Recent zbMATH articles in MSC 53Dhttps://zbmath.org/atom/cc/53D2023-12-07T16:00:11.105023ZWerkzeugQuadratic Lie algebras with 2-plectic structureshttps://zbmath.org/1522.170282023-12-07T16:00:11.105023Z"Bajo, Ignacio"https://zbmath.org/authors/?q=ai:bajo.ignacio"Benayadi, Saïd"https://zbmath.org/authors/?q=ai:benayadi.saidSummary: We study the existence of 2-plectic structures on Lie algebras which admit an \(\text{ad} \)-invariant non-degenerate symmetric bilinear form, frequently called quadratic Lie algebras. It is well-known that every centerless quadratic Lie algebra admits a 2-plectic form but not many quadratic examples with nontrivial center are known. In this paper we give several constructions to obtain large families of 2-plectic quadratic Lie algebras with nontrivial center, many of them among the class of nilpotent Lie algebras. We give some sufficient conditions to assure that certain extensions of 2-plectic quadratic Lie algebras result to be 2-plectic as well. We prove that every quadratic and symplectic Lie algebra with dimension greater than 4 also admits a 2-plectic form. Further, conditions to assure that one may find a 2-plectic which is exact on certain quadratic Lie algebras are also obtained.A new look at Lie algebrashttps://zbmath.org/1522.170372023-12-07T16:00:11.105023Z"Dobrogowska, Alina"https://zbmath.org/authors/?q=ai:dobrogowska.alina"Jakimowicz, Grzegorz"https://zbmath.org/authors/?q=ai:jakimowicz.grzegorzSummary: We present a new look at the description of real finite-dimensional Lie algebras. The basic ingredient is a pair \((F, v)\) consisting of a linear mapping \(F \in \operatorname{End}(V)\) with an eigenvector \(v\). This pair allows to build a Lie bracket on a dual space to a linear space \(V\). The Lie algebra obtained in this way is solvable. In particular, when \(F\) is nilpotent, the Lie algebra is actually nilpotent. We show that these solvable algebras are the basic bricks of the construction of all other Lie algebras. Using relations between the Lie algebra, the Lie-Poisson structure and the Nambu bracket, we show that the algebra invariants (Casimir functions) are solutions of an equation which has an interesting geometric significance. Several examples illustrate the importance of these constructions.Generalized ILW hierarchy: solutions and limit to extended lattice GD hierarchyhttps://zbmath.org/1522.370802023-12-07T16:00:11.105023Z"Takasaki, Kanehisa"https://zbmath.org/authors/?q=ai:takasaki.kanehisaThe author proposes a generalization of the integrable hierarchy of the intermediate long wave equation (ILW). This hierarchy can be viewed as a reduction of the lattice Kadomtsev-Petviashvili (KP) hierarchy and depends on a certain parameter. By setting that parameter to zero, the author proves that the generalized ILW hierarchy leads to an extended lattice Gelfand-Dickey hierarchy. In a similar manner it can be proved that the equivariant one-dimensional (resp., bigraded) Toda hierarchy reduces to the extended one-dimensional (resp., bigraded) Toda hierarchy.
Another important result in the paper concerns the integration of the considered hierarchy. Starting from the soliton solutions of the lattice KP hierarchy, the author obtains certain conditions that lead to soliton solutions of the generalized ILW hierarchy. An alternative approach of integration is based upon a factorization problem of difference solution-generating operators related to the lattice KP hierarchy with some reduction imposed. The described procedure allows one to construct all the solutions to the generalized ILW hierarchy. A special attention is paid to the case when the solution-generating operators are analogous to those appearing in the fermionic description of the equivariant Gromov-Witten theory of \(\mathbb{C}P^1\).
Reviewer: Tihomir Valchev (Sofia)Symplectic groupoids for Poisson integratorshttps://zbmath.org/1522.370852023-12-07T16:00:11.105023Z"Cosserat, Oscar"https://zbmath.org/authors/?q=ai:cosserat.oscarSummary: We use local symplectic Lie groupoids to construct Poisson integrators for generic Poisson structures. More precisely, recursively obtained solutions of a Hamilton-Jacobi-like equation are interpreted as Lagrangian bisections in a neighborhood of the unit manifold, that, in turn, give Poisson integrators. We also insist on the role of the Magnus formula, in the context of Poisson geometry, for the backward analysis of such integrators.Contact instantons and partial connectionshttps://zbmath.org/1522.530142023-12-07T16:00:11.105023Z"Udomlertsakul, Nathapon"https://zbmath.org/authors/?q=ai:udomlertsakul.nathapon"Wang, Shuguang"https://zbmath.org/authors/?q=ai:wang.shuguang|wang.shuguang.1Summary: We study instanton equations on contact manifolds from the point of view of partial connections. Results of \textit{H. Urakawa} [Math. Z. 216, No. 4, 541--573 (1994; Zbl 0815.32008)] and \textit{S. Dragomir} and \textit{H. Urakawa} [Interdiscip. Inf. Sci. 6, No. 1, 41--52 (2000; Zbl 0959.58020)] are generalized from strongly pseudo-convex CR manifolds to contact manifolds.Anti-quasi-Sasakian manifoldshttps://zbmath.org/1522.530362023-12-07T16:00:11.105023Z"Di Pinto, D."https://zbmath.org/authors/?q=ai:di-pinto.dario"Dileo, G."https://zbmath.org/authors/?q=ai:dileo.giuliaThe authors introduce and investigate a new class of almost contact metric manifolds and call them anti-quasi-Saskian manifolds (aqS manifolds for short). Concretely, an almost contact metric manifold \((M,\varphi, \xi, \eta, g)\) with fundamental \(2\)-form \(\Phi(X,Y) = g(X,\varphi(Y))\) is said to be aqS if
\[
d\Phi = 0,\qquad N_\varphi = 2d\eta\otimes \xi,
\]
where \(N_\varphi=[\varphi,\varphi]+d\eta\otimes \xi\) and \([\varphi,\varphi]\) is the Nijenhuis torsion of \(\varphi\).
This class intersects the class of quasi-Saskian manifolds (where instead \(N_\varphi=0\) and \(d\Phi=0\)) precisely in the case of co-Kähler manifolds (i.e., where additionally \(\eta\) is closed). From the point of view of transverse geometry (with respect to the foliation by Reeb orbits) aqS geometry is characterized by a Kähler structure, together with a closed \((2,0)\)-form. The authors derive a Boothby-Wang fibration with this type of base manifold, in case the Reeb vector field is regular. As an important special case, there appear hyper-Kähler manifolds as quotients. Further examples of aqS manifolds include weighted Heisenberg groups and their nilmanifolds.
The authors show that aqS manifolds with constant sectional curvature are flat and co-Kähler, and investigate more closely aqS manifolds with \(\xi\)-sectional curvatures one. Moreover, they show the existence of a canonical metric connection with torsion on any aqS manifold and explore the consequences of their geometry.
Reviewer: Oliver Goertsches (Marburg)Magnetic curves in quasi-Sasakian manifolds of product typehttps://zbmath.org/1522.530412023-12-07T16:00:11.105023Z"Munteanu, Marian Ioan"https://zbmath.org/authors/?q=ai:munteanu.marian-ioan"Nistor, Ana Irina"https://zbmath.org/authors/?q=ai:nistor.ana-irinaIn a previous work [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 214, Article ID 112571, 18 p. (2022; Zbl 1485.53033)] the authors stated the following conjecture: A contact magnetic curve in a quasi-Sasakian manifold of dimension \(2n+1\geq 5\) must be a Frenet helix of maximum order \(5\). The paper gives a positive answer to this conjecture. The main focus is on magnetic curves in a quasi-Sasakian manifold of product type \(N^{2p+1}\times B^{2k}\), where \(N\) is a Sasakian manifold and \(B\) is a Kähler one. Concrete examples of magnetic curves are provided for the manifold \(\mathbb{S}^3\times \mathbb{S}^2\).
For the entire collection see [Zbl 1508.53008].
Reviewer: Mircea Crâşmăreanu (Iaşi)The translation number and quasi-morphisms on groups of symplectomorphisms of the diskhttps://zbmath.org/1522.530652023-12-07T16:00:11.105023Z"Maruyama, Shuhei"https://zbmath.org/authors/?q=ai:maruyama.shuheiThis paper is concerned with the construction of homogenous quasi-morphisms on groups of symplectomorphisms of the disk.
Let \(D=\{ (x,y) \in \mathbb R^2\mid x^2 + y^2 \leq 1 \}\) be the unit disk in \(\mathbb R^2\) and \(\omega = dx \wedge dy \) be the standard symplectic form on \(D\). Let \(G= \mathrm{Symp}(D)\) be the group of symplectomorphisms of \(D\) (which may not be the identity on the boundary \(\partial D\)). In this paper the author first constructs a homogeneous quasi-morphism on \(G\), extending the Calabi invariant. Recall that the restriction homomorphism \(\rho: G \to \mathrm{Diff}_+ (S^1)\), where \(\mathrm{Diff}_+ (S^1)\) is the group of orientation-preserving diffeomorphisms of the unit circle \(S^1 = \partial D\), is surjective, see [\textit{T. Tsuboi}, Trans. Am. Math. Soc. 352, No. 2, 515--524 (2000; Zbl 0937.57023)]. Denote by \(G_{\mathrm{rel}}\) the kernel of \(\rho\). The Calabi invariant \(\mathrm{Cal}: G_{\mathrm{rel}} \to \mathbb R\) is defined by
\[
\mathrm{Cal}(h) = \int_D h^*\eta \wedge \eta,
\]
where \(\eta\) is a \(1\)-form satisfying \(d\eta = \omega\). It is well known that this invariant is a surjective homomorphism and it is independent of the choice of \(\eta\). The author defines the map \(\tau_\eta: G \to \mathbb R\) in the same way:
\[
\tau_\eta (g) = \int_D g^*\eta \wedge \eta.
\]
This map is not a homomorphism and does depend on \(\eta\). It turns out that it is a quasi-morphism, its homogenization \(\bar{\tau}\) does not depend on \(\eta\) and it is an extension of the Calabi invariant. Moreover, there is another extension of the Calabi invariant previously introduced by \textit{T. Tsuboi} [loc. cit.], which is a homomorphism to \(\mathbb R\) from the universal covering group of \(G\). One of the main results of this paper (Theorem 1.1) establishes a relation between the two extensions, involving the translation number introduced by \textit{H. Poincaré} [C. R. Acad. Sci., Paris 90, 673--675 (1880; JFM 12.0588.01)].
In a similar way, the author constructs a homogeneous quasi-morphism \(\bar{\sigma}\) on the subgroup \(G_o\) of \(G\), consisting of symplectomorphisms preserving the origin, which extends a version of the flux homomorphism. More precisely, let \(G_{o,\mathrm{rel}} = G_{\mathrm{rel}} \cap G_o\). The flux homomorphism \(\mathrm{Flux}_{\mathbb R}\) is defined by
\[
\mathrm{Flux}_{\mathbb R}(h) = \int_\gamma h^*\eta - \eta,
\]
where \(\gamma\) is a path from the origin \(o\) to a point in the boundary \(\partial D\). This homomorphism is surjective and it is independent of the choice of \(\eta\) and \(\gamma\). As in the previous case, the quasi-morphism \(\bar{\sigma}\) relates with another extension of the flux homomorphism through the translation number (Theorem 1.2).
Finally, in the last section, the author shows that the difference \(\bar{\tau} -\pi \bar{\sigma}: G_{o} \to \mathbb R\) is a continuous homomorphism, extending the difference \(\mathrm{Cal} -\pi \mathrm{Flux}_{\mathbb R}\), although \(\mathrm{Cal}\) and \(\mathrm{Flux}_{\mathbb R}\) cannot can be extended to homomorphisms on \(G_o\).
Reviewer: Sílvia Anjos (Lisboa)A Bangert-Hingston theorem for starshaped hypersurfaceshttps://zbmath.org/1522.530662023-12-07T16:00:11.105023Z"Pellegrini, Alessio"https://zbmath.org/authors/?q=ai:pellegrini.alessioThe aim of this paper is to prove a version of the theorem of \textit{V. Bangert} and \textit{N. Hingston} [J. Differ. Geom. 19, 277--282 (1984; Zbl 0545.53036)] for starshaped hypersurfaces. Let \(Q\) be a closed manifold with non-trivial first Betti number that admits a non-trivial \(S^1\)-action, and \(\Sigma\subseteq T^*Q\) a nondegenerate starshaped hypersurface. In this paper, the author proves that the number of geometrically distinct Reeb orbits of period at most \(T\) on \(\Sigma\) grows at least logarithmically in \(T\).
The paper is organized as follows: Section 1 is an introduction to the subject. Section 2 deals with Reeb orbits and starshaped domains. Section 3 is devoted to spectral invariants and minimax values. Here, the author introduces spectral invariants and compares them to minimax values on \(T^*Q\). Section 4 deals with pinching and Floer homologies. In this section, the author closely follows [\textit{L. Macarini} and \textit{F. Schlenk}, Math. Proc. Camb. Philos. Soc. 151, No. 1, 103--128 (2011; Zbl 1236.53063); \textit{M. Heistercamp}, ``The Weinstein conjecture with multiplicities on spherizations'', Preprint, \url{arXiv:1105.3886}; \textit{R. E. Wullschleger}, Counting Reeb chords on spherizations. Université de Neuchâtel (PhD thesis) (2014)] and constructs three sequences of non-degenerate Hamiltonians. Sections 5 and 6 are devoted to index relations and the main result respectively.
Reviewer: Ahmed Lesfari (El Jadida)Contact 3-manifolds with pseudo-parallel characteristic Jacobi operatorhttps://zbmath.org/1522.530672023-12-07T16:00:11.105023Z"Inoguchi, Jun-ichi"https://zbmath.org/authors/?q=ai:inoguchi.jun-ichi"Lee, Ji-Eun"https://zbmath.org/authors/?q=ai:lee.jieunSummary: In this article, we study contact metric 3-manifolds with pseudo-parallel characteristic Jacobi operator. Contact metric 3-manifolds with pseudo-parallel characteristic Jacobi operator are \(M_\ell\)-manifolds (contact metric 3-manifolds with vanishing characteristic Jacobi operator) or generalized contact \((\kappa, \mu, \nu)\)-spaces. Moreover, we prove that contact metric 3-manifolds with pseudo-parallel characteristic Jacobi operator are classified into four classes. In particular, we give a complete classification of homogeneous contact metric 3-manifolds with proper pseudo-parallel characteristic Jacobi operator.The geometry of some thermodynamic systemshttps://zbmath.org/1522.530682023-12-07T16:00:11.105023Z"Simoes, Alexandre Anahory"https://zbmath.org/authors/?q=ai:anahory-simoes.alexandre"Martín de Diego, David"https://zbmath.org/authors/?q=ai:martin-de-diego.david"Valcázar, Manuel Lainz"https://zbmath.org/authors/?q=ai:valcazar.manuel-lainz"de León, Manuel"https://zbmath.org/authors/?q=ai:de-leon.manuelThis paper presents an alternative approach of the geometrical framework for studying some thermodynamic systems on a \((2n+1)\)-dimensional contact manifold, by using the so-called evolution vector field \(\mathcal{E}_H\) (described in terms of a skew-symmetric bracket of functions) instead of the contact vector field \(X_H\) used by other authors, both provided by a Hamiltonian \(H : T^{\ast}Q \times\mathbb{R}\to\mathbb{R}\), defined by \((q^i, p_i, S) \to H(q^i, p_i, S)\), where \((q^i)\) is the position, \((p_i)\) is the momentum and \(S\) is the entropy. Note that the integral curves of this evolution vector field describe the trajectories of a thermodynamic system which fulfills the first and second laws of thermodynamics, being thus, in the authors' opinion, a good candidate for the study of thermodynamic processes. Therefore, the authors investigate in their way the simple mechanical systems with friction (isolated systems) or composed thermodynamic systems without friction. Finally, the authors provide some geometric integrators for their used formalism.
For the entire collection see [Zbl 1468.68006].
Reviewer: Mircea Neagu (Braşov)Characterization of Whitney spheres among Lagrangian submanifolds with conformal Maslov formhttps://zbmath.org/1522.530692023-12-07T16:00:11.105023Z"Zhao, Entao"https://zbmath.org/authors/?q=ai:zhao.en-tao"Cao, Shunjuan"https://zbmath.org/authors/?q=ai:cao.shunjuanWhitney spheres are typical Lagrangian submanifolds in complex space forms. A Lagrangian submanifold \(M\) in a Kähler manifold is said to be a Lagrangian submanifold with conformal Maslov form if \(JH\) is a conformal vector field of \(M\). In this paper, the authors give new characterizations of Whitney spheres in complex space forms by the conformal Maslov form.
Let \(\mathbf{N}^n(4c)\) be a complex space form of complex dimension \(n\) with constant holomorphic sectional curvature \(4c\) with \(c \in {0,\pm 1}\). Let \(\phi : M^n \longrightarrow \mathbf{N}^n(4c)\) be an isometric immersion from an \(n\)-dimensional manifold \(M\) into \(\mathbf{N}^n(4c)\). Let \(H\) be the mean curvature vector. Let \(h(X,Y)\) be the second fundamental form. Define a modified second fundamental form \(B\) of \(M\) in \(\mathbf{N}^n(4c)\) by
\[
B(X,Y)=h(X,Y) - \frac{n}{n+2}(\langle X,Y\rangle H+\langle JX,H\rangle JY +\langle JY,H\rangle JX)
\]
for vector fields \(X,Y\) on \(M\). Let \(\gamma (n, |H|, c)\) denote the square of the nonnegative root of the equation
\[
\frac{3}{2} x^2 + \frac{n-2}{\sqrt{n(n-1)}} |H| x - \Big( (n+1) c + \frac{n^2}{n+2} |H|^2\Big) =0.
\]
The authors mainly prove two theorems in this paper:
Theorem. Let \(M\) be an \(n\)-dimensional compact Lagrangian submanifold with conformal Maslov form in \(\mathbf{N}^n(4c)\). Suppose \(|H|^2 + (n+1)(n+2)c /n^2 > 0\) for \(c<0\). If \(|B|\leq \gamma(n, |H|, c)\), then \(M\) is one of the following:
(i) A totally geodesic submanifold in \(\mathbb{CP}^n\);
(ii) A Whitney sphere in \(\mathbf{N}^n(4c)\);
(iii) A Clifford torus in \(\mathbb{CP}^2\).
Theorem. Let \(M\) be an \(n\)-dimensional non-minimal compact Lagrangian submanifold with conformal Maslov form in \(\mathbf{N}^n(4c)\).
(i) If \(c = 1\) and \(\int_M |B|^n d\mu < C_1(n)\), where \(C_1(n)\) is a positive constant depending only on \(n\), then \(M\) is either a totally geodesic submanifold or the Whitney sphere in \(\mathbb{CP}^n\):
(ii) If \(c = 0\) and \(\int_M B|^n d\mu < C_2(n)\), where \(C_2(n)\) is a positive constant depending only on \(n\), then \(M\) is the Whitney sphere in \(\mathbb{C}^n\);
(iii) If \(c = -1\), \(|H|^2- (n+1)(n+2)/n^2 \geq \tau\) for some \(\tau >0\) and \(\int _M |B|^n d\mu <C(n,\tau)\), where \(C(n,\tau)\) is a positive constant depending only on \(n\) and \(\tau\), then \(M\) is the Whitney sphere in \(\mathbb{CH}^n\).
Reviewer: Shiquan Ren (Singapore)Invariant almost contact structures and connections on the Lobachevsky spacehttps://zbmath.org/1522.530702023-12-07T16:00:11.105023Z"Rastrepina, A. O."https://zbmath.org/authors/?q=ai:rastrepina.anastasia-o"Surina, O. P."https://zbmath.org/authors/?q=ai:surina.olga-petrovnaAuthors' abstract: In this paper, we study the existence of invariant almost contact metric structures and connections in the Lobachevsky space using the Poincaré model and the group model of the Lobachevsky space. It is established that in the Lobachevsky space there exist left-invariant almost contact structures, among which an integrable normal almost contact metric structure is distinguished. All left-invariant linear connections consistent with the given structure are found, among which connections with zero curvature tensor are distinguished. It is proved that these connections are compatible with a foliation of an integrable normal almost contact metric structure in the sense that a unique geodesic belonging to the given foliation passes through each point in each direction tangent to this foliation. In addition to the Levi-Civita connections, in the Lobachevsky space there is also a metric connection with skew-symmetric torsion invariant under the full six-dimensional group of motions, as well as the only semisymmetric contact metric connection invariant under the four-dimensional subgroup of the group of motions.
Reviewer: Mohammed Guediri (Riyadh)Hessenberg varieties and Poisson sliceshttps://zbmath.org/1522.530712023-12-07T16:00:11.105023Z"Crooks, Peter"https://zbmath.org/authors/?q=ai:crooks.peter"Röser, Markus"https://zbmath.org/authors/?q=ai:roser.markusSummary: This expository article considers a circle of Lie-theoretic ideas involving Hessenberg varieties, Poisson geometry, and wonderful compactifications. In more detail, one may associate a symplectic Hamiltonian \(G\)-variety \(\mu:G\times\mathcal{S}\longrightarrow\mathfrak{g}\) to each complex semisimple Lie algebra \(\mathfrak{g}\) with adjoint group \(G\) and fixed Kostant section \(\mathcal{S}\subseteq\mathfrak{g}\). This variety is one of Bielawski's hyperkähler slices, and it is central to \textit{G. W. Moore} and \textit{Y. Tachikawa}'s work [Proc. Sympos. Pure Math., 85, Am. Math. Soc., 191--207 (2012)] on topological quantum field theories. It also bears a close relation to two log symplectic Hamiltonian \(G\)-varieties \(\overline{\mu}_{\mathcal{S}}:\overline{G\times\mathcal{S}}\longrightarrow\mathfrak{g}\) and \(\nu:\mathrm{Hess}\longrightarrow\mathfrak{g}\). The former is a Poisson transversal in the log cotangent bundle of the wonderful compactification \(\overline{G}\), while the latter is the standard family of Hessenberg varieties. Each of \(\overline{\mu}\) and \(\nu\) is known to be a fibrewise compactification of \(\mu\).
We exploit the theory of Poisson slices to relate the fibrewise compactifications mentioned above. Our work is shown to be compatible with a Poisson isomorphism obtained by \textit{A. Bălibanu} [Represent. Theory 21, 132--150 (2017; Zbl 1428.20040)].
For the entire collection see [Zbl 1522.14005].Cosymplectic groupoidshttps://zbmath.org/1522.530722023-12-07T16:00:11.105023Z"Fernandes, Rui Loja"https://zbmath.org/authors/?q=ai:fernandes.rui-loja"Iglesias Ponte, David"https://zbmath.org/authors/?q=ai:iglesias-ponte.davidSummary: A cosymplectic groupoid is a Lie groupoid with a multiplicative cosymplectic structure. We provide several structural results for cosymplectic groupoids and we discuss the relationship between cosymplectic groupoids, Poisson groupoids of corank 1, and oversymplectic groupoids of corank 1.On symmetries of singular foliationshttps://zbmath.org/1522.530732023-12-07T16:00:11.105023Z"Louis, Ruben"https://zbmath.org/authors/?q=ai:louis.rubenIn this interesting paper the author shows that a weak symmetry action of a Lie algebra \(g\) on a singular foliation \(F\) induces a unique up to homotopy Lie \(\infty\)-morphism from \(g\) to the DGLA of vector fields on a universal Lie \(\infty\)-algebroid of \(F\). Such a morphism was called an \(L_\infty\)-algebra action in [\textit{R. A. Mehta} and \textit{M. Zambon}, Differ. Geom. Appl. 30, No. 6, 576--587 (2012; Zbl 1267.58003)]. From this general result several geometrical consequences are deduced. It is given an example of a Lie algebra action on an affine subvariety which cannot be extended to the ambient space. Finally, the notion of bi-submersion towers over a singular foliation and lift symmetries to those is introduced.
Reviewer: Liviu Popescu (Craiova)A Poisson bracket on the space of Poisson structureshttps://zbmath.org/1522.530742023-12-07T16:00:11.105023Z"Machon, Thomas"https://zbmath.org/authors/?q=ai:machon.thomasLet \(M\) be a smooth, closed and orientable manifold. The author considers the set of all Poisson structures on \(M\), denoted \(\mathcal{P}(M)\) and shows that \(\mathcal{P}(M)\) has itself a family of Poisson structures \(\{\,,\}_{\mu}\), depending on a choice of a volume form \(\mu\). The motivation for this work comes from ideal fluid dynamics. The aim is to extend the Poisson bracket on the space of Poisson structures and to define a Poisson bracket on the space of admissible functions. The bracket is explicitly given and the corresponding proofs are detailed. The author considers also the space of symplectic manifolds with a symplectic volume form. In this case, he constructs a further and related Poisson bracket. He studies the induced flow and gives a description in terms of the symplectic cohomology groups introduced by \textit{L.-S. Tseng} and \textit{S.-T. Yau} [J. Differ. Geom. 91, No. 3, 383--416 (2012; Zbl 1275.53079)].
Reviewer: Angela Gammella-Mathieu (Metz)Atiyah and Todd classes of regular Lie algebroidshttps://zbmath.org/1522.530752023-12-07T16:00:11.105023Z"Xiang, Maosong"https://zbmath.org/authors/?q=ai:xiang.maosongA dg manifold (or a Q-manifold) is a \(\mathbb{Z}\)-graded smooth manifold equipped with a homological vector field Q, i.e., a degree \(+1\) derivation of square zero on the algebra of smooth functions.
A Lie algebroid \((A, \rho_A, [-,-]_A)\) over a smooth manifold \(M\) is said to be {\em regular} if its anchor \(\rho_A\) is of constant rank. The kernel \(K = \ker(\rho_A)\) together with the restriction \([-,-]_K\) of the Lie bracket \([-,-]_A\) onto \(\Gamma (K)\) is a bundle of Lie algebras; the image \(F = \operatorname{Im}(\rho_A) \subseteq TM\) as the tangent bundle of the regular characteristic foliation, is a Lie subalgebroid of the tangent Lie algebroid \(TM\). In other words there is a short exact sequence
\[
0\rightarrow K \stackrel{i} \rightarrow A \stackrel{\rho_A} \rightarrow F\rightarrow 0
\]
of Lie algebroids over \(M\), known as the Atiyah sequence of \(A\).
In this paper, the author studies the Atiyah and Todd classes of dg manifolds arising from regular Lie algebroids and proves that these classes fit into a short exact sequence called Atiyah sequence.
Reviewer: Osman Mucuk (Kayseri)A non commutative Kähler structure on the Poincaré disk of a \(C^*\)-algebrahttps://zbmath.org/1522.530762023-12-07T16:00:11.105023Z"Andruchow, Esteban"https://zbmath.org/authors/?q=ai:andruchow.esteban"Corach, Gustavo"https://zbmath.org/authors/?q=ai:corach.gustavo"Recht, Lázaro"https://zbmath.org/authors/?q=ai:recht.lazaro-aThe article is the third in a series of articles by the authors, where they study the Poincaré disc of a unital \(C^{\ast}\)-algebra. Here they endow the Poincaré disc as such with a noncommutative homogeneous Kähler structure and study the associated moment map. The principal result is the generalization of the classical Atiyah-Guillemin-Sternberg theorem about the convexity of the image of this moment map, in the presence of a trace. To this end, the authors carry out an appropriate geometric (pre)quantization program.
For the entire collection see [Zbl 1492.47001].
Reviewer: Iakovos Androulidakis (Athína)Towers of Looijenga pairs and asymptotics of ECH capacitieshttps://zbmath.org/1522.530772023-12-07T16:00:11.105023Z"Wormleighton, Ben"https://zbmath.org/authors/?q=ai:wormleighton.benSummary: ECH capacities are rich obstructions to symplectic embeddings in 4-dimensions that have also been seen to arise in the context of algebraic positivity for (possibly singular) projective surfaces. We extend this connection to relate general convex toric domains on the symplectic side with towers of polarised toric surfaces on the algebraic side, and then use this perspective to show that the sub-leading asymptotics of ECH capacities for all convex and concave toric domains are \(O(1)\). We obtain sufficient criteria for when the sub-leading asymptotics converge in this context, generalising results of Hutchings and of the author, and derive new obstructions to embeddings between toric domains of the same volume. We also propose two invariants to more precisely describe when convergence occurs in the toric case. Our methods are largely non-toric in nature, and apply more widely to towers of polarised Looijenga pairs.A short proof of cuplength estimates on Lagrangian intersectionshttps://zbmath.org/1522.530782023-12-07T16:00:11.105023Z"Gong, Wenmin"https://zbmath.org/authors/?q=ai:gong.wenminLet \(M\) be a closed \(n\)-manifold with the canonical symplectic form \(\omega\) on the cotangent bundle \(T^*M\). Given \(H\in C^\infty([0,1]\times T^*M)\), the Hamiltonian vector field \(X_H\) is determined by the equation \(dH=-\omega(X_H,.)\). This vector field gives rise to the flow \(\varphi^t\) and its time-one map \(\varphi\). If \(H\) is asymptotically constant then \(\varphi\) is a Hamiltonian diffeomorphism with compact support. Let \(O_M\) denote the zero section of \(T^*M\). The author reproves the following cup-length estimate:
\[
\sharp \varphi (O_m)\cap O_M \geq cl(M).
\]
The method is making use of the Lagrangian properties of spectral invariants from Floer theory (see [\textit{A. Floer}, Commun. Pure Appl. Math. 42, No. 4, 335--356 (1989; Zbl 0683.58017); \textit{Y.-G. Oh}, J. Differ. Geom. 46, No. 3, 499--577 (1997; Zbl 0926.53031); Commun. Anal. Geom. 7, No. 1, 1--55 (1999; Zbl 0966.53055)]).
Reviewer: Zdzisław Dzedzej (Gdańsk)Lagrangian fields, Calabi functions, and local symplectic groupoidshttps://zbmath.org/1522.530792023-12-07T16:00:11.105023Z"Karabegov, Alexander"https://zbmath.org/authors/?q=ai:karabegov.alexander-vSummary: A Lagrangian field on a symplectic manifold \(M\) is a family \({\Lambda} = \{ {\Lambda}_x | x \in M \}\) of pointed Lagrangian submanifolds of \(M\). This notion is a generalization of a real Lagrangian polarization for which each \({\Lambda}_x\) is the leaf containing \(x\). Two Lagrangian fields \(\Lambda\) and \(\widetilde{{\Lambda}}\) are called transversal if \({\Lambda}_x\) intersects \(\widetilde{{\Lambda}}_x\) transversally at \(x\) for every \(x \in M\). Two transversal Lagrangian fields determine an almost para-Kähler structure on \(M\). We construct a local symplectic groupoid on a neighborhood of the zero section of \(T^\ast M\) from two transversal Lagrangian fields on \(M\). The Lagrangian manifold of \(n\)-cycles of this groupoid in \((T^\ast M)^n\) has a generating function whose germ around the diagonal of \(M^n\) is given by the \(n\)-point cyclic Calabi function of a closed (1,1)-form on a neighborhood of the diagonal of \(M^2\) obtained from the symplectic form on \(M\).The strong homotopy structure of BRST reductionhttps://zbmath.org/1522.530802023-12-07T16:00:11.105023Z"Esposito, Chiara"https://zbmath.org/authors/?q=ai:esposito.chiara"Kraft, Andreas"https://zbmath.org/authors/?q=ai:kraft.andreas"Schnitzer, Jonas"https://zbmath.org/authors/?q=ai:schnitzer.jonasThis paper proposes a reduction scheme for equivariant polydifferential operators. Let us recall that in the theory of deformation quantization, the phase space is described by a Poisson manifold \(M\). Quantifying \(M\) refers to the construction of a star product on \(M\). Let us recall that the existence and the classification of star products in the setting of general Poisson manifolds are obtained by Kontsevich's formality theorem. The motivation of this paper is to investigate the compatibility of deformation and phase space reduction in the Poisson setting. The authors obtain the desired reduction \(L_{\infty}\)-morphism by applying an explicit version of the homotopy transfer theorem. Finally, the authors prove that the reduced star product induced by reduction coincides with the reduced star product obtained via the formal Koszul complex.
Reviewer: Angela Gammella-Mathieu (Metz)Quantization of restricted Lagrangian subvarieties in positive characteristichttps://zbmath.org/1522.530812023-12-07T16:00:11.105023Z"Mundinger, Joshua"https://zbmath.org/authors/?q=ai:mundinger.joshuaSummary: \textit{R. Bezrukavnikov} and \textit{D. Kaledin} [J. Am. Math. Soc. 21, No. 2, 409--438 (2008; Zbl 1138.53067)] introduced quantizations of symplectic varieties \(X\) in positive characteristic which endow the Poisson bracket on \(X\) with the structure of a restricted Lie algebra. We consider deformation quantization of line bundles on Lagrangian subvarieties \(Y\) of \(X\) to modules over such quantizations. If the ideal sheaf of \(Y\) is a restricted Lie subalgebra of the structure sheaf of \(X\), we show that there is a certain cohomology class which vanishes if and only if a line bundle on \(Y\) admits a quantization.On naturality of the Ozsváth-Szabó contact invarianthttps://zbmath.org/1522.570322023-12-07T16:00:11.105023Z"Hedden, Matthew"https://zbmath.org/authors/?q=ai:hedden.matthew"Tovstopyat-Nelip, Lev"https://zbmath.org/authors/?q=ai:tovstopyat-nelip.levSummary: We discuss functoriality properties of the Ozsváth-Szabó contact invariant, and expose a number of results which seemed destined for folklore. We clarify the (in)dependence of the invariant on the basepoint, prove that it is functorial with respect to contactomorphisms, and show that it is strongly functorial under Stein cobordisms.
For the entire collection see [Zbl 1515.57005].Every real 3-manifold is real contacthttps://zbmath.org/1522.570432023-12-07T16:00:11.105023Z"Cengiz, Merve"https://zbmath.org/authors/?q=ai:cengiz.merve"Öztürk, Ferit"https://zbmath.org/authors/?q=ai:ozturk.feritThe authors re-prove several classical results from 3-dimensional contact topology in the category of real contact manifolds. A real contact 3-manifold is a closed manifold together with an orientation-preserving involution \(c_M\) such that the fixed point set of \(c_M\) is either empty or 1-dimensional, and a real contact structure, i.e., a contact structure such that
\begin{itemize}
\item \(c_M\) preserves the contact structure,
\item \(dc_M\) reverses the co-orientation of the contact structure.
\end{itemize}
The main result of the paper states that every real 3-manifold admits a real contact structure. The most important tools developed in this paper to prove this result are a \(c_M\)-equivariant version of contact Dehn-surgery and a Lickorish-Wallace theorem for real manifolds, stating that any closed orientable real 3-manifold can be obtained via equivariant Dehn-surgery along a certain link in the standard real \(S^3\). As an application it is shown that the tight contact structures on \(S^2\times S^1\) and on certain lens spaces are real.
Reviewer: Jakob Hedicke (Montréal)Systoles and Lagrangians of random complex algebraic hypersurfaceshttps://zbmath.org/1522.600212023-12-07T16:00:11.105023Z"Gayet, Damien"https://zbmath.org/authors/?q=ai:gayet.damienAs it is well known, the systole \(\mathrm{Sys}(M,g)\) of a complete Riemannian manifold \((M,g)\) is the infimum of the length of non-contractible loops in \((M,g)\). When \(M\) is a closed manifold, the systole is realized by the shortest non-contractible geodesic loop. The classical isosystolic inequalities take the form \(\mathrm{Vol}(M,g)\geq k (\mathrm{Sys}(M,g))^{n}\), for any metric \(g\) on the \(n\)-dimensional manifold \(M\). The first of such type of inequalities was obtained by Loewner in an unpublished paper written in 1949 where he proved that for any surface \(S\) of the topological type of a torus, \(\textit{Area}(S,g) \geq \frac{\sqrt{3}}{2} (\mathrm{Sys}(S,g))^{2}\).
One of the goals of the paper under review is to obtain a uniform positive lower bound for the probability that a projective complex curve in \(\mathbb{C}P^{2}\) of given degree equipped with the restriction of the ambient metric has a systole of small size, which is an analog of a similar bound for hyperbolic curves given by \textit{M. Mirzakhani} [J. Differ. Geom. 94, No. 2, 267--300 (2013; Zbl 1270.30014)]. In order to prove that, the author obtains some results about Lagrangian submanifolds on a smooth complex projective hypersurface in \(\mathbb{C}P^{n}\). Although these are deterministic results, they are obtained as consequences of a more precise probabilistic theorem proved in the present paper, which is inspired by a 2014 result by \textit{D. Gayet} and \textit{J.-Y. Welschinger} [J. Lond. Math. Soc., II. Ser. 90, No. 1, 105--120 (2014; Zbl 1326.14139)] on random real algebraic geometry, together with quantitative Moser-type constructions.
Reviewer: Fernando Etayo Gordejuela (Santander)Constraining Weil-Petersson volumes by universal random matrix correlations in low-dimensional quantum gravityhttps://zbmath.org/1522.830822023-12-07T16:00:11.105023Z"Weber, Torsten"https://zbmath.org/authors/?q=ai:weber.torsten"Haneder, Fabian"https://zbmath.org/authors/?q=ai:haneder.fabian"Richter, Klaus"https://zbmath.org/authors/?q=ai:richter.klaus-jurgen"Urbina, Juan Diego"https://zbmath.org/authors/?q=ai:urbina.juan-diegoSummary: Based on the discovery of the duality between Jackiw-Teitelboim quantum gravity and a double-scaled matrix ensemble by \textit{P. Saad} et al. in [``JT gravity as a matrix integral'', Preprint, \url{arXiv:1903.11115}], we show how consistency between the two theories in the universal random matrix theory (RMT) limit imposes a set of constraints on the volumes of moduli spaces of Riemannian manifolds. These volumes are given in terms of polynomial functions, the Weil-Petersson (WP) volumes, solving a celebrated nonlinear recursion formula that is notoriously difficult to analyse. Since our results imply \textit{linear} relations between the coefficients of the WP volumes, they therefore provide both a stringent test for their symbolic calculation and a possible way of simplifying their construction. In this way, we propose a long-term program to improve the understanding of mathematically hard aspects concerning moduli spaces of hyperbolic manifolds by using universal RMT results as input.