Compatible Complex Structures Research Articles

Integration of generalized complex structures

We solve the integration problem for generalized complex manifolds, obtaining as the natural integrating object a holomorphic stack with a shifted symplectic structure; in other words, a real symplectic groupoid with a compatible complex structure is defined only up to Morita equivalence. We explain how such objects differentiate to give generalized complex manifolds, and we show that a generalized complex manifold is integrable in this sense if and only if its underlying real Poisson structure is integrable. We describe several concrete examples of these integrations. Crucial to our solution are new technical tools, which are of independent interest, namely, a reduction procedure for Lie groupoid actions on Courant algebroids, as well as certain local-to-global extension results for multiplicative forms on local Lie groupoids.

The commutative nonassociative algebra of metric curvature tensors

AbstractThe space of tensors of metric curvature type on a Euclidean vector space carries a two-parameter family of orthogonally invariant commutative nonassociative multiplications invariant with respect to the symmetric bilinear form determined by the metric. For a particular choice of parameters these algebras recover the polarization of the quadratic map on metric curvature tensors that arises in the work of Hamilton on the Ricci flow. Here these algebras are studied as interesting examples of metrized commutative algebras and in low dimensions they are described concretely in terms of nonstandard commutative multiplications on self-adjoint endomorphisms. The algebra of curvature tensors on a 3-dimensional Euclidean vector space is shown isomorphic to an orthogonally invariant deformation of the standard Jordan product on$3 \times 3$symmetric matrices. This algebra is characterized up to isomorphism in terms of purely algebraic properties of its idempotents and the spectra of their multiplication operators. On a vector space of dimension at least 4, the subspace of Weyl (Ricci-flat) curvature tensors is a subalgebra for which the multiplication endomorphisms are trace-free and the Killing type trace-form is a multiple of the nondegenerate invariant metric. This subalgebra is simple when the Euclidean vector space has dimension greater than 4. In the presence of a compatible complex structure, the analogous result is obtained for the subalgebra of Kähler Weyl curvature tensors. It is shown that the anti-self-dual Weyl tensors on a 4-dimensional vector space form a simple 5-dimensional ideal isometrically isomorphic to the trace-free part of the Jordan product on trace-free$3 \times 3$symmetric matrices.

Momentum polytopes of projective spherical varieties and related Kähler geometry

We apply the combinatorial theory of spherical varieties to characterize the momentum polytopes of polarized projective spherical varieties. This enables us to derive a classification of these varieties, without specifying the open orbit, as well as a classification of all Fano spherical varieties. In the setting of multiplicity free compact and connected Hamiltonian manifolds, we obtain a necessary and sufficient condition involving momentum polytopes for such manifolds to be K\"ahler and classify the invariant compatible complex structures of a given K\"ahler multiplicity free compact and connected Hamiltonian manifold.

The geometric quantizations and the measured Gromov–Hausdorff convergences

On a compact symplectic manifold $(X,\omega)$ with a prequantum line bundle $(L,\nabla,h)$, we consider the one-parameter family of $\omega$-compatible complex structures which converges to the real polarization coming from the Lagrangian torus fibration. There are several researches which show that the holomorphic sections of the line bundle localize at Bohr-Sommerfeld fibers. In this article we consider the one-parameter family of the Riemannian metrics on the frame bundle of $L$ determined by the complex structures and $\nabla,h$, and we can see that their diameters diverge. If we fix a base point in some fibers of the Lagrangian fibration we can show that they measured Gromov-Hausdorff converge to some pointed metric measure spaces with the isometric $S^1$-actions, which may depend on the choice of the base point. We observe that the properties of the $S^1$-actions on the limit spaces actually depend on whether the base point is in the Bohr-Sommerfeld fibers or not.

Spinor modules for Hamiltonian loop group spaces

Let $LG$ be the loop group of a compact, connected Lie group $G$. We show that the tangent bundle of any proper Hamiltonian $LG$-space $\mathcal{M}$ has a natural completion $\overline{T}\mathcal{M}$ to a strongly symplectic $LG$-equivariant vector bundle. This bundle admits an invariant compatible complex structure within a natural polarization class, defining an $LG$-equivariant spinor bundle $\mathsf{S}_{\overline{T}\mathcal{M}}$, which one may regard as the Spin$_c$-structure of $\mathcal{M}$. We describe two procedures for obtaining a finite-dimensional version of this spinor module. In one approach, we construct from $\mathsf{S}_{\overline{T}\mathcal{M}}$ a twisted Spin$_c$-structure for the quasi-Hamiltonian $G$-space associated to $\mathcal{M}$. In the second approach, we describe an `abelianization procedure', passing to a finite-dimensional $T\subset LG$-invariant submanifold of $\mathcal{M}$, and we show how to construct an equivariant Spin$_c$-structure on that submanifold.

LCK metrics on toric LCS manifolds

We show a bijective correspondence between compact toric locally conformally symplectic manifolds which admit a compatible complex structure and pairs (C,a), where C is a good cone in the dual Lie algebra of the torus and a is a positive real number. Moreover, we prove that any toric locally conformally Kähler metric on a compact manifold admits a positive potential.

Integrable Complex Structures on Twistor Spaces

Abstract We introduce integrable complex structures on twistor spaces fibered over complex manifolds. We then show, in particular, that the twistor spaces associated with generalized Kahler, SKT and strong HKT manifolds all naturally admit complex structures. Moreover, in the strong HKT case, we construct a metric and three compatible complex structures on the twistor space that have equal torsions.

Special Bohr–Sommerfeld Lagrangian submanifolds of algebraic varieties

In this paper we continue our study of special Bohr– Sommerfeld submanifolds in the case when the ambient symplectic manifold possesses a compatible integrable complex structure (and is thus an algebraic variety). In this case we show how to reduce the special Bohr– Sommerfeld geometry to Morse theory on the complements of ample divisors. This gives rise to a construction of Lagrangian shadows of ample divisors in algebraic varieties, which is an example of `algebraic v. symplectic' duality. We suggest a condition for the existence of a Lagrangian shadow and give examples of Lagrangian shadows of ample divisors on the projective plane, complex quadrics and flag manifolds.

Finite-energy pseudoholomorphic planes with multiple asymptotic limits

It’s known from (Ann Inst H Poincaré Anal Non Linéaire 13(3):337–379 [11], Properties of Pseudoholomorphic Curves in Symplectisation. IV. Asymptotics with Degeneracies. In: Contact and symplectic geometry (Cambridge, 1994), vol 8 Publ. Newton Inst., pp 78–117. Cambridge Univ. Press, Cambridge [10], A Morse-Bott Approach to Contact Homology. PhD thesis, Stanford University [2]) that in a contact manifold equipped with either a nondegenerate or Morse-Bott contact form, a finite-energy pseudoholomorphic curve will be asymptotic at each of its nonremovable punctures to a single periodic orbit of the Reeb vector field and that the convergence is exponential. We provide examples here to show that this need not be the case if the contact form is degenerate. More specifically, we show that on any contact manifold $$(M, \xi )$$ with cooriented contact structure one can choose a contact form $$\lambda $$ with $$\ker \lambda =\xi $$ and a compatible complex structure J on $$\xi $$ so that for the associated $$\mathbb {R}$$ -invariant almost complex structure $$\tilde{J}$$ on $$\mathbb {R}\times M$$ there exist families of embedded finite-energy $$\tilde{J}$$ -holomorphic cylinders and planes having embedded tori as limit sets.

Comparing Poisson Sigma Model with A-model

We discuss the A-model as a gauge fixing of the Poisson Sigma Model with target a symplectic structure. We complete the discussion in [arXiv:0706.3164], where a gauge fixing defined by a compatible complex structure was introduced, by showing how to recover the A-model hierarchy of observables in terms of the AKSZ observables. Moreover, we discuss the off-shell supersymmetry of the A-model as a residual BV symmetry of the gauge-fixed PSM action.

On rotation of complex structures

We put in a general framework the situations in which a Riemannian manifold admits a family of compatible complex structures, including hyperkähler metrics and the Spin -rotations of Muñoz (2014). We determine the (polystable) holomorphic bundles which are rotable, i.e., they remain holomorphic when we change a complex structure by a different one in the family.

Quaternion-Kähler four-manifolds and Przanowski's function

Quaternion-Kähler four-manifolds, or equivalently anti-self-dual Einstein manifolds, are locally determined by one scalar function subject to Przanowski's equation. Using twistorial methods, we construct a Lax Pair for Przanowski's equation, confirming its integrability. The Lee form of a compatible local complex structure, which one can always find, gives rise to a conformally invariant differential operator acting on sections of a line bundle. Special cases of the associated generalised Laplace operator are the conformal Laplacian and the linearised Przanowski operator. We provide recursion relations that allow us to construct cohomology classes on twistor space from solutions of the generalised Laplace equation. Conversely, we can extract such solutions from twistor cohomology, leading to a contour integral formula for perturbations of Przanowski's function. Finally, we illuminate the relationship between Przanowski's function and the twistor description, in particular, we construct an algorithm to retrieve Przanowski's function from twistor data in the double-fibration picture. Using a number of examples, we demonstrate this procedure explicitly.

GEOMETRIC PHASES IN THE QUANTISATION OF BOSONS AND FERMIONS

AbstractAfter reviewing geometric quantisation of linear bosonic and fermionic systems, we study the holonomy of the projectively flat connection on the bundle of Hilbert spaces over the space of compatible complex structures and relate it to the Maslov index and its various generalisations. We also consider bosonic and fermionic harmonic oscillators parametrised by compatible complex structures and compare Berry’s phase with the above holonomy.

Weak mirror symmetry of Lie algebras

Weak mirror symmetry relates a manifold with complex structure to another manifold equipped with a symplectic structure through a quasiisomorphism of associated differential Gerstenhaber algebras. The two manifolds are then mirror partners. In this paper we consider the analogous problem on Lie algebras. In particular we show that the semi-direct product of a Lie algebra equipped with a torsion-free flat connection with itself is a mirror partner of a semi-direct product of the same Lie algebra with its dual space. For nilpotent algebras this analysis on Lie algebras can be applied to the compact quotients of the underlying nilpotent group. We classify mirror pairs among 6-dimensional nilpotent Lie algebras that have the semi-direct product structure as well as mirror pairs admitting the more involved special Lagrangian structure, namely compatible complex and symplectic structures on the same space.

Polytopes with Mass Linear Functions, Part 1

Let Δ be an n-dimensional polytope that is simple, that is, exactly n facets meet at each vertex. An affine function is “mass linear” on Δ if its value on the center of mass of Δ depends linearly on the positions of the supporting hyperplanes. On the one hand, we show that certain types of symmetries of Δ give rise to nonconstant mass linear functions on Δ. On the other hand, we show that most polytopes do not admit any nonconstant mass linear functions. Further, if every affine function is mass linear on Δ, then Δ is a product of simplices. Our main result is a classification of all smooth polytopes of dimension ≤ 3 which admit nonconstant mass linear functions. In particular, there is only one family of smooth three-dimensional polytopes—and no polygons—that admit “essential mass linear functions,” that is, mass linear functions that do not arise from the symmetries described above. In part II, we will complete this classification in the four-dimensional case. These results have geometric implications. Fix a symplectic toric manifold (M,ω,T,Φ) with moment polytope Δ = Φ(M). Let denote the identity component of the group of symplectomorphisms of (M,ω). Any linear function H on Δ generates a Hamiltonian ℝ action on M whose closure is a subtorus T H of T. We show that if the map has finite image, then H is mass linear. Combining this fact and the claims described above, we prove that in most cases, the induced map is an injection. Moreover, the map does not have finite image unless M is a product of projective spaces. Note also that there is a natural maximal compact connected subgroup ; there is a natural compatible complex structure J on M, and is the identity component of the group of symplectomorphisms that also preserve this structure. We prove that if the polytope Δ supports no essential mass linear functions, then the induced map is injective. Therefore, this map is injective for all four-dimensional symplectic toric manifolds and is injective in the six-dimensional case unless M is a bundle over .

Compatible complex structures on symplectic rational ruled surfaces

In this article, we study the topology of the space Iω of complex structures compatible with a fixed symplectic form ω, using the framework of Donaldson. By comparing our analysis of the space Iω with results of McDuff on the space Jω of compatible almost complex structures on rational ruled surfaces, we find that Iω is contractible in this case. We then apply this result to study the topology of the symplectomorphism group of a rational ruled surface, extending results of Abreu and McDuff

A personal tour through symplectic topology and geometry

(i) Gromov’s Compactness Theorem for pseudo-holomorphic curves in symplectic manifolds ([23]) and the topology of symplectomorphism groups of rational ruled surfaces (sections 2 and 3, references [1, 2]). (ii) Atiyah-Guillemin-Sternberg’s Convexity Theorem for the moment map of Hamiltonian torus actions ([9, 25]) and Kahler geometry of toric orbifolds in symplectic coordinates (sections 4 and 5, references [3, 4, 5]). (iii) Donaldson’s moment map framework for the action of the symplectomorphism group on the space of compatible almost complex structures ([17]) and the topology of the space of compatible integrable complex structures of a rational ruled surface (sections 6 and 7, references [6, 7]).

Remarks on symplectic twistor spaces

We consider some classical fibre bundles furnished with almost complex structures of twistor type, deduce their integrability in some cases and study \textit{self-holomorphic} sections of a \textit{symplectic} twistor space. With these we define a moduli space of $\omega$-compatible complex structures. We recall the theory of flag manifolds in order to study the Siegel domain and other domains alike, which is the fibre of the referred twistor space. Finally the structure equations for the twistor of a Riemann surface with the canonical symplectic-metric connection are deduced, based on a given conformal coordinate on the surface. We then relate with the moduli space defined previously.

Geometric Quantization, Parallel Transport and the Fourier Transform

In quantum mechanics, the momentum space and position space wave functions are related by the Fourier transform. We investigate how the Fourier transform arises in the context of geometric quantization. We consider a Hilbert space bundle \(\mathcal{H}\) over the space \(\mathcal{J}\) of compatible complex structures on a symplectic vector space. This bundle is equipped with a projectively flat connection. We show that parallel transport along a geodesic in the bundle \(\mathcal{H} \to \mathcal{J}\) is a rescaled orthogonal projection or Bogoliubov transformation. We then construct the kernel for the integral parallel transport operator. Finally, by extending geodesics to the boundary (for which the metaplectic correction is essential), we obtain the Segal-Bargmann and Fourier transforms as parallel transport in suitable limits.

Complex product structures on Lie algebras

A study is made of real Lie algebras admitting compatible complex and product structures, including numerous 4-dimensional examples. If g is a Lie algebra with such a structure then its complexification has a hypercomplex structure. It is shown in addition that g splits into the sum of two left-symmetric subalgebras. Interpretations of these results are obtained that are relevant to the theory of both hypercomplex and hypersymplectic manifolds and their associated connections.