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Mapping-space Gromov--Witten classes and the Fano variety of lines
Abstract
Let Y be a smooth cubic 3-fold. We show that S2 × Y admits an infinite sequence of cohomologous symplectic forms which are all deformation equivalent but pairwise non-isotopic; McDuff showed the corresponding result on (S2)2 × T2, but these appear to be the first simply-connected and monotone examples. The results are obtained from McDuff's bordism-flavoured Gromov--Witten invariant, re-interpreted cohomologically in terms of an evaluation map valued in the space of maps from the 2-sphere to a symplectic target. The application then relies on classical properties of the Fano space of lines on Y. We also discuss higher-dimensional generalisations.
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BibTeX
@misc{smith2026mappingspace,
title = {{Mapping-space Gromov--Witten classes and the Fano variety of lines}},
author = {Ivan Smith and Abigail Ward},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05324},
eprint = {2611.05324},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05324},
}
RIS
TY - UNPB
TI - Mapping-space Gromov--Witten classes and the Fano variety of lines
AU - Ivan Smith
AU - Abigail Ward
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05324
UR - https://arxiv.org/abs/2611.05324
AB - Let Y be a smooth cubic 3-fold. We show that S^2 × Y admits an infinite sequence of cohomologous symplectic forms which are all deformation equivalent but pairwise non-isotopic; McDuff showed the corresponding result on (S^2)^2 × T^2, but these appear to be the first simply-connected and monotone examples. The results are obtained from McDuff's bordism-flavoured Gromov--Witten invariant, re-interpreted cohomologically in terms of an evaluation map valued in the space of maps from the 2-sphere to a symplectic target. The application then relies on classical properties of the Fano space of lines on Y. We also discuss higher-dimensional generalisations.
ER -