Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userUnbendable rational curves of Goursat type and Cartan type
110
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
We study unbendable rational curves, i.e., nonsingular rational curves in a complex manifold of dimension $n$ with normal bundles isomorphic to $mathcal{O}_{mathbb{P}^1}(1)^{oplus p} oplus mathcal{O}_{mathbb{P}^1}^{oplus (n-1-p)}$ for some nonnegative integer $p$. Well-known examples arise from algebraic geometry as general minimal rational curves of uniruled projective manifolds. After describing the relations between the differential geometric properties of the natural distributions on the deformation spaces of unbendable rational curves and the projective geometric properties of their varieties of minimal rational tangents, we concentrate on the case of $p=1$ and $n leq 5$, which is the simplest nontrivial situation. In this case, the families of unbendable rational curves fall essentially into two classes: Goursat type or Cartan type. Those of Goursat type arise from ordinary differential equations and those of Cartan type have special features related to contact geometry. We show that the family of lines on any nonsingular cubic 4-fold is of Goursat type, whereas the family of lines on a general quartic 5-fold is of Cartan type, in the proof of which the projective geometry of varieties of minimal rational tangents plays a key role.
rate research
Read More
We give an introduction to the theory of varieties of minimal rational tangents, emphasizing its aspect as a fusion of algebraic geometry and differential geometry, more specifically, a fusion of Mori geometry of minimal rational curves and Cartan geometry of cone structures.
A nonsingular rational curve $C$ in a complex manifold $X$ whose normal bundle is isomorphic to $${mathcal O}_{{mathbb P}^1}(1)^{oplus p} oplus {mathcal O}_{{mathbb P}^1}^{oplus q}$$ for some nonnegative integers $p$ and $q$ is called an unbendable rational curve on $X$. Associated with it is the variety of minimal rational tangents (VMRT) at a point $x in C,$ which is the germ of submanifolds ${mathcal C}^C_x subset {mathbb P} T_x X$ consisting of tangent directions of small deformations of $C$ fixing $x$. Assuming that there exists a distribution $D subset TX$ such that all small deformations of $C$ are tangent to $D$, one asks what kind of submanifolds of projective space can be realized as the VMRT ${mathcal C}^C_x subset {mathbb P} D_x$. When $D subset TX$ is a contact distribution, a well-known necessary condition is that ${mathcal C}_x^C$ should be Legendrian with respect to the induced contact structure on ${mathbb P} D_x$. We prove that this is also a sufficient condition: we construct a complex manifold $X$ with a contact structure $D subset TX$ and an unbendable rational curve $C subset X$ such that all small deformations of $C$ are tangent to $D$ and the VMRT ${mathcal C}^C_x subset {mathbb P} D_x$ at some point $xin C$ is projectively isomorphic to an arbitrarily given Legendrian submanifold. Our construction uses the geometry of contact lines on the Heisenberg group and a technical ingredient is the symplectic geometry of distributions the study of which has originated from geometric control theory.
In a joint work with N. Mok in 1997, we proved that for an irreducible representation $G subset {bf GL}(V),$ if a holomorphic $G$-structure exists on a uniruled projective manifold, then the Lie algebra of $G$ has nonzero prolongation. We tried to generalize this to an arbitrary connected algebraic subgroup $G subset {bf GL}(V)$ and a complex manifold containing an immersed rational curve, but the proposed proof had a flaw.
In this paper, we will give a precise formula to compute delta invariants of projective bundles and projective cones of Fano type.
We provide an algorithm to check whether two rational space curves are related by a similarity. The algorithm exploits the relationship between the curvatures and torsions of two similar curves, which is formulated in a computer algebra setting. Helical curves, where curvature and torsion are proportional, need to be distinguished as a special case. The algorithm is easy to implement, as it involves only standard computer algebra techniques, such as greatest common divisors and resultants, and Grobner basis for the special case of helical curves. Details on the implementation and experimentation carried out using the computer algebra system Maple 18 are provided.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
