论文标题
最小的有理曲线和1板不可还原的G结构
Minimal rational curves and 1-flat irreducible G-structures
论文作者
论文摘要
等效地，不可还原的G结构不可减少的G结构已被广泛研究，以差异几何形状进行了广泛的研究，尤其是与仿射构想的理论有关。我们建议在代数几何形状的环境中研究它们，它们是由与未释放的投影歧管上的最小理性曲线家族相关的最小有理切线（VMRT）产生的。 我们证明，当未释放的投影歧管的尺寸至少为5时，这种结构是局部对称的。通过Merkulov和Schwachhöfer的分类结果，对不可约的仿射自律的分类结果，当VMRT在Uniruled Projective Projective歧管的一般角度下，问题就会降低到该案例中。在后一种情况下，我们证明了一个更强的结果，即在没有1-铂度的情况下，VMRT产生的结构始终是局部平坦的。该证明采用了cartan连接的方法。一个有趣的特征是，cartan连接不是针对G结构本身的，而是针对最小理性曲线空间上的某些几何结构。
1-flat irreducible G-structures, equivalently, irreducible G-structures admitting torsion-free affine connections, have been studied extensively in differential geometry, especially in connection with the theory of affine holonomy groups. We propose to study them in a setting in algebraic geometry, where they arise from varieties of minimal rational tangents (VMRT) associated to families of minimal rational curves on uniruled projective manifolds. We prove that such a structure is locally symmetric when the dimension of the uniruled projective manifold is at least 5. By the classification result of Merkulov and Schwachhöfer on irreducible affine holonomy, the problem is reduced to the case when the VMRT at a general point of the uniruled projective manifold is isomorphic to a subadjoint variety. In the latter situation, we prove a stronger result that, without the assumption of 1-flatness, the structure arising from VMRT is always locally flat. The proof employs the method of Cartan connections. An interesting feature is that Cartan connections are considered not for the G-structures themselves, but for certain geometric structures on the spaces of minimal rational curves.
