辛Lefschetz纤维化和有理爆破的地理学
Geography of symplectic Lefschetz fibrations and rational blowdowns
摘要：
我们生产简单连接，最小的，辛Lefschetz纤维实现所有的晶格点在辛地理平面下的Noether线。这提供了一个辛扩展的经典作品填充复杂的地理平面与全纯Lefschetz纤维。我们的例子是通过合理地吹落Lefschetz纤维与集群节点纤维，其中的全空间是潜在的新同伦椭圆曲面。同样，集群节点纤维上的高属Lefschetz纤维化的标准理性的表面，我们得到合理的排污配置，产生新的建设小辛异国情调的$4$ -流形。我们提出了一个例子，通过这个程序适用于属- $3$纤维化的最小辛异国情调的$\mathbb{CP} \# 5\,\overline{\mathbb{CP}}$建设。
英文摘要：
We produce simply connected, minimal, symplectic Lefschetz fibrations realizing all the lattice points in the symplectic geography plane below the Noether line. This provides a symplectic extension of the classical works populating the complex geography plane with holomorphic Lefschetz fibrations. Our examples are obtained by rationally blowing down Lefschetz fibrations with clustered nodal fibers, the total spaces of which are potentially new homotopy elliptic surfaces. Similarly, clustering nodal fibers on higher genera Lefschetz fibrations on standard rational surfaces, we get rational blowdown configurations that yield new constructions of small symplectic exotic $4$-manifolds. We present an example of a construction of a minimal symplectic exotic $\mathbb{CP} \# 5\,\overline{\mathbb{CP}}$ through this procedure applied to a genus-$3$ fibration.
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