Calabi quasimorphism and quantum homology.

*(English)*Zbl 1047.53055Let \(G=\text{Ham}(M,\omega)\) be the group of Hamiltonian diffeomorphisms of a closed connected symplectic manifold \(M\). The Calabi homomorphism is defined on \(G_{U}\), where \(U\subset M\) is any open subset such that \(\omega| _{U}\) is exact, as
\[
\text{Cal}_{U} (F_{1})= \int_{M\times [0,1]} F_{t}\omega^{n} \,dt,
\]
for every time-dependent Hamiltonian \(F_{t}\) supported on \(U\) [see E. Calabi, Problems in analysis, Symposium in Honor of Salomon Bochner, Princeton University Press, New Jersey, 1–26 (1970; Zbl 0209.25801)].

This paper extends the Calabi homomorphism to a quasimorphism \(\mu:G \rightarrow \mathbb{R}\) (this means that \(| \mu(fg)-\mu(f)-\mu(g)| \leq r\), for all \(f,g\) with some uniform \(r\)) for the following symplectic manifolds: \(S^{2}\), \(S^{2}\times S^{2}\) and \({\mathbb{C}\mathbb{P}}^{n}\).

Actually, the Calabi quasimorphism \(\mu\) is constructed as a map from the universal cover \(\widetilde{G}\) of \(G\) to \(\mathbb{R}\), for any spherically monotone symplectic manifold \((M,\omega)\) (i.e., those satisfying \(\omega| _{\pi_{2}} = \lambda c_{1}| _{\pi_{2}}\), for some \(\lambda>0\)) which has the property that the even part of the quantum homology \(QH_{ev}(M)\) is semisimple (i.e., a direct sum of fields).

The result about the map \(\mu:G\rightarrow \mathbb{R}\) follows from the finiteness of \(\pi_{1}(G)\) for \(M\) being \(S^{2}\), \(S^{2}\times S^{2}\) or \({\mathbb{C}\mathbb{P}}^{2}\). For \(M={\mathbb{C}\mathbb{P}}^{n}\), \(n\geq 3\), \(\mu\) is shown to vanish on \(\pi_{1}(G)\) by a careful study of the Seidel action [P. Seidel, Geom. Funct. Anal. 7, 1046–1095 (1997; Zbl 0928.53042)] of \(\pi_{1}(G)\) on \(QH(M)\).

This result is applied to bound below the commutator norm of any \(h\in G_{U}\) for any \(U\) and \(M\) as above, and also to bound below the minimum \(r\) such that any \(f\in G\) can be written as \(f=f_{1}\cdots f_{r}\), where \(f_{i}\in G_{U_{i}}\) and \(U_{i}\) and \(M\) are as above.

Let \(\rho\) be the Hofer metric on \(G\) [H. Hofer, Proc. R. Soc. Edinb., Sect. A 115, 25–38 (1990; Zbl 0713.58004)]. This paper also shows that for any generic Hamiltonian \(F\) on the \(2\)-sphere, \(\rho(1,\psi_{t})\) grows as \(ct| | F| | _{C^{0}}\), for some \(c>0\), where \(\psi_{t}\) is the \(1\)-parameter subgroup generated by \(F\).

This paper extends the Calabi homomorphism to a quasimorphism \(\mu:G \rightarrow \mathbb{R}\) (this means that \(| \mu(fg)-\mu(f)-\mu(g)| \leq r\), for all \(f,g\) with some uniform \(r\)) for the following symplectic manifolds: \(S^{2}\), \(S^{2}\times S^{2}\) and \({\mathbb{C}\mathbb{P}}^{n}\).

Actually, the Calabi quasimorphism \(\mu\) is constructed as a map from the universal cover \(\widetilde{G}\) of \(G\) to \(\mathbb{R}\), for any spherically monotone symplectic manifold \((M,\omega)\) (i.e., those satisfying \(\omega| _{\pi_{2}} = \lambda c_{1}| _{\pi_{2}}\), for some \(\lambda>0\)) which has the property that the even part of the quantum homology \(QH_{ev}(M)\) is semisimple (i.e., a direct sum of fields).

The result about the map \(\mu:G\rightarrow \mathbb{R}\) follows from the finiteness of \(\pi_{1}(G)\) for \(M\) being \(S^{2}\), \(S^{2}\times S^{2}\) or \({\mathbb{C}\mathbb{P}}^{2}\). For \(M={\mathbb{C}\mathbb{P}}^{n}\), \(n\geq 3\), \(\mu\) is shown to vanish on \(\pi_{1}(G)\) by a careful study of the Seidel action [P. Seidel, Geom. Funct. Anal. 7, 1046–1095 (1997; Zbl 0928.53042)] of \(\pi_{1}(G)\) on \(QH(M)\).

This result is applied to bound below the commutator norm of any \(h\in G_{U}\) for any \(U\) and \(M\) as above, and also to bound below the minimum \(r\) such that any \(f\in G\) can be written as \(f=f_{1}\cdots f_{r}\), where \(f_{i}\in G_{U_{i}}\) and \(U_{i}\) and \(M\) are as above.

Let \(\rho\) be the Hofer metric on \(G\) [H. Hofer, Proc. R. Soc. Edinb., Sect. A 115, 25–38 (1990; Zbl 0713.58004)]. This paper also shows that for any generic Hamiltonian \(F\) on the \(2\)-sphere, \(\rho(1,\psi_{t})\) grows as \(ct| | F| | _{C^{0}}\), for some \(c>0\), where \(\psi_{t}\) is the \(1\)-parameter subgroup generated by \(F\).

Reviewer: Vicente Muñoz (Madrid)