通过有理SFT研究接触流形的景观
A landscape of contact manifolds via rational SFT
AI总结：
本文基于有理辛场论的$BL_\infty$形式定义了从精确辛配边范畴到全序集的层级函子，开发了计算工具并证明其满射性，进而用于研究精确配边存在性及探讨相关几何实例。
摘要：
我们利用有理辛场论（RSFT）的$BL_\infty$（双李）形式，定义了一个从精确辛配边范畴到全序集的层级函子。该层级函子包含三层结构，即代数平面挠、半膨胀阶和平面性，取值均在$\mathbb{N}\cup \{\infty\}$中，其中代数平面挠可理解为RSFT背景下Latschev-Wendl代数挠的类比。该层级函子通过RSFT的部分构造得以良定义，且处于既有的虚技术范畴内。我们开发了这些函子的计算工具，并证明它们三者均为满射。特别地，平面性函子在所有$\ge 3$的维度中均为满射。随后，我们利用该层级函子研究精确配边的存在性。我们讨论了相关例子，包括迭代平面开书、脊开书、具有单有理紧化的仿射簇以及奇点链接。
英文摘要：
We define a hierarchy functor from the exact symplectic cobordism category to a totally ordered set from a $BL_\infty$ (Bi-Lie) formalism of the rational symplectic field theory (RSFT). The hierarchy functor consists of three levels of structures, namely algebraic planar torsion, order of semi-dilation and planarity, all taking values in $\mathbb{N}\cup \{\infty\}$, where algebraic planar torsion can be understood as the analogue of Latschev-Wendl's algebraic torsion in the context of RSFT. The hierarchy functor is well-defined through a partial construction of RSFT and is within the scope of established virtual techniques. We develop computation tools for those functors and prove all three of them are surjective. In particular, the planarity functor is surjective in all dimension $\ge 3$. Then we use the hierarchy functor to study the existence of exact cobordisms. We discuss examples including iterated planar open books, spinal open books, affine varieties with uniruled compactification and links of singularities.
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