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Semipositivity and complete subvarieties of projectivised strata of holomorphic differentials
Abstract
We show that if the projectivised stratum ℙHg(μ) associated to a partition μ=(m1,…,mn) of 2g−2 contains a complete curve, then κμ:=∑i=1n(mi(mi+2))/(mi+1)>12. The weak inequality follows from semipositivity applied to a quotient of the Hodge bundle, while equality is excluded using the second Fujita decomposition and the geometry of fibred surfaces with maximal unitary rank. Consequently, no projectivised holomorphic stratum in genus g≤5 contains a positive-dimensional complete subvariety. In genus g=6 the same conclusion holds for (10), (9,1), (8,2), (8,1,1), (7,3), (6,4), (5,5).
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BibTeX
@misc{mullane2026semipositivity,
title = {{Semipositivity and complete subvarieties of projectivised strata of holomorphic differentials}},
author = {Scott Mullane},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41287},
eprint = {2610.41287},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41287},
}
RIS
TY - UNPB
TI - Semipositivity and complete subvarieties of projectivised strata of holomorphic differentials
AU - Scott Mullane
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41287
UR - https://arxiv.org/abs/2610.41287
AB - We show that if the projectivised stratum ℙH_g(μ) associated to a partition μ=(m_1,…,m_n) of 2g−2 contains a complete curve, then κ_μ:=∑_i=1^n(m_i(m_i+2))/(m_i+1)>12. The weak inequality follows from semipositivity applied to a quotient of the Hodge bundle, while equality is excluded using the second Fujita decomposition and the geometry of fibred surfaces with maximal unitary rank. Consequently, no projectivised holomorphic stratum in genus g≤5 contains a positive-dimensional complete subvariety. In genus g=6 the same conclusion holds for (10), (9,1), (8,2), (8,1,1), (7,3), (6,4), (5,5).
ER -