Real Symplectic Manifold Research Articles

On the invariance of Welschinger invariants

We collect in this note some observations about original Welschinger invariants of real symplectic fourfolds. None of their proofs is difficult, nevertheless these remarks do not seem to have been made before. Our main result is that when $X$ is a real rational algebraic surface, Welschinger invariants only depend on the number of real interpolated points, and some homological data associated to $X$. This strengthened the invariance statement initially proved by Welschinger. This main result follows easily from a formula relating Welschinger invariants of two real symplectic manifolds differing by a surgery along a real Lagrangian sphere. In its turn, once one believes that such formula may hold, its proof is a mild adaptation of the proof of analogous formulas previously obtained by the author on the one hand, and by Itenberg, Kharlamov and Shustin on the other hand. We apply the two aforementioned results to complete the computation of Welschinger invariants of real rational algebraic surfaces, and to obtain vanishing, sign, and sharpness results for these invariants that generalize previously known statements. We also discuss some hypothetical relations of our work with tropical refined invariants defined by Block-G\ottsche and G\ottsche-Schroeter.

Representation formula for symmetrical symplectic capacity and applications

This is the second installment in a series of papers aimed at generalizing symplectic capacities and homologies. We study symmetric versions of symplectic capacities for real symplectic manifolds, and obtain corresponding results for them to those of the first [ 19 ] of this series (such as representation formula, a theorem by Evgeni Neduv, Brunn-Minkowski type inequality and Minkowski billiard trajectories proposed by Artstein-Avidan-Ostrover).

Volume growth in the component of fibered twists

For a Liouville domain [Formula: see text] whose boundary admits a periodic Reeb flow, we can consider the connected component [Formula: see text] of fibered twists. In this paper, we investigate an entropy-type invariant, called the slow volume growth, in the component [Formula: see text] and give a uniform lower bound of the growth using wrapped Floer homology. We also show that [Formula: see text] has infinite order in [Formula: see text] if there is an admissible Lagrangian [Formula: see text] in [Formula: see text] whose wrapped Floer homology is infinite dimensional. We apply our results to fibered twists coming from the Milnor fibers of [Formula: see text]-type singularities and complements of a symplectic hypersurface in a real symplectic manifold. They admit so-called real Lagrangians, and we can explicitly compute wrapped Floer homology groups using a version of Morse–Bott spectral sequences.

Real Gromov-Witten theory in all genera and real enumerative geometry: Construction

We construct positive-genus analogues of Welschinger's invariants for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold. In some cases, our invariants provide lower bounds for counts of real positive-genus curves in real algebraic varieties. Our approach to the orientability problem is based entirely on the topology of real bundle pairs over symmetric surfaces; the previous attempts involved direct computations for the determinant lines of Fredholm operators over bordered surfaces. We use the notion of real orientation introduced in this paper to obtain isomorphisms of real bundle pairs over families of symmetric surfaces and then apply the determinant functor to these isomorphisms. This allows us to endow the uncompactified moduli spaces of real maps from symmetric surfaces of all topological types with natural orientations and to verify that they extend across the codimension-one boundaries of these spaces, thus implementing a far-reaching proposal from C.-C.Liu's thesis for a fully fledged real Gromov-Witten theory. The second and third parts of this work concern applications: they describe important properties of our orientations on the moduli spaces, establish some connections with real enumerative geometry, provide the relevant equivariant localization data for projective spaces, and obtain vanishing results in the spirit of Walcher's predictions.

A recursion for counts of real curves in ℂℙ2n−1: Another proof

In a recent paper, we obtained a WDVV-type relation for real genus 0 Gromov–Witten invariants with conjugate pairs of insertions; it specializes to a complete recursion in the case of odd-dimensional projective spaces. This paper provides another, more complex-geometric, proof of the latter. The main part of this approach readily extends to real symplectic manifolds with empty real locus, but not to the general case.

Surgery of real symplectic fourfolds and Welschinger invariants

A surgery of a real symplectic manifold $X_{\mathbb R}$ along a real Lagrangian sphere $S$ is a modification of the symplectic and real structure on $X_{\mathbb R}$ in a neigborhood of $S$. Genus 0 Welschinger invariants of two real symplectic $4$-manifolds differing by such a surgery have been related in a previous work in collaboration with N. Puignau. In the present paper, we explore some particular situations where these general formulas greatly simplify. As an application, we complete the computation of genus 0 Welschinger invariants of all del~Pezzo surfaces, and of all $\mathbb R$-minimal real conic bundles. As a by-product, we establish the existence of some new relative Welschinger invariants. We also generalize our results to the enumeration of curves of higher genus, and give relations between hypothetical invariants defined in the same vein as a previous work by Shustin.

Enumeration of real curves in CP2n−1 and a Witten–Dijkgraaf–Verlinde–Verlinde relation for real Gromov–Witten invariants

We establish a homology relation for the Deligne-Mumford moduli spaces of real curves which lifts to a WDVV-type relation for real Gromov-Witten invariants of real symplectic manifolds; we also obtain a vanishing theorem for these invariants. For many real symplectic manifolds, these results reduce all genus 0 real invariants with conjugate pairs of constraints to genus 0 invariants with a single conjugate pair of constraints. In particular, we give a complete recursion for counts of real rational curves in odd-dimensional projective spaces with conjugate pairs of constraints and specify all cases when they are nonzero and thus provide non-trivial lower bounds in high-dimensional real algebraic geometry. We also show that the real invariants of the three-dimensional projective space with conjugate point constraints are congruent to their complex analogues modulo 4.

Real orientations, real Gromov-Witten theory, and real enumerative geometry

The present note overviews our recent construction of real Gromov-Witten theory in arbitrary genera for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold, its properties, and its connections with real enumerative geometry. Our construction introduces the principle of orienting the determinant of a differential operator relative to a suitable base operator and a real setting analogue of the (relative) spin structure of open Gromov-Witten theory. Orienting the relative determinant, which in the now-standard cases is canonically equivalent to orienting the usual determinant, is naturally related to the topology of vector bundles in the relevant category. This principle and its applications allow us to endow the uncompactified moduli spaces of real maps from symmetric surfaces of all topological types with natural orientations and to verify that they extend across the codimension-one boundaries of these spaces, thus implementing a far-reaching proposal from C.-C. Liu's thesis.

Local analytic geometry of generalized complex structures

A generalized complex manifold is locally gauge-equivalent to the product of a holomorphic Poisson manifold with a real symplectic manifold, but in possibly many different ways. In this paper we show that the isomorphism class of the holomorphic Poisson structure occurring in this local model is independent of the choice of gauge equivalence, and is hence the unique local invariant of generalized complex manifolds. We use this result to prove that the complex locus of a generalized complex manifold naturally inherits the structure of a complex analytic space.

Open Gromov–Witten disk invariants in the presence of an anti-symplectic involution

For a symplectic manifold with an anti-symplectic involution having non-empty fixed locus, we construct a model of the moduli space of real sphere maps out of moduli spaces of decorated disk maps and give an explicit expression for its first Stiefel–Whitney class. As a corollary, we obtain a large number of examples, which include all odd-dimensional projective spaces and many complete intersections, for which many types of real moduli spaces are orientable. For these manifolds, we define open Gromov–Witten invariants with no restriction on the dimension of the manifolds or the type of the constraints if there are no boundary marked points; a WDVV-type recursion obtained in a sequel computes these invariants for many real symplectic manifolds. If there are boundary marked points, we define the invariants under some restrictions on the allowed boundary constraints, even though the moduli spaces are not orientable in these cases.

Symplectic geometry of semisimple orbits

Let G be a complex semisimple group, T ⊂ G a maximal torus and B a Borel subgroup of G containing T. Let Ω be the Kostant-Kirillov holomorphic symplectic structure on the adjoint orbit O = Ad( G) c ≃ G/ Z( c), where c ∈ Lie( T), and Z( c) is the centralizer of c in G. We prove that the real symplectic form Re Ω (respectively, Im Ω) on O is exact if and only if all the eigenvalues ad( c) are real (respectively, purely imaginary). Furthermore, each of these real symplectic manifolds is symplectomorphic to the cotangent bundle of the partial flag manifold G/Z( cc) B, equipped with the Liouville symplectic form. The latter result is generalized to hyperbolic adjoint orbits in a real semisimple Lie algebra.

On the Dynamics of finite-gap solutions in classical string theory

We study the dynamics of finite-gap solutions in classical string theory on R x S^3. Each solution is characterised by a spectral curve, \Sigma, of genus g and a divisor, \gamma, of degree g on the curve. We present a complete reconstruction of the general solution and identify the corresponding moduli-space, M^(2g)_R, as a real symplectic manifold of dimension 2g. The dynamics of the general solution is shown to be equivalent to a specific Hamiltonian integrable system with phase-space M^(2g)_R. The resulting description resembles the free motion of a rigid string on the Jacobian torus J(\Sigma). Interestingly, the canonically-normalised action variables of the integrable system are identified with certain filling fractions which play an important role in the context of the AdS/CFT correspondence.

Invariant hyperkaehler structures on the cotangent bundles of Hermitian symmetric spaces

Let G/K be an irreducible Hermitian symmetric space of compact type with standard homogeneous complex structure. Then the real symplectic manifold (T*(G/K),{omega}) has the natural complex structure J{sup -}. All G-invariant Kehler structures (J,{omega}) on G-invariant subdomains of T*(G/K) anticommuting with J{sup -} are constructed. Each hypercomplex structure of this kind, equipped with a suitable metric, defines a hyperkehler structure. As an application, a new proof of the theorem of Harish-Chandra and Moore for Hermitian symmetric spaces is obtained.

Invariant hyperkähler structures on the cotangent bundles of Hermitian symmetric spaces

Let be an irreducible Hermitian symmetric space of compact type with standard homogeneous complex structure. Then the real symplectic manifold has the natural complex structure . All -invariant Kéhler structures on -invariant subdomains of anticommuting with are constructed. Each hypercomplex structure of this kind, equipped with a suitable metric, defines a hyperkéhler structure. As an application, a new proof of the theorem of Harish-Chandra and Moore for Hermitian symmetric spaces is obtained.

Differential geometry on Grassmann algebras

H. C. Lee [1] developed the analogue of Riemannian geometry on a real symplectic manifold — the fundamental skew two-form taking the place of the symmetric tensor. The usual Riemannian concepts do not adapt themselves very well, thus ‘curvature’ is represented by a tensor of the third rank and ‘Killing's equations’ now involve this ‘curvature tensor’. The immediate reason for this is that otherwise familiar terms appear with the wrong sign. We have found that these unaesthetic features disappear, and formal elegance is marvellously restored, when the manifold is replaced by a Grassmann algebra. The connection with supersymmetry is explained but applications are not reported here.