论文标题
ECH嵌入理性表面的障碍物
ECH embedding obstructions for rational surfaces
论文作者
论文摘要
令$（y，a）$为平滑的理性表面或可能具有丰富除数$ a $的奇异曲面表面。我们表明，一个基于ECH的家族，代数几何不变性$ C^{\ text {alg}} _ k（y，a）$由Wormleighton facks facks symplectic嵌入到$ y $中。确切地说，如果$（x，ω_x）$是$ 4 $二维星形的域，$ω_y$是符合形式的poincaréDualdual至$ [a] $ [a] $ the \ [（x，ω_x） c^{\ text {ech}} _ k（x，ω_x）\ le c^{\ text {alg}} _ k（y，a）\]我们为折磨嵌入问题提供了三个应用程序：（1）这些障碍物是尖锐的孔子域中的凹入式毛刺域中的障碍物， （2）Gromov宽度和几个概括是关于包含光滑（和许多单数）曲面表面的矩多边形的单调的； （3）这种圆磨表面的gromov宽度受矩多边形的晶格宽度的界定，解决了averkov的猜想 - hofscheier-nill。
Let $(Y,A)$ be a smooth rational surface or a possibly singular toric surface with ample divisor $A$. We show that a family of ECH-based, algebro-geometric invariants $c^{\text{alg}}_k(Y,A)$ proposed by Wormleighton obstruct symplectic embeddings into $Y$. Precisely, if $(X,ω_X)$ is a $4$-dimensional star-shaped domain and $ω_Y$ is a symplectic form Poincaré dual to $[A]$ then \[(X,ω_X)\text{ embeds into }(Y,ω_Y)\text{ symplectically } \implies c^{\text{ECH}}_k(X,ω_X) \le c^{\text{alg}}_k(Y,A)\] We give three applications to toric embedding problems: (1) these obstructions are sharp for embeddings of concave toric domains into toric surfaces; (2) the Gromov width and several generalizations are monotonic with respect to inclusion of moment polygons of smooth (and many singular) toric surfaces; and (3) the Gromov width of such a toric surface is bounded by the lattice width of its moment polygon, addressing a conjecture of Averkov--Hofscheier--Nill.
