Equivariant Quotients of Derived Symplectic Spaces and Legendrian Intersection Theorem
Abstract
The classical transversality lemma of contact geometry constructs a contact structure on a hypersurface transverse to a Liouville vector field using point-set topology and local flows. This paper translates the classical transversality lemma into the context of derived algebraic geometry and provides the derived Legendrian intersection theorem, along with various applications to moduli theory. In brief, we first prove that taking the quotient of a derived symplectic space descends the symplectic data to a contact structure, avoiding the transverse hypersurface requirement, such that the fundamental vector field of a weight 1 𝔾m-action in the derived setting replaces the classical Liouville vector field. Second, we establish the derived Legendrian intersection theorem using base change, an ∞-categorical descent cube, and 𝔾m-equivariant lifts along the symplectification projection. As applications of the main results, we first examine the derived geometry of the discriminant loci of 1-jet bundles and show that these loci carry a (−1)-shifted contact structure. In addition, we show that our results apply to certain moduli problems, including projective Higgs bundles, ℓ-adic local systems, and Lie 2-groups, and produce further examples of contact derived moduli stacks.
Related
- Equivariant Poisson structures on moment-angle manifolds and their quotients
- Quantitative symplectic topology of Katok's examples and equivariant symplectic embeddings
- Equivariant Contact Darboux Quotients
- Higher Braided Mixed Sums, Intersection Lattices, and Symplectic Four-Manifolds
- A Derived Legendrian Category for Shifted Contact Stacks
BibTeX
@misc{izbudak2026equivariant,
title = {{Equivariant Quotients of Derived Symplectic Spaces and Legendrian Intersection Theorem}},
author = {Efe İzbudak and Kadri İlker Berktav},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.08394},
eprint = {2605.08394},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.08394},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Equivariant Quotients of Derived Symplectic Spaces and Legendrian Intersection Theorem
AU - Efe İzbudak
AU - Kadri İlker Berktav
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2605.08394
UR - https://arxiv.org/abs/2605.08394
AB - The classical transversality lemma of contact geometry constructs a contact structure on a hypersurface transverse to a Liouville vector field using point-set topology and local flows. This paper translates the classical transversality lemma into the context of derived algebraic geometry and provides the derived Legendrian intersection theorem, along with various applications to moduli theory. In brief, we first prove that taking the quotient of a derived symplectic space descends the symplectic data to a contact structure, avoiding the transverse hypersurface requirement, such that the fundamental vector field of a weight 1 𝔾_m-action in the derived setting replaces the classical Liouville vector field. Second, we establish the derived Legendrian intersection theorem using base change, an ∞-categorical descent cube, and 𝔾_m-equivariant lifts along the symplectification projection. As applications of the main results, we first examine the derived geometry of the discriminant loci of 1-jet bundles and show that these loci carry a (−1)-shifted contact structure. In addition, we show that our results apply to certain moduli problems, including projective Higgs bundles, ℓ-adic local systems, and Lie 2-groups, and produce further examples of contact derived moduli stacks.
ER -