Symplectic Dolbeault operators on Kähler manifolds

For a Kähler manifold $M$ , the “symplectic Dolbeault operators” are defined using the symplectic spinors and associated Dirac operators, in complete analogy to how the usual Dolbeault operators, $\bar{\partial }$ and $\bar{\partial }^*$ , arise from Dirac operators on the canonical complex spinors on $M$ . We give special attention to two special classes of Kähler manifolds: Riemann surfaces and flag manifolds ( $G/T$ for $G$ a simply-connected compact semisimple Lie group and $T$ a maximal torus). For Riemann surfaces, the symplectic Dolbeault operators are elliptic and we compute their indices. In the case of flag manifolds, we will see that the representation theory of $G$ plays a role and that these operators can be used to distinguish (as Kähler manifolds) between the flag manifolds corresponding to the Lie algebras $B_n$ and $C_n$ . We give a thorough analysis of these operators on $\mathbb{C } P^1$ (the intersection of these classes of spaces), where the symplectic Dolbeault operators have an especially interesting structure.     

Generalized quaternionic manifolds

We initiate the study of the generalized quaternionic manifolds by classifying the generalized quaternionic vector spaces, and by giving two classes of nonclassical examples of such manifolds. Thus, we show that any complex symplectic manifold is endowed with a natural (nonclassical) generalized quaternionic structure, and the same applies to the heaven space of any three-dimensional Einstein–Weyl space. In particular, on the product \(Z\) of any complex symplectic manifold \(M\) and the sphere, there exists a natural generalized complex structure, with respect to which \(Z\) is the twistor space of  \(M\) .     

EXOTIC SYMMETRIC SPACE OVER A FINITE FIELD,III

Let \( \mathcal{X}=G/H\times V \) , where V is a symplectic space such that G = GL(V) and H = Sp(V). In previous papers, the authors constructed character sheaves on \( \mathcal{X} \) , based on the explicit data. On the other hand, there exists a conceptual definition of character sheaves on \( \mathcal{X} \) based on the idea of Ginzburg in the case of symmetric spaces. Our character sheaves form a subset of Ginzburg type character sheaves. In this paper we show that these two definitions actually coincide, which implies a classification of Ginzburg type character sheaves on \( \mathcal{X} \) .     

Construction of error-tolerance pooling designs in symplectic spaces

The paper provides the construction of error-correcting pooling designs with the incidence matrix of two types of subspaces of symplectic spaces over finite fields. As a generalization of Guo et al.’s matrix, we use the general subspaces of type $(m,s)$ to substitute special subspaces of type $(2s,s)$ . If $\nu $ is big enough, we can get the higher degree of error-tolerant property.     

Invariant Hilbert schemes and desingularizations of symplectic reductions for classical groups

Let $G \subset GL(V)$ be a reductive algebraic subgroup acting on the symplectic vector space $W=(V \oplus V^*)^{\oplus m}$ , and let $\mu :\ W \rightarrow Lie(G)^*$ be the corresponding moment map. In this article, we use the theory of invariant Hilbert schemes to construct a canonical desingularization of the symplectic reduction $\mu ^{-1}(0)/\!/G$ for classes of examples where $G=GL(V)$ , $O(V)$ , or $Sp(V)$ . For these classes of examples, $\mu ^{-1}(0)/\!/G$ is isomorphic to the closure of a nilpotent orbit in a simple Lie algebra, and we compare the Hilbert–Chow morphism with the (well-known) symplectic desingularizations of $\mu ^{-1}(0)/\!/G$ .     

Self-adjoint curl operators

We study the exterior derivative as a symmetric unbounded operator on square integrable 1-forms on a 3D bounded domain D. We aim to identify boundary conditions that render this operator self-adjoint. By the symplectic version of the Glazman-Krein-Naimark theorem, this amounts to identifying complete Lagrangian subspaces of the trace space of H(curl, D) equipped with a symplectic pairing arising from the ${\wedge}$ -product of 1-forms on ${\partial D}$ . Substantially generalizing earlier results, we characterize Lagrangian subspaces associated with closed and co-closed traces. In the case of non-trivial topology of the domain, different contributions from co-homology spaces also distinguish different self-adjoint extensions. Finally, all self-adjoint extensions discussed in the paper are shown to possess a discrete point spectrum, and their relationship with curl curl-operators is discussed.     

Symplectic homology of displaceable Liouville domains and leafwise intersection points

In this note we prove that the symplectic homology of a Liouville domain $W$ displaceable in the symplectic completion vanishes. Nevertheless if the Euler characteristic of $(W,\partial W)$ is odd, the filtered symplectic homologies of $W$ do not vanish and give rise to leafwise intersection points on the symplectic completion of $W$ for a perturbation displacing $W$ from itself. In contrast to the existing results we can find a leafwise intersection point for a given period but its energy varies by period instead.     

Semi-Invariants of Symmetric Quivers of Tame Type

A symmetric quiver (Q, σ) is a finite quiver without oriented cycles Q?=?(Q ₀, Q ₁) equipped with a contravariant involution σ on $Q_0\sqcup Q_1$ . The involution allows us to define a nondegenerate bilinear form $\langle -,-\rangle_V$ on a representation V of Q. We shall say that V is orthogonal if $\langle -,-\rangle_V$ is symmetric and symplectic if $\langle -,-\rangle_V$ is skew-symmetric. Moreover, we define an action of products of classical groups on the space of orthogonal representations and on the space of symplectic representations. So we prove that if (Q, σ) is a symmetric quiver of tame type then the rings of semi-invariants for this action are spanned by the semi-invariants of determinantal type c ^V and, when the matrix defining c ^V is skew-symmetric, by the Pfaffians pf ^V . To prove it, moreover, we describe the symplectic and orthogonal generic decomposition of a symmetric dimension vector.     

Proportionality principle for the simplicial volume of families of \mathbb Q -rank 1 locally symmetric spaces

We establish the proportionality principle between the Riemannian volume and locally finite simplicial volume for $\mathbb Q $ -rank 1 locally symmetric spaces covered by products of hyperbolic spaces, giving the first examples for manifolds whose cusp groups are not necessarily amenable. Also, we give a simple direct proof of the proportionality principle for the locally finite simplicial volume and the relative simplicial volume of $\mathbb Q $ -rank $1$ locally symmetric spaces with amenable cusp groups established by Löh and Sauer [26].     

Special symplectic Lie groups and hypersymplectic Lie groups

A special symplectic Lie group is a triple ${(G,\omega,\nabla)}$ such that G is a finite-dimensional real Lie group and ω is a left invariant symplectic form on G which is parallel with respect to a left invariant affine structure ${\nabla}$ . In this paper starting from a special symplectic Lie group we show how to “deform” the standard Lie group structure on the (co)tangent bundle through the left invariant affine structure ${\nabla}$ such that the resulting Lie group admits families of left invariant hypersymplectic structures and thus becomes a hypersymplectic Lie group. We consider the affine cotangent extension problem and then introduce notions of post-affine structure and post-left-symmetric algebra which is the underlying algebraic structure of a special symplectic Lie algebra. Furthermore, we give a kind of double extensions of special symplectic Lie groups in terms of post-left-symmetric algebras.     

Symplectic Twistor Operator on \mathbb R ^{2n} and the Segal–Shale–Weil Representation

The aim of our article is the study of solution space of the symplectic twistor operator $T_s$ in symplectic spin geometry on standard symplectic space $(\mathbb R ^{2n},\omega )$ , which is the symplectic analogue of the twistor operator in (pseudo) Riemannian spin geometry. In particular, we observe a substantial difference between the case $n=1$ of real dimension $2$ and the case of $\mathbb R ^{2n}, n>1$ . For $n>1$ , the solution space of $T_s$ is isomorphic to the Segal–Shale–Weil representation.     

Symplectic polarities of buildings of type E 6

A symplectic polarity of a building Δ of type E ₆ is a polarity whose fixed point structure is a building of type F ₄ containing residues isomorphic to symplectic polar spaces. In this paper, we present two characterizations of such polarities among all dualities. Firstly, we prove that, if a duality θ of Δ never maps a point to a neighbouring symp, and maps some element to a non-opposite element, then θ is a symplectic duality. Secondly, we show that, if a duality θ never maps a chamber to an opposite chamber, then it is a symplectic polarity. The latter completes the programme for dualities of buildings of type E ₆ of determining all domestic automorphisms of spherical buildings, and it also shows that symplectic polarities are the only polarities in buildings of type E ₆ for which the Phan geometry is empty.     

Torsion homology of arithmetic lattices and K_2 of imaginary fields

Let \(X = G/K\) be a symmetric space of noncompact type. A result of Gelander provides exponential upper bounds in terms of the volume for the torsion homology of the noncompact arithmetic locally symmetric spaces \(\Gamma \backslash X\) . We show that under suitable assumptions on \(X\) this result can be extended to the case of nonuniform arithmetic lattices \(\Gamma \subset G\) that may contain torsion. Using recent work of Calegari and Venkatesh we deduce from this upper bounds (in terms of the discriminant) for \(K_2\) of the ring of integers of totally imaginary number fields \(F\) . More generally, we obtain such bounds for rings of \(S\) -integers in  \(F\) .     

Fischer decomposition in symplectic harmonic analysis

In the framework of quaternionic Clifford analysis in Euclidean space \(\mathbb {R}^{4p}\) , which constitutes a refinement of Euclidean and Hermitian Clifford analysis, the Fischer decomposition of the space of complex valued polynomials is obtained in terms of spaces of so-called (adjoint) symplectic spherical harmonics, which are irreducible modules for the symplectic group Sp \((p)\) . Its Howe dual partner is determined to be \(\mathfrak {sl}(2,\mathbb {C}) \oplus \mathfrak {sl}(2,\mathbb {C}) = \mathfrak {so}(4,\mathbb {C})\) .     

Dagger Categories of Tame Relations

Within the context of an involutive monoidal category the notion of a comparison relation ${\textsf{cp} : \overline{X} \otimes X \rightarrow \Omega}$ is identified. Instances are equality = on sets, inequality ${\leq}$ on posets, orthogonality ${\perp}$ on orthomodular lattices, non-empty intersection on powersets, and inner product ${\langle {-}|{-} \rangle}$ on vector or Hilbert spaces. Associated with a collection of such (symmetric) comparison relations a dagger category is defined with “tame” relations as morphisms. Examples include familiar categories in the foundations of quantum mechanics, such as sets with partial injections, or with locally bifinite relations, or with formal distributions between them, or Hilbert spaces with bounded (continuous) linear maps. Of one particular example of such a dagger category of tame relations, involving sets and bifinite multirelations between them, the categorical structure is investigated in some detail. It turns out to involve symmetric monoidal dagger structure, with biproducts, and dagger kernels. This category may form an appropriate universe for discrete quantum computations, just like Hilbert spaces form a universe for continuous computation.     

A Topologist’s View of Chu Spaces

For a symmetric monoidal-closed category $\mathcal{X}$ and any object K, the category of K-Chu spaces is small-topological over $\mathcal{X}$ and small cotopological over $\mathcal{X}^{{{\text{op}}}}$ . Its full subcategory of $\mathcal{M}$ -extensive K-Chu spaces is topological over $\mathcal{X}$ when $\mathcal{X}$ is $\mathcal{M}$ -complete, for any morphism class $\mathcal{M}$ . Often this subcategory may be presented as a full coreflective subcategory of Diers’ category of affine K-spaces. Hence, in addition to their roots in the theory of pairs of topological vector spaces (Barr) and their connections with linear logic (Seely), the Dialectica categories (Hyland, de Paiva), and with the study of event structures for modeling concurrent processes (Pratt), Chu spaces seem to have a less explored link with algebraic geometry. We use the Zariski closure operator to describe the objects of the *-autonomous category of $\mathcal{M}$ -extensive and $\mathcal{M}$ -coextensive K-Chu spaces in terms of Zariski separation and to identify its important subcategory of complete objects.     

New Tools for Classifying Hamiltonian Circle Actions with Isolated Fixed Points

For every compact almost complex manifold \((\mathsf {M},\mathsf {J})\) equipped with a \(\mathsf {J}\) -preserving circle action with isolated fixed points, a simple algebraic identity involving the first Chern class is derived. This enables us to construct an algorithm to obtain linear relations among the isotropy weights at the fixed points. Suppose that \(\mathsf {M}\) is symplectic and the action is Hamiltonian. If the manifold satisfies an extra so-called positivity condition, then this algorithm determines a family of vector spaces that contain the admissible lattices of weights. When the number of fixed points is minimal, this positivity condition is necessarily satisfied whenever \(\dim (\mathsf {M})\le 6\) and, when \(\dim (\mathsf {M})=8\) , whenever the \(S^1\) -action extends to an effective Hamiltonian \(T^2\) -action, or none of the isotropy weights is \(1\) . Moreover, there are no known examples with a minimal number of fixed points contradicting this condition, and their existence is related to interesting questions regarding fake projective spaces. We run the algorithm for \(\dim (\mathsf {M})\le 8\) , quickly obtaining all the possible families of isotropy weights. In particular, we simplify the proofs of Ahara and Tolman for \(\dim (\mathsf {M})=6\) and, when \(\dim (\mathsf {M})=8\) , we prove that the equivariant cohomology ring, Chern classes, and isotropy weights agree with those of \({\mathbb {C}}P^4\) with the standard \(S^1\) -action (thereby proving the symplectic Petrie conjecture in this setting).     

Virtual Betti numbers and virtual symplecticity of 4-dimensional mapping tori

In this note, we compute the virtual first Betti numbers of 4-manifolds fibering over $S^1$ with prime fiber. As an application, we show that if such a manifold is symplectic with nonpositive Kodaira dimension, then the fiber itself is a sphere or torus bundle over $S^1$ . In a different direction, we prove that if the 3-dimensional fiber of such a 4-manifold is virtually fibered then the 4-manifold is virtually symplectic unless its virtual first Betti number is 1.     

On the group sheaf of \mathcal{A}-symplectomorphisms

This is a part of a further undertaking to affirm that most of classical module theory may be retrieved in the framework of Abstract Differential Geometry (à la Mallios). More precisely, within this article, we study some defining basic concepts of symplectic geometry on free \(\mathcal{A}\) -modules by focussing in particular on the group sheaf of \(\mathcal{A}\) -symplectomorphisms, where \(\mathcal{A}\) is assumed to be a torsion-free PID ?-algebra sheaf. The main result arising hereby is that \(\mathcal{A}\) -symplectomorphisms locally are products of symplectic transvections, which is a particularly well-behaved counterpart of the classical result.