Canonical and canonoid transformations for Hamiltonian systems on locally conformal symplectic manifolds

Abstract In previous work, we have studied the moment polytope of the closure of the complex subtorus orbit in a symplectic toric manifold associated to an affine subspace when the closure is a smooth complex submanifold. In this paper, we clarify the condition for nonsingularity of the closure of the codimension-one complex subtorus orbit in terms of polytopes. The main result is a generalization of the Delzant theorem.

Abstract We generalise the notions of scalar-valued holomorphic p -contact and s -symplectic structures introduced recently on compact complex manifolds by the second-named author jointly with H. Kasuya and L. Ugarte to their analogues with values in a holomorphic line bundle. We then study the resulting holomorphic p -contact and s -symplectic manifolds which, unlike their scalar counterparts that are never Kähler, can even be projective. In particular, we investigate the (lack of) positivity properties of the canonical bundle of these manifolds when it is given a possibly singular Hermitian fibre metric. One of the tools used is a very recent regularisation result for m -psh functions obtained jointly by S. Dinew and the second-named author.

We present a novel extension of Hamiltonian mechanics to nonconservative systems built upon the Schwinger-Keldysh-Galley double-variable action principle. Departing from Galley's initial-value action, we clarify important subtleties regarding boundary conditions, the emergence of the physical-limit trajectory, and the decomposition of the Lagrangian into conservative and dissipative sectors. Importantly, we demonstrate that the redundant doubled configuration space admits a gauge freedom at the level of the canonical momenta that leaves the physical dynamics unchanged. From a Legendre transform, we construct the corresponding family of gauge-related nonconservative Hamiltonians; we show that virtually any classical initial-value problem can be embedded on our enlarged symplectic manifold, supplying the associated Hamiltonian and Lagrangian functions explicitly. As a further contribution, we derive a completely equivalent linear "Lie" formulation of the double-variable action and Hamiltonian, which streamlines computations and renders transparent many structural properties of the formalism.

Abstract We prove contact big fiber theorems, analogous to the symplectic big fiber theorem by Entov and Polterovich, using symplectic cohomology with support. Unlike in the symplectic case, the validity of the statements requires conditions on the closed contact manifold. One such condition is to admit a Liouville filling with non-zero symplectic cohomology. In the case of Boothby-Wang contact manifolds, we prove the result under the condition that the Euler class of the circle bundle, which is the negative of an integral lift of the symplectic class, is not an invertible element in the quantum cohomology of the base symplectic manifold. As applications, we obtain new examples of rigidity of intersections in contact manifolds and also of contact non-squeezing.

Abstract The initial value spaces of the Painlevé equations are proposed by Okamoto. They are symplectic manifolds in which the Painlevé equations are described as polynomial Hamiltonian systems on all coordinates. In this article, we construct an initial value space of the four dimensional Painlevé system with affine Weyl group symmetry of type $$(A_5+A_1)^{(1)}$$ ( A 5 + A 1 ) ( 1 ) .

Given a monotone Lagrangian $L$ in a compact symplectic manifold $X$, we construct a commutative diagram relating the closed-open string map $\mathcal{CO}_\lambda \colon \operatorname{QH}^*(X) \to \operatorname{HH}^*(\mathcal{F} (X)_\lambda)$ to a variant of the length-zero closed-open map on $L$ incorporating $\mathbf{k}[\operatorname{H}_1(L; \mathbb{Z})]$ coefficients, denoted $\mathcal{CO}^0_\mathbf{L}$. The former is categorically important but very difficult to compute, whilst the latter is geometrically natural and amenable to calculation. We further show that, after a suitable completion, injectivity of $\mathcal{CO}^0_\mathbf{L}$ implies injectivity of $\mathcal{CO}_\lambda$. Via Sheridan's version of Abouzaid's generation criterion, this gives a powerful tool for proving split-generation of the Fukaya category. We illustrate this by showing that the real part of a monotone toric manifold (of minimal Chern number at least 2) split-generates the Fukaya category in characteristic 2. We also give a short new proof (modulo foundational assumptions in the non-monotone case) that the Fukaya category of an arbitrary compact toric manifold is split-generated by toric fibres.

We determine all possible exact noncommutative symplectic structures for certain path algebras with relations. These algebras are the path algebras of a cyclic quiver with r+1 arrows quotiented by the ideal generated by the paths of length l. The main result is that there are exact noncommutative symplectic structures only when l=s(r+1)+1 with s≥1. In this case a description of the open subset of one-forms z such that dz is a non-degenerate symplectic form is given, by reducing it to the case s=1.

On projective deformations of Lagrangian fibrations on primitive symplectic varieties

Abstract Homological mirror symmetry predicts that there is a relation between autoequivalence groups of derived categories of coherent sheaves on Calabi–Yau varieties and the symplectic mapping class groups of symplectic manifolds. In this paper, as an analogue of Dehn twists for closed oriented real surfaces, we study spherical twists for dg‐enhanced triangulated categories. We introduce the intersection number and relate it to group‐theoretic properties of spherical twists. Using an inequality analogous to a fundamental one in the theory of mapping class groups about the behavior of the intersection number via iterations of Dehn twists, we classify the subgroups generated by two spherical twists using the intersection number. As an application, we compute the center of autoequivalence groups of derived categories of K3 surfaces.

Isotrivial Lagrangian fibrations of compact hyper-Kähler manifolds

To a simple polarized hyperplane arrangement (not necessarily cyclic) V , one can associate a stopped Liouville manifold (equivalently, a Liouville sector) .M.V /; /, where M.V / is the complement of finitely many hyperplanes in C d , obtained as the complexifications of the real hyperplanes in V .The Liouville structure on M.V / comes from a very affine embedding, and the stop is determined by the polarization.In this article, we study the symplectic topology of .M.V /; /.In particular, we prove that their partially wrapped Fukaya categories are generated by Lagrangian submanifolds associated to the bounded and feasible chambers of V .A computation of the Fukaya A 1 -algebra of these Lagrangians then enables us to identify the partially wrapped Fukaya categories W.M.V /; / with the G d m -equivariant hypertoric convolution algebras z B.V / associated to V .This confirms a conjecture of Lauda, Licata and Manion ( 2024) and provides evidence for the general conjecture of Lekili and Segal (2023) on the equivariant Fukaya categories of symplectic manifolds with Hamiltonian torus actions.53D40 1.

We present a proof of a result on displaceability of subsets of symplectic manifolds satisfying certain conditions one of which is that the subset is precompact in a connected neighborhood that symplectically embeds into $\mathbb{R}^{2n}$. The proof utilizes an inequality between the displacement energy and the cylindrical capacity for subsets of $\mathbb{R}^{2n}$ to obtain an inequality for subsets of the symplectic manifold. We also state a corollary which utilizes other results on nondisplaceable Lagrangians.

In this paper we study compact monotone tall complexity one T T -spaces. We use the classification of Karshon and Tolman, and the monotone condition, to prove that any two such spaces are isomorphic if and only if they have equal Duistermaat-Heckman measures. Moreover, we show that the moment polytope is Delzant and reflexive, and provide a complete description of the possible Duistermaat-Heckman measures. Whence we obtain a finiteness result that is analogous to that for compact monotone symplectic toric manifolds. Furthermore, we show that any such T T -action can be extended to a toric ( T × S 1 ) (T \times S^1) -action. Motivated by a conjecture of Fine and Panov, we prove that any compact monotone tall complexity one T T -space is equivariantly symplectomorphic to a Fano manifold endowed with a suitable symplectic form and a complexity one T T -action.

To a symplectic Lefschetz pencil on a monotone symplectic manifold, we associate an algebraic structure, which is a pencil of categories in the sense of noncommutative geometry.

In this paper, we will show that certain types of symplectic homology can be used as an invariant of 3-dimensional Besse manifolds, which are strict contact manifolds with periodic Reeb flow. For simplicity, we will assume our Besse structures to be a trivial plane bundle. To identify Besse manifolds with such a condition, we actually compute the first Chern class of each Besse structure and classify the Besse manifolds with vanishing first Chern class. We will also compute Robbin-Salamon indices of periodic Reeb orbits in Besse manifolds, and symplectic homology (of its filling) of certain cases. From its definition, Besse manifolds naturally become Seifert fibration, and thus one can extract invariants such as the orbifold Euler characteristic and the Euler number of this Seifert fibration. These invariants will become important in our computation.

Using the global Kuranishi charts constructed by Hirschi–Swaminathan, we define gravitational descendants and equivariant Gromov–Witten invariants for general symplectic manifolds. We prove that these invariants satisfy the axioms of Kontsevich and Manin and their generalisations. A virtual localisation formula holds in this setting; we use it to derive an explicit formula for the equivariant Gromov–Witten invariants of Hamiltonian GKM manifolds. In particular, the symplectic Gromov–Witten invariants of smooth toric varieties agree with their algebro-geometric counterpart. In the semipositive case, the invariants studied here recover those of Ruan and Tian.

Abstract Given a strictly unbounded toric symplectic 4-manifold, we explicitly construct complete toric scalar-flat Kähler metrics on the complement of a toric divisor. These symplectic 4-manifolds correspond to a specific class of non-compact Kähler surfaces. We also provide an alternative construction of toric scalar-flat Kähler metrics with conical singularity along the toric divisor, following the approach of Abreu and Sena-Dias.

We introduce operations with p -adic integer coefficients, associated to idempotents in the quantum cohomology of a monotone symplectic manifold, and apply them to the structure of the quantum connection.

We extend the cohomological setting developed by Batalin, Fradkin and Vilkovisky (BFV), which produces a resolution of coisotropic reduction in terms of hamiltonian dg manifolds, to the case of nested coisotropic embeddings Chookrightarrow C_circ hookrightarrow F inside a symplectic manifold F. To this, we naturally assign underline{C} and underline{C_circ }, as well as the respective BFV dg manifolds. We show that the data of a nested coisotropic embedding defines a natural graded coisotropic embedding inside the BFV dg manifold assigned to underline{C}, whose reduction can further be resolved using the BFV prescription. We call this construction double BFV resolution, and we use it to prove that “resolution commutes with reduction” for a large class of nested coisotropic embeddings. We then deduce a quantisation of underline{C}, from the (graded) geometric quantisation of the double BFV Hamiltonian dg manifold (when it exists), following the quantum BFV prescription. As an application, we provide a well defined candidate space of (physical) quantum states of three-dimensional Einstein–Hilbert theory, which is thought of as a partial reduction of the Palatini–Cartan model for gravity.