复射影平面上的马氏数与拉格朗日胞腔复形
Markov numbers and Lagrangian cell complexes in the complex projective plane
摘要：
研究了一类二维胞腔复形$L_{p,q}$在复射影平面上的拉格朗日嵌入。这些胞腔复形，我们称之为风车，在代数几何中自然出现，作为$\frac{1}{p^2}(pq-1,1)$型商奇点（Wahl奇点）的消失环。我们表明，如果风车承认拉格朗日嵌入到$\mathbf{CP}^2$，那么$p$是一个马尔可夫数，我们完全$q$。我们还表明，收集拉格朗日风车$L_{p_i,q_i}$，$i=1,\ldots,N$，不能不相交，除非$N\leq 3$和$p_i$形成的马尔可夫三元组的一部分。这些结果是辛模拟定理的哈金和Prokhorov，其中分类复杂的表面与商奇异承认$\mathbf{Q}$ -Gorenstein平滑，其一般纤维是$\mathbf{CP}^2$。
英文摘要：
We study Lagrangian embeddings of a class of two-dimensional cell complexes $L_{p,q}$ into the complex projective plane. These cell complexes, which we call pinwheels, arise naturally in algebraic geometry as vanishing cycles for quotient singularities of type $\frac{1}{p^2}(pq-1,1)$ (Wahl singularities). We show that if a pinwheel admits a Lagrangian embedding into $\mathbf{CP}^2$ then $p$ is a Markov number and we completely characterise $q$. We also show that a collection of Lagrangian pinwheels $L_{p_i,q_i}$, $i=1,\ldots,N$, cannot be made disjoint unless $N\leq 3$ and the $p_i$ form part of a Markov triple. These results are the symplectic analogue of a theorem of Hacking and Prokhorov, which classifies complex surfaces with quotient singularities admitting a $\mathbf{Q}$-Gorenstein smoothing whose general fibre is $\mathbf{CP}^2$.
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