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 a transverse hypersurface, where the fundamental vector field of a weight 1 $\mathbb{G}_m$-action, in the derived setting, replaces the classical Liouville vector field. Secondly, the derived Legendrian intersection theorem is proven using base change, an $\infty$-categorical descent cube, and $\mathbb{G}_m$-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, $\ell$-adic local systems, and Lie 2-groups, and we provide further examples of contact derived moduli stacks.