论文标题
识别$ {\ rm g} _2 $ - PICARD数字1的Horospherical歧管。
Recognizing the ${\rm G}_2$-horospherical manifold of Picard number 1 by varieties of minimal rational tangents
论文作者
论文摘要
Pasquier和Perrin发现，Picard数字1的$ {\ rm g} _2 $ - hor磷歧管$ {\ bf x} $可以实现为合理同质空间的平稳专业化，以5维超质量的空间在其他语言上可以降低5维超过的空间，从而可以在5维超强的空间上进行贬值。我们证明$ {\ bf x} $是该属性唯一的光滑投射品种。这是由于我们的主要结果获得的结果是，$ {\ bf x} $可以通过其VMRT认可，即，Picard编号1的Fano歧管是双重的，而仅当其VMRT在总体上是$ {\ bf x} $的折线，从总体上讲，它的VMRT是Prognal Phoct Project to Projecty to isomorphic to isomorphic to isomorphic to isomorphic to;我们采用了作者开发的方法来解决符号拉格曼尼亚人的相应问题，该问题在一般最小的理性曲线附近建立了扁平的cartan连接。在将这种方法调整为$ {\ bf x} $时，我们需要对矢量捆绑包的阳性/负面性的复杂研究相对于一个理性曲线的家庭，这比符合性格拉曼尼亚人的案例微妙，因为$ {\ bf x} $ vmrt的差异几何结构的性质。
Pasquier and Perrin discovered that the ${\rm G}_2$-horospherical manifold ${\bf X}$ of Picard number 1 can be realized as a smooth specialization of the rational homogeneous space parameterizing the lines on the 5-dimensional hyperquadric, in other words, it can be deformed nontrivially to the rational homogeneous space. We show that ${\bf X}$ is the only smooth projective variety with this property. This is obtained as a consequence of our main result that ${\bf X}$ can be recognized by its VMRT, namely, a Fano manifold of Picard number 1 is biregular to ${\bf X}$ if and only if its VMRT at a general point is projectively isomorphic to that of ${\bf X}$. We employ the method the authors developed to solve the corresponding problem for symplectic Grassmannians, which constructs a flat Cartan connection in a neighborhood of a general minimal rational curve. In adapting this method to ${\bf X}$, we need an intricate study of the positivity/negativity of vector bundles with respect to a family of rational curves, which is subtler than the case of symplectic Grassmannians because of the nature of the differential geometric structure on ${\bf X}$ arising from VMRT.
