Geometry of conic connections

Abstract: A cone structure on a complex manifold $M$ is a closed submanifold $\mathcal C\subset \mathbb P TM$ of the projectivized tangent bundle of $M$ that is submersive over $M$. So this defines a set $\mathcal C_x$ of distinguished directions in each point $x\in M$. A conic connection on $\mathcal C$ is then a family of unparametrized curves on $M$ that comprises exactly one curve through each point $x\in M$ in each direction in $\mathcal C_x\subset \mathbb PT_xM$. This can be encoded by a line subbundle $\mathcal F\subset T\mathcal C$. The subclass of characteristic conic connections is defined by the vanishing of a simple invariant, called characteristic torsion. For those, one has a much more subtle and slightly mysterious invariant called the cubic torsion. The first aim of this article is to provide a new approach to the cubic torsion, which also leads to a geometric condition characterizing its vanishing. We then specialize to the case of isotrivial cone structures, for which the fibers of $\mathcal C$ are assumed to be of some fixed type. Such a structure induces a first-order $G$-structure on $M$ whose structure group is the projective automorphism group of the model fiber. Moreover, any connection $γ$ on the associated $G$-structure induces a conic connection $\mathcal F<sup>γ$ on $\mathcal C$. Assuming that the model fiber is homogeneous, we study the relation between the torsion and curvature of a connection $γ$ and the characteristic and cubic torsion of $\mathcal F<sup>γ$. As an application we discuss cone structures of subadjoint type, showing in particular that there are such structures admitting conic connections with vanishing characteristic and cubic torsion that are not locally flat.

• The paper establishes a new framework for cone structures on complex manifolds, highlighting conic connections and cubic torsion as key geometric invariants.
• It characterizes characteristic conic connections by linking the vanishing of torsion to principal connection properties in isotrivial settings.
• The study reveals that for subadjoint cone structures, vanishing cubic torsion does not imply local flatness, refining classical rigidity results.

Summary of "Geometry of Conic Connections" (2607.03604)

Introduction and Context

This paper defines a systematic framework for the study of cone structures on complex manifolds and their associated conic connections, focusing on conceptual clarity and geometric invariants, particularly cubic torsion. The analysis is situated at the interface of differential geometry, algebraic geometry, and the theory of $G$-structures, with special attention to isotrivial cone structures whose local fibers are projectively equivalent to a fixed model variety.

The key motivation is to understand how geometric data (such as families of unparametrized curves constrained to prescribed directions in tangent spaces) are encoded and characterized by canonical invariants: characteristic torsion and cubic torsion. The paper also refines the connection between the local geometry of cone structures and homogeneous spaces, especially for subadjoint varieties arising in the representation theory of simple Lie algebras.

Cone Structures and Conic Connections

A cone structure on a complex manifold $M$ is defined as a surjective submanifold $of the projectivized tangent bundle$\mathbb{P}TM$such that fibers over$x\in M$select a distinguished set of directions in$T_xM$$. Isotrivial cone structures are the subclass where these fibers are all projectively equivalent to a fixed model$$Z\subset \mathbb{P}W$. A conic connection on a cone structure$ is a family of unparametrized holomorphic curves through each point and direction in $_x$, formalized as a line subbundle of $T^{-1}$ (the first nontrivial bundle in a canonical filtration of $T \mathcal{C}$). The notion generalizes path geometry and is an abstraction of various geometric structures, including those arising from holomorphic conformal structures and varieties of minimal rational tangents (VMRTs).

The paper develops a detailed intrinsic description of the natural vector bundle filtrations that encode the geometry of cone structures. These filtrations play a central role in defining the local invariants (characteristic torsion and cubic torsion) and understanding their geometric implications.

Characteristic and Cubic Torsion

A conic connection is called characteristic if its characteristic torsion vanishes. The characteristic torsion is a primary invariant expressed as a component of the Levi bracket associated to the natural filtration. Characteristic conic connections, where this torsion vanishes, are often unique (modulo certain projective degeneracies in the fibers).

For these characteristic connections, a finer invariant—the cubic torsion—arises. The cubic torsion is subtle: for instance, its vanishing in the context of VMRT structures leads to powerful rigidity theorems (e.g., the local recognition of rational homogeneous varieties from the geometry of their VMRTs). Previous definitions of cubic torsion relied on formulas from parabolic geometry; this paper gives a novel, geometric interpretation, relating it to the canonical (Bott) partial connection on a natural bundle associated with the foliation by the family of distinguished curves.

The main technical result in this direction is a precise description of when the Bott connection preserves a canonical split quaternionic structure on a specific associated vector bundle, showing that this occurs if and only if the cubic torsion vanishes. This result generalizes structural properties of path geometries and links the torsion invariants to projective geometric data.

Isotrivial Cone Structures, $G$-Structures, and Principal Connections

In the isotrivial case, where the model fiber $Z$ is homogeneous under a group $G$, the cone structure induces a $G$-structure on $M$0. Principal connections on this $M$1-structure then induce conic connections on the cone structure. The paper systematically relates the torsion and curvature of the principal connection to the characteristic and cubic torsion of the induced conic connection via explicit vector bundle morphisms and correspondences.

A main result equates the vanishing of characteristic torsion for the induced conic connection to the principal connection's torsion lying in an explicit projectively defined subbundle. Furthermore, under mild conditions (e.g., the model cone $M$2 homogeneous), the cubic torsion is expressed explicitly in terms of both the torsion and the curvature of the principal connection. Vanishing of the cubic torsion thereby becomes equivalent to the curvature taking values in a subspace described by the projective geometry of $M$3.

Applications to Subadjoint Cone Structures

The results are then applied to cone structures modeled on subadjoint varieties—special homogeneous spaces $M$4 associated to the minimal nilpotent orbits in simple Lie algebras not of types $M$5 or $M$6. These include a broad class of compact Hermitian symmetric spaces in their minimal embeddings.

The authors demonstrate that for such subadjoint cone structures (excluding type $M$7), the existence of a characteristic conic connection is equivalent to the existence of a torsion-free connection on the associated first-order $M$8-structure (called a parabolic almost conformally symplectic structure, or PACS). They further prove that for these structures, all characteristic conic connections have identically vanishing cubic torsion.

Crucially, the paper establishes that the vanishing of both characteristic and cubic torsion does not generally imply the local flatness of the cone structure: specifically, for several subadjoint types, there exist non-flat cone structures (their holonomy is special but not flat) admitting characteristic conic connections with vanishing cubic torsion. Thus, the class of such cone structures is strictly broader than the class of locally flat or VMRT-cone structures.

Implications and Future Perspectives

The clarified geometric definition of cubic torsion advances the theoretical understanding of the local geometry of cone structures and ties the vanishing of this invariant to the existence of canonical geometric structures (Segre and split quaternionic structures) on natural foliated leaf spaces. These results unify and extend techniques from parabolic geometry, $M$9-structures, and the theory of minimal rational tangents.

The linkage of torsion/curvature constraints to flatness and rigidity results suggest avenues for further research, including:

• Precise classification of structures for which vanishing torsion and cubic torsion imply local flatness versus the existence of more flexible geometry (as shown for subadjoint types).
• Analysis of moduli and deformation theory of cone structures with vanishing invariants but non-flat geometry.
• Application to the theory of special holonomy and exotic symplectic holonomies, leveraging the connection with classification results in the theory of linear holonomy groups.
• Further exploration of the role of these invariants in algebraic and complex geometric settings, particularly in characterizing and reconstructing varieties with specified projective or rational curve geometry.

Conclusion

The paper provides a solid geometric foundation for the theory of conic connections on isotrivial cone structures, sharpening the role of characteristic and cubic torsion as diagnostic invariants for the local and global geometry. The results have implications for both the structure of highly symmetric geometric models (including rational homogeneous spaces) and their possible deformations. For subadjoint types, the existence of characteristic, cubic torsion-free conic connections is necessary but not sufficient for local flatness, demonstrating the need for refined invariants and deeper analysis in this domain.