The action homomorphism, quasimorphisms and moment maps  on the space of compatible almost complex structures

Shelukhin, Egor

doi:10.4171/cmh/313

Cited by 16 publications

(17 citation statements)

References 46 publications

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“…The proof follows directly from Shelukhin's proof in [33]. The only difference comes from the fact that we need to use the fundamental Theorem of differential calculus for smooth maps from an interval into R[[ν]] and Stokes Theorem for integration on spheres of the 2-form…”

Section: Alsomentioning

confidence: 99%

“…We obtain a formal symplectic form on the space of symplectic connections and derive a formal moment map for the action of the group of Hamiltonian diffeormorphisms. We then relate the underlying action homomorphism from Shelukhin [33] to an invariant from [16].…”

Section: Introductionmentioning

confidence: 99%

“…We show that the parallel lift of a Hamiltonian diffeomorphisms path gives Hamiltonian automorphisms of the star product [25]. Next, we apply the construction of Shelukhin [33], to derive a Weinstein action homomorphism which is an invariant on the π 1 of the group of Hamiltonian diffeomorphisms. On Kähler manifold, when restricting this action homomorphism to Hamiltonian biholomorphisms we recover the invariant from [16] obstructing the closedness of the Fedosov star product * ∇ .…”

Section: Introductionmentioning

confidence: 99%

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The formal moment map geometry of the space of symplectic connections

La Fuente-Gravy

2021

Preprint

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We deform the moment map picture on the space of symplectic connections on a symplectic manifold. To do that, we study a vector bundle of Fedosov star product algebra on the space of symplectic connections E. We describe a natural formal connection on this bundle adapted to the star product algebras on the fibers. We study its curvature and show the * -product trace of the curvature is a formal symplectic form on E. The action of Hamiltonian diffeomorphisms on E preserves the formal symplectic structure and we show the * -product trace can be interpreted as a formal moment map for this action. Finally, we apply this picture to study automorphisms of star products and Hamiltonian diffeomorphisms.

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Section: Alsomentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The formal moment map geometry of the space of symplectic connections

La Fuente-Gravy

2021

Preprint

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“…Let us simply mention that similar invariants can also be defined on the groups of symplectomorphisms of more general symplectic manifolds. For the state of the art on this topic, we recommend Borman and Zapolsky [28] or Shelukhin [149]. Some of these invariants are related to Floer homology.…”

Section: éTienne Ghys and Andrew Ranickimentioning

confidence: 99%

Six papers on signatures, braids and Seifert surfaces

Ghys¹,

Ranicki²

2016

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+ U (v) = Constant and this implies stability: any trajectory stays in a bounded neighborhood of the origin since U (v) remains bounded. Recall that the differential equation d 2 v/dt 2 +ω 2 v = 0 has solutions v = a exp(±iωt). These solutions can only be bounded (as the time runs over R) if ω is a real number. Indeed exp(i(α + iβ)t) = exp(iαt) exp(−bt) is bounded if an only if b = 0. "Therefore", S has only real (and positive) eigenvalues. If S is not positive definite, just consider S + λId with a large positive real number λ.Au reste, quoiqu'il soit difficile, peut-être impossible, de déterminer en général les racines de l'équation P = 0, on peut cependant s'assurer, par la nature même du problème, que ces racines sont nécessairement toutes réelles [...]; car sans cela les valeurs de y , y , y , . . . pourraient croître à l'infini, ce qui serait absurde [99].Cauchy's proof (1829) [38]. Let λ be a complex eigenvalue of S and let v be a non zero eigenvector in C n . Of course,v is an eigenvector with eigenvalueλ. We still denote by the same symbol , the extension to C n of the scalar product as a bilinear form (not as a hermitian form). We haveSince v = 0, we have v,v = 0. This implies that λ =λ so that λ is indeed real. Amazingly, this proof, which looks crystal clear today, was not so convincing at the beginning of the nineteenth century, probably because complex numbers still sounded mysterious (they were called imaginary or even impossible).A more algebraic proof. If n 3, one uses the fact that any matrix with real entries has an invariant 2 dimensional subspace E. Of course, since S is symmetric, the orthogonal of E is also invariant and the proof goes by induction. One could argue that the standard proof of the existence of an invariant 2 dimensional subspace uses complex numbers so that this proof is very close to Cauchy's proof.Just for fun, let us describe a magic proof given by Sylvester (1852) [157]. Consider the characteristic polynomial P (X) = det(XI n − S). Note that P (X)P (−X) = det(X 2 I n − S 2 ). Since S is symmetric, all diagonal entries of S 2 are sums of squares and are therefore non negative. Let us write P (X)P (−X) = (−X 2 ) n + a 1 (−X 2 ) n−1 + a 2 (−X 2 ) n−2 + . . . + a n .

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“…One is the group Symp(M, ω) of symplectomorphisms, and the other is the group Ham(M, ω) of Hamiltonian diffeomorphisms. It is known that there are various Symp(M, ω)invariant quasimorphisms on Ham(M, ω) (see [6], [14], and [16] for example). n exists and we call scl G (x) the stable commutator length of x. Bavard's duality theorem gives a connection between quasimorphisms and commutator lengths in the following form: Theorem 1.1 (Bavard [1]).…”

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confidence: 99%

Bavard’s duality theorem on conjugation-invariant norms

Kawasaki

2017

Pacific J. Math.

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Bavard proved a duality theorem between commutator length and quasimorphisms. Burago, Ivanov and Polterovich introduced the notion of a conjugation-invariant norm which is a generalization of commutator length. Entov and Polterovich proved that Oh-Schwarz spectral invariants are subset-controlled quasimorphisms which are geralizations of quasimorphisms. In the present paper, we prove a Bavard-type duality theorem between conjugation-invariant (pseudo-)norms and subset-controlled quasimorphisms on stable groups. We also pose a generalization of our main theorem and prove that "stably non-displaceable subsets of symplectic manifolds are heavy" in a rough sense if that generalization holds.

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The action homomorphism, quasimorphisms and moment maps on the space of compatible almost complex structures

Cited by 16 publications

References 46 publications

The formal moment map geometry of the space of symplectic connections

The formal moment map geometry of the space of symplectic connections

Six papers on signatures, braids and Seifert surfaces

Bavard’s duality theorem on conjugation-invariant norms

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