论文标题
通过复曲面变性的Hofer几何形状
Hofer geometry via toric degeneration
论文作者
论文摘要
本文的主要主题是使用复曲面的变性来在汉密尔顿的差异性群体上产生独特的均匀畸形。我们专注于（复杂的$ n $二维）二次超表面和del pezzo表面，并研究了两类杰出的拉格朗日式submanifolds，它们自然而然地出现在感谢您的变性中，即Lagrangian Torus，即lagrangian torus-firus-fibibiration fibibration和Lagrangian sperssceres canderes canthing bandy vanshing vanshing vanshing vanshish vanshing van的单调纤维。对于四边形，我们证明，哈密顿二型差异性群体接受了两个不同的均匀的准畸形，并获得了一些超级措施。在此过程中，我们表明，感谢您的变性与Biran分解兼容。这意味着对于$ n = 2 $，拉格朗日纤维圆环（gelfand-zeitlin torus）是chekanov torus的汉密尔顿同位素，它回答了Y. Kim的问题。我们将申请给出$ C^0 $ -Symplectic拓扑，其中包括四边形高度表面的Entov-Polterovich- -py问题。我们还证明了Del Pezzo表面的类似结果。
The main theme of this paper is to use toric degeneration to produce distinct homogeneous quasimorphisms on the group of Hamiltonian diffeomorphisms. We focus on the (complex $n$-dimensional) quadric hypersurface and the del Pezzo surfaces, and study two classes of distinguished Lagrangian submanifolds that appear naturally in a toric degeneration, namely the Lagrangian torus which is the monotone fiber of a Lagrangian torus fibration, and the Lagrangian spheres that appear as vanishing cycles. For the quadrics, we prove that the group of Hamiltonian diffeomorphisms admits two distinct homogeneous quasimorphisms and derive some superheaviness results. Along the way, we show that the toric degeneration is compatible with the Biran decomposition. This implies that for $n=2$, the Lagrangian fiber torus (Gelfand--Zeitlin torus) is Hamiltonian isotopic to the Chekanov torus, which answers a question of Y. Kim. We give applications to $C^0$-symplectic topology which include the Entov--Polterovich--Py question for the quadric hypersurface. We also prove analogous results for the del Pezzo surfaces.
