Leafrians, Frobenius splittings, and moduli spaces over Poisson surfaces
Abstract
Let X be a smooth projective Poisson surface over an algebraically closed field of characteristic p>2, and let s∈ H0(X,ωX−1) be nonzero. We construct an anticanonical section associated to s, called the leafrian, on moduli spaces M of stable objects on X, including the moduli of stable sheaves, of principal G-bundles, and of instantons. The leafrian splits M if and only if sp−1 Frobenius splits the base. It is invariant under the Hamiltonian flows of Bottacin's Poisson structure on M, so every compatibly split subvariety of M is a Poisson subvariety. Furthermore, the leafrian extends to an anticanonical section on the Gieseker compactification M̅, and splits M̅ under the same criterion. As a consequence, for X a rational surface, a ruled surface over an elliptic curve, a K3 surface, or an abelian surface, and ρ>0, the Gieseker compactification M̅ satisfies Kodaira-type cohomology vanishing, including singular M̅, and the same holds in characteristic zero.
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BibTeX
@misc{peng2026leafrians,
title = {{Leafrians, Frobenius splittings, and moduli spaces over Poisson surfaces}},
author = {Jun Peng},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31619},
eprint = {2610.31619},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31619},
}
RIS
TY - UNPB
TI - Leafrians, Frobenius splittings, and moduli spaces over Poisson surfaces
AU - Jun Peng
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.31619
UR - https://arxiv.org/abs/2610.31619
AB - Let X be a smooth projective Poisson surface over an algebraically closed field of characteristic p>2, and let s∈ H^0(X,ω_X^−1) be nonzero. We construct an anticanonical section associated to s, called the leafrian, on moduli spaces M of stable objects on X, including the moduli of stable sheaves, of principal G-bundles, and of instantons. The leafrian splits M if and only if s^p−1 Frobenius splits the base. It is invariant under the Hamiltonian flows of Bottacin's Poisson structure on M, so every compatibly split subvariety of M is a Poisson subvariety. Furthermore, the leafrian extends to an anticanonical section on the Gieseker compactification M̅, and splits M̅ under the same criterion. As a consequence, for X a rational surface, a ruled surface over an elliptic curve, a K3 surface, or an abelian surface, and ρ>0, the Gieseker compactification M̅ satisfies Kodaira-type cohomology vanishing, including singular M̅, and the same holds in characteristic zero.
ER -