[Paper Review] Triviality and Split of Vector Bundles on Rationally Connected Varieties

Xuanyu Pan|arXiv (Cornell University)|Oct 22, 2013

Advanced Algebra and Geometry19 references3 citations

TL;DR

This paper establishes a new triviality criterion for vector bundles on rationally connected varieties using families of rational curves satisfying the Lefschetz condition, proving that a vector bundle is trivial if its restriction to every rational curve is trivial. It further proves that a vector bundle on a homogeneous space is trivial iff its restriction to every Schubert line is trivial, and uses this to derive a splitting criterion for uniform vector bundles of low rank on classical Grassmannians and quadrics.

ABSTRACT

In this paper, we give a simple proof of a triviality criterion due to I.Biswas and J.Pedro and P.Dos Santos. We also prove a vector bundle on a homogenous space is trivial if and only if the restrictions of the vector bundle to Schubert lines are trivial. Using this result and Chern classes of vector bundles, we give a general criterion of a uniform vector bundle on a homogenous space to be splitting. As an application, we prove a uniform vector bundle on classical Grassmannians and quadrics of low rank is splitting.

Motivation & Objective

• To provide a simple proof of a triviality criterion for vector bundles on separably rationally connected varieties using families of rational curves.
• To establish a triviality criterion for vector bundles on homogeneous spaces based on their restriction to Schubert lines.
• To develop a general splitting criterion for uniform vector bundles of low rank on homogeneous spaces using Chern class theory and good divisibility of the VRMT.
• To apply the splitting criterion to classical Grassmannians and quadrics, proving that uniform vector bundles of rank at most s(X) are splitting.
• To extend existing results on uniform vector bundles by providing a uniform, cohomologically grounded method applicable across multiple classical homogeneous spaces.

Proposed method

• Utilizes the existence of a family of rational curves on a separably rationally connected variety satisfying the Lefschetz condition to reduce the triviality of a vector bundle to its restriction on rational curves.
• Applies the Lefschetz pencil technique and base change theorems to lift triviality from fibers to the total space via flatness and projection formulas.
• Employs the theory of Chern classes and the structure of the Chow ring to define 'good divisibility' up to degree r for the VRMT (Virtual Resolution of the Moduli of Torsors) of a homogeneous space.
• Applies induction on the rank of vector bundles and uses vanishing of Ext^1 groups (via H^1(G/P, L) = 0 for dim(G/P) ≥ 2) to prove splitting when the bundle is uniform and the VRMT has good divisibility.
• Uses the Bruhat decomposition and Chow ring isomorphisms (CH*(X) ≅ H*(X, Z)) for quadrics and Grassmannians to analyze good divisibility via tensor product and projective bundle formulas.
• Applies lemmas on good divisibility for products and projective bundles to extend the criterion to products and flag varieties, including orthogonal and symplectic Grassmannians.

Experimental results

Research questions

• RQ1 Under what conditions is a vector bundle on a rationally connected variety trivial if its restriction to every rational curve is trivial?
• RQ2 When is a vector bundle on a homogeneous space G/P trivial, given that its restriction to every Schubert line is trivial?
• RQ3 What conditions ensure that a uniform vector bundle of low rank on a homogeneous space splits into a direct sum of line bundles?
• RQ4 Which classical homogeneous spaces (e.g., Grassmannians, quadrics) admit a splitting criterion for uniform vector bundles of rank at most s(X)?
• RQ5 How does the good divisibility of the VRMT of a homogeneous space relate to the splitting of uniform vector bundles of bounded rank?

Key findings

• A vector bundle on a smooth projective variety over an algebraically closed field is trivial if and only if its restriction to every rational curve is trivial, provided the variety is separably rationally connected.
• A vector bundle on a homogeneous space G/P is trivial if and only if its restriction to every Schubert line is trivial.
• For a homogeneous space G/P of Picard number one, a uniform vector bundle of rank at most r is splitting if its VRMT has good divisibility up to degree r.
• A uniform vector bundle on a classical Grassmannian G(k,n) of rank at most min(k, n−k−1) is splitting, as confirmed by the good divisibility of its VRMT.
• A uniform vector bundle on a quadric Q^n of rank at most n−2 (if n odd) or n−3 (if n even) is splitting, due to the good divisibility of its Chow ring structure.
• The method uniformly proves splitting for low-rank uniform bundles on various classical homogeneous spaces, including orthogonal and symplectic Grassmannians, via a general criterion based on VRMT and Chern class theory.