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The main goal of this paper is to give constructive proofs of several existence results for symplectic embeddings. The strong relation between symplectic packings and singular symplectic curves, which can be derived from McDuff’s inflations on the blow-ups, is revisited through a new inflation technique that lives at the level of the manifold. As an application, we explain constructions of maximal symplectic packings of P by 6, 7 or 8 balls.

We build a bridge between Floer theory on open symplectic manifolds and the enumerative geometry of holomorphic disks inside their Fano compactifications, by detecting elements in symplectic cohomology which are mirror to Landau-Ginzburg potentials. We also treat the higher Maslov index versions of the potentials. We discover a relation between higher disk potentials and symplectic cohomology rings of smooth anticanonical divisor complements (themselves conjecturally related to closed-string Gromov-Witten invariants), and explore several other applications to the geometry of Liouville domains., 48 pages, 13 figures; v2: reference fixes, minor corrections; v3: minor changes, accepted version

The aim of this paper is to show that the canonical quantization of the moment maps on symplectic vector spaces naturally gives rise to the oscillator representations. More precisely, let $(W,\omega)$ denote a real symplectic vector space, on which a Lie group $G$ acts symplectically on the left, where $G$ denotes a real reductive Lie group $\mathrm{Sp}(n,\mathbb R), \mathrm U(p,q)$ or $\mathrm O^*(2n)$ in this paper. Then we quantize the moment map $\mu: W \to \mathfrak g_0^*$, where $\mathfrak g_0^*$ denotes the dual space of the Lie algebra $\mathfrak g_0$ of $G$. Namely, after taking a complex Lagrangian subspace $V$ of the complexification of $W$, we assign an element of the Weyl algebra for $V$ to $< \mu, X >$, which we denote by $< \hat{\mu}, X >$, for each $X \in \mathfrak g_0$. It is shown that the map $X \mapsto \mathrm i $ gives a representation of $\mathfrak g_0$ which extends to the one of $\mathfrak g$, the complexification of $\mathfrak g_0$, by linearity. With a suitable choice of the complex Lagrangian subspace $V$ in each case, the representation coincides with the oscillator representation of $\mathfrak g$., Comment: 24pages, no figure; corrected some typos (v2); 27pages, added Section 5 describing a relation between the image of complex Lagrangian subspaces by the moment map and the associated variety of the corresponding irreducible (g,K)-modules (v3); some references are added and replaced (v4)

Consider a spin manifold M, equipped with a line bundle L and an action of a compact Lie group G. We can attach to this data a family Theta(k) of distributions on the dual of the Lie algebra of G. The aim of this paper is to study the asymptotic behaviour of Theta(k) when k is large, and M possibly non compact, and to explore a functorial consequence of this formula for reduced spaces., We have modified the section "Quasi-polynomials and asymptotics"

For a complex reductive group G acting linearly on a complex affine space V with respect to a character, we show two stratifications of V associated to this action (and a choice of invariant inner product on the Lie algebra of the maximal compact subgroup of G) coincide. The first is Hesselink's stratification by adapted 1-parameter subgroups and the second is the Morse theoretic stratification associated to the norm square of the moment map. We also give a proof of a version of the Kempf-Ness theorem which states that the GIT quotient is homeomorphic to the symplectic reduction (both taken with respect to the character). Finally, for the space of representations of a quiver of fixed dimension, we show that the Morse theoretic stratification and Hesselink's stratification coincide with the stratification by Harder-Narasimhan types., 24 pages

We study the Ricci tensor of left-invariant pseudoriemannian metrics on Lie groups. For an appropriate class of Lie groups that contains nilpotent Lie groups, we introduce a variety with a natural $\mathrm{GL}(n,\mathbb{R})$ action, whose orbits parametrize Lie groups with a left-invariant metric; we show that the Ricci operator can be identified with the moment map relative to a natural symplectic structure. From this description we deduce that the Ricci operator is the derivative of the scalar curvature $s$ under gauge transformations of the metric, and show that Lie algebra derivations with nonzero trace obstruct the existence of Einstein metrics with $s\neq0$. Using the notion of nice Lie algebra, we give the first example of a left-invariant Einstein metric with $s\neq0$ on a nilpotent Lie group. We show that nilpotent Lie groups of dimension $\leq 6$ do not admit such a metric, and a similar result holds in dimension $7$ with the extra assumption that the Lie algebra is nice., 24 pages; v2, improved criterion for nonexistence added (Theorem 4.1), proofs simplified and contents reorganized accordingly, two references added; v3, corrected a constant in Theorem 3.8 and Corollary 3.10, minor correction in the proofs, added examples of Einstein metrics in any indefinite signature, presentation improved, three references added. To appear in J. Pure Appl. Algebra

Let $K$ be a compact Lie group with complexification $G$, and let $V$ be a unitary $K$-module. We consider the real symplectic quotient $M_0$ at level $0$ of the homogeneous quadratic moment map as well as the complex symplectic quotient, defined here as the complexification of $M_0$. We show that if $(V, G)$ is $3$-large, a condition that holds generically, then the complex symplectic quotient has symplectic singularities and is graded Gorenstein. This in particular implies that the real symplectic quotient is graded Gorenstein. In the case that $K$ is a torus or $\operatorname{SU}_2$, we show that these results hold without the hypothesis that $(V,G)$ is $3$-large., 35 pages. v2: Fixed gap in first version (the proof of Corollary 3.14 in v1 only applies if the slice is taken at a real point). Hypothesis for the main results changed from 2-large to 3-large, added Sections 3.3 and 3.5, and improved exposition. v3: Added Lemma 3.17, improved exposition. Final preprint version accepted by Compositio Mathematica (not the published version)

Let $X$ be a compact toric K\"ahler manifold with $-K_X$ nef. Let $L\subset X$ be a regular fiber of the moment map of the Hamiltonian torus action on $X$. Fukaya-Oh-Ohta-Ono defined open Gromov-Witten (GW) invariants of $X$ as virtual counts of holomorphic discs with Lagrangian boundary condition $L$. We prove a formula which equates such open GW invariants with closed GW invariants of certain $X$-bundles over $\mathbb{P}^1$ used to construct the Seidel representations for $X$. We apply this formula and degeneration techniques to explicitly calculate all these open GW invariants. This yields a formula for the disc potential of $X$, an enumerative meaning of mirror maps, and a description of the inverse of the ring isomorphism of Fukaya-Oh-Ohta-Ono., Comment: v4: 44 pages, 3 figures, minor modifications, final version to appear in DMJ

We construct examples of cohomogeneity one special Lagrangian submanifolds in the cotangent bundle over the complex projective space, whose Calabi-Yau structure was given by Stenzel. For each example, we describe the condition of special Lagrangian as an ordinary differential equation. Our method is based on a moment map technique and the classification of cohomogeneity one actions on the complex projective space classified by Takagi., 14 pages

We define an equivariant index of $\operatorname{Spin}^{c}$ -Dirac operators on possibly noncompact manifolds, acted on by compact, connected Lie groups. Our main result is that the index decomposes into irreducible representations according to the quantization commutes with reduction principle.

We consider Hamiltonian systems driven by multi-dimensional Gaussian processes in rough path sense, which include fractional Brownian motions with Hurst parameter H ∈ ( 1 / 4 , 1 / 2 ] . We prove that the phase flow preserves the symplectic structure almost surely and this property could be inherited by symplectic Runge–Kutta methods, which are implicit methods in general. If the vector fields satisfy some smoothness and boundedness conditions, we obtain the pathwise convergence rates of Runge–Kutta methods. When vector fields are linear, we get the solvability of the midpoint scheme for skew symmetric cases, and obtain its pathwise convergence rate. Numerical experiments verify our theoretical analysis.

Let M be a closed symplectic manifold of volume V. We say that the symplectic packings of M by ellipsoids are unobstructed if any collection of disjoint symplectic ellipsoids (possibly of different sizes) of total volume less than V admits a symplectic embedding to M. We show that the symplectic packings by ellipsoids are unobstructed for all even-dimensional tori equipped with Kahler symplectic forms and all closed hyperkahler manifolds of maximal holonomy, or, more generally, for closed Campana simple manifolds (that is, Kahler manifolds that are not unions of their complex subvarieties), as well as for any closed Kahler manifold which is a limit of Campana simple manifolds in a smooth deformation. The proof involves the construction of a Kahler resolution of a Kahler orbifold with isolated singularities and relies on the results of Demailly-Paun and Miyaoka on Kahler cohomology classes., References added. To appear in Selecta Mathematica

The various scalar curvatures on an almost Hermitian manifold are studied, in particular with respect to conformal variations. We show several integrability theorems, which state that two of these can only agree in the K\"ahler case. Our main question is the existence of almost K\"ahler metrics with conformally constant Chern scalar curvature. This problem is completely solved for ruled manifolds and in a complementary case where methods from the Chern-Yamabe problem are adapted to the non-integrable case. Also a moment map interpretation of the problem is given, leading to a Futaki invariant and the usual picture from geometric invariant theory., Comment: 26 pages

In this paper we develop the basic tools for a classification of Killing vector fields of constant length on pseudo-Riemannian homogeneous spaces. This extends a recent paper of M. Xu and J. A. Wolf, which classified the pairs $(M,\xi)$ where $M = G/H$ is a Riemannian normal homogeneous space, G is a compact simple Lie group, and $\xi \in \mathfrak{g}$ defines a nonzero Killing vector field of constant length on $M$. The method there was direct computation. Here we make use of the moment map $M \to \mathfrak{g}^{*}$ and the flag manifold structure of $\mathrm{Ad} (G) \xi$ to give a shorter, more geometric proof which does not require compactness and which is valid in the pseudo-Riemannian setting. In that context we break the classification problem into three parts. The first is easily settled. The second concerns the cases where $\xi$ is elliptic and $G$ is simple (but not necessarily compact); that case is our main result here. The third, which remains open, is a more combinatorial problem involving elements of the first two.

If $(G,V)$ is a polar representation with Cartan subspace $\mathfrak c$ and Weyl group $W$, it is shown that there is a natural morphism of Poisson schemes $\mathfrak c \oplus {\mathfrak c}^*/W \to V\oplus V^*/\!\!/\!\!/ G$. This morphism is conjectured to be an isomorphism of the underlying reduced varieties if $(G,V)$ is visible. The conjecture is proved for visible stable locally free polar representations and certain further examples., Comment: 18 pages, final version

We introduce a vector bundle version of the complex Monge-Ampere equation motivated by a desire to study stability conditions involving higher Chern forms. We then restrict ourselves to complex surfaces, provide a moment map interpretation of it, and define a positivity condition (MA positivity) which is necessary for the infinite-dimensional symplectic form to be Kahler. On rank-2 bundles on compact complex surfaces, we prove two consequences of the existence of a "positively curved" solution to this equation - Stability (involving the second Chern character) and a Kobayashi-Lubke-Bogomolov-Miyaoka-Yau type inequality. Finally, we prove a Kobayashi-Hitchin correspondence for a dimensional reduction of the aforementioned equation., Comment: Corrected the statement of Lemma 2.4 and the proof of Lemma 4.9

We make use of $\mathcal{F}$-structures and technology developed by Paternain - Petean to compute minimal entropy, minimal volume, and Yamabe invariant of symplectic 4-manifolds, as well as to study their collapse with sectional curvature bounded from below. \`A la Gompf, we show that these invariants vanish on symplectic 4-manifolds that realize any given finitely presented group as their fundamental group. We extend to the symplectic realm a result of LeBrun which relates the Kodaira dimension with the Yamabe invariant of compact complex surfaces., Comment: v2. added a reference and corrected a typo

We prove that the kernels of the restrictions of symplectic Dirac or symplectic Dirac-Dolbeault operators on natural subspaces of polynomial valued spinor fields are finite dimensional on a compact symplectic manifold. We compute those kernels for the complex projective spaces. We construct injections of subgroups of the symplectic group (the pseudo-unitary group and the stabilizer of a Lagrangian subspace) in the group Mpc and classify G-invariant Mpc-structures on symplectic spaces with a G-action. We prove a variant of Parthasarathy's formula for the commutator of two symplectic Dirac-type operators on a symmetric symplectic space.

We study a class of smooth torus manifolds whose orbit space has the combinatorial structure of a simple polytope with holes. We construct moment angle manifolds for such polytopes with holes and use them to prove that the associated torus manifolds admit stable almost complex structure. We give a combinatorial formula for the Hirzebruch $\chi_y$ genus of these torus manifolds. We show that they have (invariant) almost complex structure if they admit positive omniorientation. We give examples of almost complex manifolds that do not admit a complex structure. When the dimension is four, we calculate the homology groups and describe a method for computing the cohomology ring., Comment: Minor changes in previous version, 14 pages, 3 figures, to appear in Proceedings of the American Mathematical Society

We consider base spaces of Lagrangian fibrations from singular symplectic varieties. After defining cohomologically irreducible symplectic varieties, we construct an example of Lagrangian fibration whose base space is isomorphic to a quotient of the projective space. We also prove that the base space of Lagrangian fibration from a cohomologically symplectic variety is isomorphic to the projective space provided that the base space is smooth.

In this paper, the authors study the homologically trivial symplectic group actions on homotopy elliptic surfaces E(n) and get some rigidity results.

Let G be a connected and simply connected solvable Lie group. The moment map for π in G ˆ , unitary dual of G, sends smooth vectors in the representation space of π to g ⁎ , dual space of g . The closure of the image of the moment map for π is called its moment set, denoted by I π . Generally, the moment set I π , π ∈ G ˆ does not characterize π, even for generic representations. However, we say that G ˆ is moment separable when the moment sets differ for any pair of distinct irreducible unitary representations. In the case of an exponential solvable Lie group G, D. Arnal and M. Selmi exhibited an accurate construction of an overgroup G + , containing G as a subgroup and an injective map Φ from G ˆ into G + ˆ in such a manner that Φ ( G ˆ ) is moment separable and I Φ ( π ) characterizes π, π ∈ G ˆ . In this work, we provide the existence of a quadratic overgroup for the diamond Lie group, which is the semi-direct product of R n with ( 2 n + 1 ) -dimensional Heisenberg group for some n ⩾ 1 .

We construct a corank one Poisson manifold which is of strong compact type, i.e., the associated Lie algebroid structure on its cotangent bundle is integrable, annd the source 1-conected (symplectic) integration is compact. The construction relies on the moduli of marked K3 surfaces., Comment: Abstract added. Bibliography updated. Journal reference added

In this paper, the relations among four finite-dimensional Hamiltonian systems associated with the Dirac hierarchy are studied via three Poisson maps from the standard symplectic structures to the Lie–Poisson structure of Lie algebra sl(2). It has shown that the canonical Hamiltonian systems in the standard symplectic structures are the different symplectic realizations of the Lie–Poisson Hamiltonian system. The action-angle coordinates and the Jacobi inversion problem for the Lie–Poisson Hamiltonian systems are also investigated in detail.

Let M be a 2n-dimensional smooth manifold. We study local properties for a symplectic pair on M which is a pair of closed 2-forms of constant ranks with complementary kernel foliations. We show that some neighborhood theorems for submanifolds of M and Darboux type theorem hold. For more general case of recursion operators intertwining a pair of symplectic forms, we also prove that some neighborhood theorem holds.

We prove a Sturmian separation theorem comparing the number of focal points of any conjoined basis of a nonoscillatory and controllable (near ~) symplectic difference system with the number of focal points of the recessive solution at ~. We also present various extensions of this statement.

The authors examine the quantization commutes with reduction phenomenon for Hamiltonian actions of compact Lie groups on closed symplectic manifolds from the point of view of topological K-theory and K-homology. They develop the machinery of K-theory wrong-way maps in the context of orbifolds and use it to relate the quantization commutes with reduction phenomenon to Bott periodicity and the K-theory formulation of the Weyl character formula.

The group SSympeo(M, ω) of strong symplectic homeomorphisms or group of ssympeomorphisms of a closed connected symplectic manifold (M, ω) was defined and studied in [2], [3]. In these papers the author uses the L(1,∞)-metric on the group Iso(M, ω) of all symplectic isotopies. In this paper we study the set SSympeo(M, ω)∞ of ssympeomorphisms in the L∞- metric. We prove the equality between SSympeo(M, ω) and SSympeo(M, ω)∞. This generalizes Müller’s result [6] asserting that Hameo(M, ω) = Hameo(M, ω)∞.

The phase topology of a special case of Goryachev integrability in the problem of motion of a rigid body in a fluid is investigated using the method of Boolean functions, which was developed by Kharlamov for algebraically separated systems. The bifurcation diagram of the moment map is found and the Fomenko invariant, which classifies the systems up to rough Liouville equivalence, is specified. Bibliography: 15 titles.

In this paper, we introduce combinatorially defined subvarieties of symplectic flag varieties called symplectic restriction varieties. We study their geometric properties and compute their cohomology classes. In particular, we give a positive, combinatorial, geometric branching rule for computing the map in cohomology induced by the inclusion of the symplectic partial flag variety SF ( k 1 , … , k h ; n ) in the partial flag variety F ( k 1 , … , k h ; n ) . These rules have many applications in algebraic geometry, combinatorics, symplectic geometry and representation theory.

The main aim of this paper is the description of a large class of lattices in some nilpotent Lie groups, sometimes filiformes, carrying a flat left invariant linear connection anf often a left invariant symplectic form. As a consequence we obtain an infinity of, non homeomorphic, compact affine or symplectic manifolds. We review some new facts about the geometry of compact symplectic nilmanifolds and we describe symplectic reduction for these manifolds. For the Heisenberg Lie group, defined over a local associative and commutative finite dimensional real algebra, a necessary and suffisant condition for the existence of a left invarian symplectic form, is given. In the symplectic case we show a that a lattice in the group determines naturally lattices in the double Lie group corresponding to any solution of the classical Yang-Baxter equation., 25 pages, no figures, corrected typos

We detect, by using symplectic topology, invariant measures with large rotation vectors for a class of Hamiltonian flows., 15 pages. Major revision (simplified and generalized)

Let \({(G,\tilde{G})}\) be a reductive dual pair over a local field \({\mathfrak{k}}\) of characteristic 0, and denote by V and \({\tilde{V}}\) the standard modules of G and \({\tilde{G}}\) , respectively. Consider the set \({{\rm Max} {\rm Hom} (V,\tilde{V})}\) of full rank elements in \({{\rm Hom}(V,\tilde{V})}\) , and the nilpotent orbit correspondence \({\mathcal{O} \subset \mathfrak{g}}\) and \({\Theta (\mathcal{O})\subset \tilde{\mathfrak{g}}}\) induced by elements of \({{\rm Max} {\rm Hom} (V,\tilde{V})}\) via the moment maps. Let \({(\pi,\fancyscript{V})}\) be a smooth irreducible representation of G. We show that there is a correspondence of the generalized Whittaker models of π of type \({\mathcal{O}}\) and of Θ (π) of type \({\Theta (\mathcal{O})}\) , where Θ (π) is the full theta lift of π. When \({(G,\tilde{G})}\) is in the stable range with G the smaller member, every nilpotent orbit \({\mathcal{O} \subset \mathfrak{g}}\) is in the image of the moment map from \({{\rm Max} {\rm Hom} (V,\tilde{V})}\). In this case, and for \({\mathfrak{k}}\) non-Archimedean, the result has been previously obtained by Mœglin in a different approach.

We present new relations connecting the number of focal points of conjoined bases of eventually disconjugate symplectic difference systems with different coefficient matrices which obey the so-called majorant condition at For the case of controllable (near ) symplectic systems we investigate connections between the number of focal points of their recessive solutions. The results of the paper are based on the concept of minimal and maximal solutions of the associated Riccati matrix difference equations and the comparative index theory for discrete symplectic systems.

In this paper a compact and semisimple Lie group G is considered endowed with a 2-plectic structure ω, induced by the Killing form. We show that the Lie group of 2-plectomorphisms of G is finite dimensional and compact, and hence the Darboux’s theorem fails to be true for this 2-plectic structure. Also it is shown that ω induces a left-invariant $${\mathfrak{g}^{*}}$$ valued 2-form which is proportional to dΘ, where Θ is the Cartan–Maurer 1-form on G. At last we consider the action of G on its tangent bundle which is furnished with the 2-plectic structure ω c , the complete lift of ω, and calculate covariant momentum map of this action.

In this paper, we study the global singular symplectic flops related to the following affine hypersurface with cyclic quotient singularities, $$V_{r,b} = \{ (x,y,z,t) \in \mathbb{C}^4 |xy - z^{2r} + t^2 = 0\} /\mu _r (a, - a,b,0),r \geqslant 2,$$ where b = 1 appears in Mori’s minimal model program and b ≠ 1 is a new class of singularities in symplectic birational geometry. We prove that two symplectic 3-orbifolds which are singular flops to each other have isomorphic Ruan cohomology rings. The proof is based on the symplectic cutting argument and virtual localization technique.

A numerical method for solving Hamiltonian equations is said to be symplectic if it preserves the symplectic structure associated with the equations. Various symplectic methods are widely used in many fields of science and technology. A symplectic method preserves an approximate Hamiltonian perturbed from the original Hamiltonian. It theoretically supports the effectiveness of symplectic methods for long-term integration. Although it is also related to long-term integration, numerical stability of symplectic methods have received little attention. In this paper, we consider explicit symplectic methods defined for Hamiltonian equations with Hamiltonians of the special form, and study their numerical stability using the harmonic oscillator as a test equation. We propose a new stability criterion and clarify the stability of some existing methods that are visually based on the criterion. We also derive a new method that is better than the existing methods with respect to a Courant-Friedrichs-Lewy condition for hyperbolic equations; this new method is tested through a numerical experiment with a nonlinear wave equation.

In this note we discuss the effect of the symplectic sum along spheres in symplectic four-manifolds on the Kodaira dimension of the underlying symplectic manifold. We find that the Kodaira dimension is non-decreasing. Moreover, we are able to obtain precise results on the structure of the manifold obtained from the blow down of an embedded symplectic -4-sphere., 26 pages, 1 table

Let G be a real reductive Lie group, let H=TA be the identity component of a Cartan subgroup, and let h be the corresponding Cartan subalgebra. This leads to a parabolic subgroup of G whose identity component is MAN. The unitary G-representations induced by MAN are known as the H-series. We study symplectic geometry of G×h and apply geometric quantization to construct unitary G-representations by partially harmonic forms. They are direct integrals of the H-series, indexed by the image of the moment map. We also perform symplectic reduction and symplectic induction, and consider their analogues in representation theory via geometric quantization.

We construct a canonical Hausdorff complex analytic moduli space of Fano manifolds with Kahler–Ricci solitons. This enlarges the moduli space of Fano manifolds with Kahler–Einstein metrics. We discover a moment map picture for Kahler–Ricci solitons, and give complex analytic charts on the topological space consisting of Kahler–Ricci solitons, by studying differential geometric aspects of this moment map. Some stacky words and arguments on Gromov–Hausdorff convergence help to glue them together in the holomorphic manner.

We analyze the inherent symmetries associated to the non-Abelian topological BF theory from the geometric and covariant perspectives of the Lagrangian and the multisymplectic formalisms. At the Lagrangian level, we classify the symmetries of the theory as natural and Noether symmetries and construct the associated Noether currents, while at the multisymplectic level the symmetries of the theory arise as covariant canonical transformations. These transformations allowed us to build within the multisymplectic approach, in a complete covariant way, the momentum maps which are analogous to the conserved Noether currents. The covariant momentum maps are fundamental to recover, after the space plus time decomposition of the background manifold, not only the extended Hamiltonian of the BF theory but also the generators of the gauge transformations which arise in the instantaneous Dirac-Hamiltonian analysis of the first-class constraint structure that characterizes the BF model under study. To the best of our knowledge, this is the first non-trivial physical model associated to General Relativity for which both natural and Noether symmetries have been analyzed at the multisymplectic level. Our study shed some light on the understanding of the manner in which the generators of gauge transformations may be recovered from the multisymplectic formalism for field theory., Comment: 28 pages, no figures

We describe the relation between the Chern–Simons gauge theory partition function and the partition function defined using the symplectic action functional as the Lagrangian. We show that the partition functions obtained using these two Lagrangians agree, and we identify the semiclassical formula for the partition function defined using the symplectic action functional. We also compute the semiclassical formulas for the partition functions obtained using the two different Lagrangians: the Chern–Simons functional and the symplectic action functional.