13055},
eprint = {2606.13055},
archivePrefix = {arXiv},
url = {https://u-pec.hal.science/hal-05652259v2},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - BALANCING THE 2-SYSTOLES OF SOME KÄHLER MANIFOLDS WITH POSITIVE SCALAR CURVATURE
AU - Hugues Auvray
AU - Thomas Richard
PY - 2026
DA - 2026/09/18
PB - HAL
DO - 10.48550/arxiv.2606.13055
UR - https://u-pec.hal.science/hal-05652259v2
AB - Theorems by Bray-Brendle-Neves and Zhu about positive scalar curvature metrics on products of a 2-sphere and an n-torus suggests that positive scalar curvature suggests and appropriate topological assumptions should lead to the existence of a topologically non trivial 2-spheres of small area, which can be stated as upper bound on the 2-systole of such manifolds. Recent progress have been made in this direction by Sha and Tsiamis under the additional Kähler assumption while Checcini-Hirsh-Ziedler, Stryker and Tsiamis showed similar upper bounds on the stable 2-systole using index theoretic methods. We prove here similar inequalities for some Kähler manifold which control the relative sizes of the representatives of a well chosen set of homology classes. For instance on (ℂℙ^1×ℂℙ^1 , ω) with a positive scalar curvature Kähler metric we quantitavely show that largeness of one factor imposes smallness of the other one.
ER -