论文标题
尺寸在四维投影空间上的两个全态分布
Dimension two holomorphic distributions on four-dimensional projective space
论文作者
论文摘要
我们在$ \ mathbb {p}^4 $上研究二维全体形状分布。我们将尺寸分配给两个分布，最多为$ 2 $，具有本地免费的切线捆或本地自由的圆锥形捆，其单数方案具有纯尺寸。我们证明相应的滑轮是分开的。接下来，我们研究此类分布的几何形状，从最大不可融合到可集成分布的研究。在最大不可融合的情况下，我们表明该分布是洛伦兹型的分布，或者是通过cartan延长的cartan延长图的合理图，在加权投影仪上的cartan延长3倍。我们研究了$ \ mathbb {p}^4 $中尺寸的分布，其圆形或骨是horrocks-Mumford Sheaves，描述了其单数方案的数值不变性，这些方案是平稳且连接的。这样的分布是最大不可汇总的，由它们的单数方案确定，并且由$ h_5 \ rtimes sl（2，\ mathbb {z} _5）\ subset sp（4，\ mathbb {q}）$不变，其中$ h_5 $是Heisenberg Group of Lever $ 5 $ 5 $ 5 $。我们证明，蒙福德分布的模量空间是不可约的准标准品种，我们确定了它们的尺寸。最后，我们观察到，在$ \ mathbb {p}^4 $上有一个$ d \ geq 6 $的condimension One分布的空间，有一个退化的平坦全态riemannian指标。此外，此类指标的退化除数由$ H_5 \ rtimes sl（2，\ Mathbb {z} _5）$不变的编态度分布组成，并沿着堕落的亚伯利亚表面（$（1,5）$（1,5）$ - 极性和级别 - 级别的$ 5 $ - $ 5 $ - $ 5美元。
We study two-dimensional holomorphic distributions on $\mathbb{P}^4$. We classify dimension two distributions, of degree at most $2$, with either locally free tangent sheaf or locally free conormal sheaf and whose singular scheme has pure dimension one. We show that the corresponding sheaves are split. Next, we investigate the geometry of such distributions, studying from maximally non-integrable to integrable distributions. In the maximally non-integrable case, we show that the distribution is either of Lorentzian type or a push-forward by a rational map of the Cartan prolongation of a singular contact structure on a weighted projective 3-fold. We study distributions of dimension two in $\mathbb{P}^4$ whose the conormal sheaves are the Horrocks-Mumford sheaves, describing the numerical invariants of their singular schemes which are smooth and connected. Such distributions are maximally non-integrable, uniquely determined by their singular schemes and invariant by a group $H_5 \rtimes SL(2,\mathbb{Z}_5) \subset Sp(4, \mathbb{Q})$, where $H_5$ is the Heisenberg group of level $5$. We prove that the moduli spaces of Horrocks-Mumford distributions are irreducible quasi-projective varieties and we determine their dimensions. Finally, we observe that the space of codimension one distributions, of degree $d\geq 6$, on $\mathbb{P}^4$ have a family of degenerated flat holomorphic Riemannian metrics. Moreover, the degeneracy divisors of such metrics consist of codimension one distributions invariant by $H_5 \rtimes SL(2,\mathbb{Z}_5)$ and singular along a degenerate abelian surface with $(1,5)$-polarization and level-$5$-structure.
