Adopting the omni-Lie algebroid approach to Dirac-Jacobi structures, we propose and investigate a notion of weak dual pairs in Dirac-Jacobi geometry. Their main motivating examples arise from the theory of multiplicative precontact structures on Lie groupoids. Among other properties of weak dual pairs, we prove two main results. 1) We show that the property of fitting in a weak dual pair defines an equivalence relation for Dirac-Jacobi manifolds. So, in particular, we get the existence of self-dual pairs and this immediately leads to an alternative proof of the normal form theorem around Dirac-Jacobi transversals. 2) We prove the characteristic leaf correspondence theorem for weak dual pairs paralleling and extending analogous results for symplectic and contact dual pairs. Moreover, the same ideas of this proof apply to get a presymplectic leaf correspondence for weak dual pairs in Dirac geometry (not yet present in literature).
We develop explicit local operations that may be applied to Liouville domains, with the goal of simplifying the dynamics of the Liouville vector field. These local operations, which are Liouville homotopies, are inspired by the techniques used by Honda and Huang in [HH19] to show that convex hypersurfaces are $C^0$-generic in contact manifolds. As an application, we use our operations to show that Mitsumatsus well-known Liouville-but-not-Weinstein domains are stably Weinstein, answering a question asked by Huang in [Hua20].
The mixed equation, defined as a combination of the anti-self-duality equation in gauge theory and Cauchy-Riemann equation in symplectic geometry, is studied. In particular, regularity and Fredholm properties are established for the solutions of this equation, and it is shown that the moduli spaces of solutions to the mixed equation satisfy a compactness property which combines Uhlenbeck and Gormov compactness theorems. The results of this paper are used in a sequel to study the Atiyah-Floer conjecture.
A version of the Atiyah-Floer conjecture, adapted to admissible SO(3)-bundles, is established.
We introduce a notion of compatibility between (almost) Dirac structures and (1,1)-tensor fields extending that of Poisson-Nijenhuis structures. We study several properties of the Dirac-Nijenhuis structures thus obtained, including their connection with holomorphic Dirac structures, the geometry of their leaves and quotients, as well as the presence of hierarchies. We also consider their integration to Lie groupoids, which includes the integration of holomorphic Dirac structures as a special case.
We present some computations of relative symplectic cohomology, with the help of an index bounded contact form. For a Liouville domain with an index bounded boundary, we construct a spectral sequence which starts from its classical symplectic cohomology and converges to its relative symplectic cohomology inside a Calabi-Yau manifold.
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.
We show that in general the gradient of the twisted Rabinowitz action functional does not exist without further modifications.
In this work we show that the Weil-Petersson volume (which coincides with the CM degree) in the case of weighted points in the projective line is continuous when approaching the Calabi-Yau geometry from the Fano geometry. More specifically, the CM volume computed via localization converges to the geometric volume, computed by McMullen with different techniques, when the sum of the weights approaches the Calabi-Yau geometry.
We show that it is possible to define the contribution of degree one covers of a disk to open Gromov-Witten invariants. We build explicit sections of obstruction bundles in order to extend the algebro-geometric techniques of Pandharipande to the case of domains with boundary.
