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Calabi quasimorphism and quantum homology. (English) Zbl 1047.53055
Let $$G=\text{Ham}(M,\omega)$$ 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 $$F$$.

##### MSC:
 53D35 Global theory of symplectic and contact manifolds 53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds 57R17 Symplectic and contact topology in high or arbitrary dimension
##### Keywords:
symplectic; quantum homology; quasimorphism
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