[PDF] Pseudo-Holomorphic Hamiltonian Systems-论文阅读讨论-ReadPaper

摘要

In this paper, we first explore holomorphic Hamiltonian systems. Inparticular, we define action functionals for those systems and show thatholomorphic trajectories obey an action principle, i.e., that they can beunderstood - in some sense - as critical points of these action functionals. Asan application, we use holomorphic Hamiltonian systems to establish a relationbetween Lefschetz fibrations and almost toric fibrations. During theinvestigation of action functionals for holomorphic Hamiltonian systems, weobserve that the complex structure $J$ corresponding to a holomorphicHamiltonian system poses strong restrictions on the existence of certaintrajectories. For instance, no non-trivial holomorphic trajectories withcomplex tori as domains can exist in $\mathbb{C}^{2n}$ due to the maximumprinciple. To lift this restriction, we generalize the notion of holomorphicHamiltonian systems to systems with non-integrable almost complex structures$J$ leading us to the definition of pseudo-holomorphic Hamiltonian systems. Weshow that these systems exhibit properties very similar to their holomorphiccounterparts, notably, that they are also subject to an action principle.Furthermore, we prove that the integrability of $J$ is equivalent to theclosedness of the "pseudo-holomorphic symplectic" form. Lastly, we show that,aside from dimension four, the set of proper pseudo-holomorphic Hamiltoniansystems is open and dense in the set of pseudo-holomorphic Hamiltonian systemsby considering deformations of holomorphic Hamiltonian systems. This impliesthat proper pseudo-holomorphic Hamiltonian systems are generic.

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