No Arabic abstract
We introduce the critical Weinstein category - the result of stabilizing the category of Weinstein sectors and inverting subcritical morphisms - and construct localizing `P-flexibilization endofunctors indexed by collections $P$ of Lagrangian disks in the stabilization of a point $T^*D^0$. Like the classical localization of topological spaces studied by Quillen, Sullivan, and others, these functors are homotopy-invariant and localizing on algebraic invariants like the Fukaya category. Furthermore, these functors generalize the `flexibilization operation introduced by Cieliebak-Eliashberg and Murphy and the `homologous recombination construction of Abouzaid-Seidel. In particular, we give an h-principle-free proof that flexibilization is idempotent and independent of presentation, up to subcriticals and stabilization. In fact, we show that $P$-flexibilization is a multiplicative localization of the critical Weinstein category, and hence gives rise to a new way of constructing commutative algebra objects from symplectic geometry.
The paper is devoted to study of Massey products in symplectic manifolds. Theory of generalized and classical Massey products and a general construction of symplectic manifolds with nontrivial Massey products of arbitrary large order are exposed. The construction uses the symplectic blow-up and is based on the author results, which describe conditions under which the blow-up of a symplectic manifold X along its submanifold Y inherits nontrivial Massey products from X ot Y. This gives a general construction of nonformal symplectic manifolds.
We study the geometry of manifolds carrying symplectic pairs consisting of two closed 2-forms of constant ranks, whose kernel foliations are complementary. Using a variation of the construction of Boothby and Wang we build contact-symplectic and contact pairs from symplectic pairs.
Recently, Tsai-Tseng-Yau constructed new invariants of symplectic manifolds: a sequence of Aoo-algebras built of differential forms on the symplectic manifold. We show that these symplectic Aoo-algebras have a simple topological interpretation. Namely, when the cohomology class of the symplectic form is integral, these Aoo-algebras are equivalent to the standard de Rham differential graded algebra on certain odd-dimensional sphere bundles over the symplectic manifold. From this equivalence, we deduce for a closed symplectic manifold that Tsai-Tseng-Yaus symplectic Aoo-algebras satisfy the Calabi-Yau property, and importantly, that they can be used to define an intersection theory for coisotropic/isotropic chains. We further demonstrate that these symplectic Aoo-algebras satisfy several functorial properties and lay the groundwork for addressing Weinstein functoriality and invariance in the smooth category.
We prove a version of the Arnold conjecture for Lagrangian submanifolds of conformal symplectic manifolds: a Lagrangian $L$ which has non-zero Morse-Novikov homology for the restriction of the Lee form $beta$ cannot be disjoined from itself by a $C^0$-small Hamiltonian isotopy. Furthermore for generic such isotopies the number of intersection points equals at least the sum of the free Betti numbers of the Morse-Novikov homology of $beta$. We also give a short exposition of conformal symplectic geometry, aimed at readers who are familiar with (standard) symplectic or contact geometry.
We give characterizations of a finite group $G$ acting symplectically on a rational surface ($mathbb{C}P^2$ blown up at two or more points). In particular, we obtain a symplectic version of the dichotomy of $G$-conic bundles versus $G$-del Pezzo surfaces for the corresponding $G$-rational surfaces, analogous to a classical result in algebraic geometry. Besides the characterizations of the group $G$ (which is completely determined for the case of $mathbb{C}P^2# Noverline{mathbb{C}P^2}$, $N=2,3,4$), we also investigate the equivariant symplectic minimality and equivariant symplectic cone of a given $G$-rational surface.