Surfaces of general type with extremal cotangent dimension
Abstract
We study the geography of surfaces of general type with extremal cotangent dimension, in other words surfaces which have either no global holomorphic symmetric differentials at all or the maximal asymptotic growth of their number, i.e. big cotangent bundle. We are mainly interested in surfaces with low slope K^2 /χ, which, as far as maximal cotangent dimension is concerned, were out of reach of previous methods. We prove vanishing theorems for symmetric logarithmic differentials on minimal rational surfaces and for differentials on their double covers and extend a bigness criterion of Sakai to fibrations of general type in the sense of Campana. As a consequence, we prove, on the one hand that generic Horikawa surfaces have no nontrivial symmetric differentials and on the other hand that there exist Horikawa surfaces with big cotangent bundle.
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BibTeX
@misc{brotbek2026surfaces,
title = {{Surfaces of general type with extremal cotangent dimension}},
author = {Damian Brotbek and Bruno de Oliveira and Erwan Rousseau},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18798},
eprint = {2609.18798},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05750626v1},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - Surfaces of general type with extremal cotangent dimension
AU - Damian Brotbek
AU - Bruno de Oliveira
AU - Erwan Rousseau
PY - 2026
DA - 2026/09/15
PB - HAL
DO - 10.48550/arxiv.2609.18798
UR - https://hal.science/hal-05750626v1
AB - We study the geography of surfaces of general type with extremal cotangent dimension, in other words surfaces which have either no global holomorphic symmetric differentials at all or the maximal asymptotic growth of their number, i.e. big cotangent bundle. We are mainly interested in surfaces with low slope K^2 /χ, which, as far as maximal cotangent dimension is concerned, were out of reach of previous methods. We prove vanishing theorems for symmetric logarithmic differentials on minimal rational surfaces and for differentials on their double covers and extend a bigness criterion of Sakai to fibrations of general type in the sense of Campana. As a consequence, we prove, on the one hand that generic Horikawa surfaces have no nontrivial symmetric differentials and on the other hand that there exist Horikawa surfaces with big cotangent bundle.
ER -