Extension of Ulrich bundles

Abstract: We study extension of Ulrich bundles from a smooth nondegenerate subvariety $X$ of $\mathbb{P}^n$. If $X$ is a complete intersection of dimension $\geq 2$, we show that the extension is not possible except in the trivial case. For an arbitrary $X$, we characterize when the extension is possible, assuming some condition on the extended vector bundle. As an application, we generalize previous results of L{ó}pez and Zamora. We also give several classes of examples of Ulrich bundles on curves that extend to the ambient projective space.

• The paper establishes that, for smooth, nondegenerate complete intersections of dimension ≥2, nontrivial Ulrich bundles cannot extend beyond trivial cases.
• It characterizes extension possibilities for arbitrary subvarieties using the T(E) invariant and Chern class restrictions, identifying five distinct geometric configurations.
• Applications include determining when tangent, wedge, and symmetric powers of projective space bundles restrict to Ulrich bundles, impacting syzygy theory and moduli questions.

Extension Properties of Ulrich Bundles

Introduction

The study of Ulrich bundles has been central in understanding deep interactions between commutative algebra, syzygy theory, and projective algebraic geometry. This paper examines the conditions under which an Ulrich bundle on a smooth, nondegenerate subvariety $X$ of projective space $\mathbb{P}^r$ can be extended to a vector bundle over the ambient projective space. The work establishes nonexistence and existence results for such extensions in various geometric settings, characterizes extension possibilities under Chern class constraints, and provides applications to symmetrizations and wedge powers of tangent bundles. The results subsume and generalize several earlier theorems, notably those of López and Zamora.

Non-Extension over Complete Intersections

The primary result asserts that, except for the trivial case, Ulrich bundles on smooth, nondegenerate complete intersections $X \subset \mathbb{P}^r$ of dimension at least two cannot be extended to vector bundles on $\mathbb{P}^r$. The proof leverages the Arithmetically Cohen-Macaulay (ACM) property, cohomology vanishing, and Horrocks' criterion to restrict possible extensions to trivial bundles. That is, for such $X$, any vector bundle $\mathcal{E}$ with $\mathcal{E}|_X$ Ulrich must be trivial, and $X$ must in fact be the entire projective space.

Characterization of Extensions for Arbitrary Subvarieties

Moving beyond complete intersections, the paper develops a detailed characterization for the extension of Ulrich bundles from arbitrary smooth nondegenerate subvarieties under Chern class and positivity assumptions. The key invariant is $T(\mathcal{E})$, defined in terms of the first Chern class and the minimal twist required for global generation. For bundles $\mathcal{E}$ on $\mathbb{P}^r$0 with $\mathbb{P}^r$1 Ulrich and $\mathbb{P}^r$2, only five classes of extensions are possible, corresponding to specific geometric configurations, such as:

• $\mathbb{P}^r$3 and $\mathbb{P}^r$4
• $\mathbb{P}^r$5 and arbitrary ranks divisible by the degree
• Cases involving projective bundles over $\mathbb{P}^r$6
• The trivial bundle case
• Quadric cases (excluded in dimension at least two by the previous result)

The global generation requirement imposed is notably strong and distinguishes these results from earlier extension theorems, ensuring optimal extension positivity.

Applications to Tangent and Symmetric Powers

The theorems are applied to determine for which subvarieties and which twists the wedge, symmetric, and tensor powers of the tangent bundle of $\mathbb{P}^r$7 restrict to Ulrich bundles. The characterization is precise: outside of trivial and exceptional cases, the only possibilities occur for rational normal curves ($\mathbb{P}^r$8 embedded via degree $\mathbb{P}^r$9), and for certain cases with $X \subset \mathbb{P}^r$0. This generalizes the work of López and Zamora ["On the Ulrichness of twisted syzygies and dual syzygies bundles" (López et al., 6 Jan 2026)], showing that such Ulrichness is highly constrained.

Examples and Positive Extension Results on Curves

While negative results dominate for high-dimensional varieties, the paper demonstrates the existence of nontrivial Ulrich bundles on smooth projective curves (including complete intersections and rational normal curves) that can be extended to globally generated vector bundles on the ambient projective space. This covers the classical case $X \subset \mathbb{P}^r$1 of rational normal curves and establishes explicit extension and positivity properties using detailed cohomological and vector bundle techniques. The results in this direction offer a contrasting account, showing the geometry-dependent subtlety of Ulrich bundle extension.

Implications and Future Directions

These results consolidate and generalize a suite of extension phenomena for Ulrich bundles, clarifying exactly when and how such bundles are compatible with the projective ambient structure. On the theoretical level, they reinforce the rigidity of Ulrich bundles on higher-dimensional complete intersections and illuminate the delicate structure on curves and certain special varieties. The novel characterization using the $X \subset \mathbb{P}^r$2 invariant may inspire further investigation into boundedness, stability, and moduli of Ulrich bundles constrained by their (non-)extendability. The positive results on globally generated extensions in the curve case refocus attention on rational and elliptic curves as a testing ground for vector bundle theory in projective geometry.

From a broader perspective, these findings provide new constraints and tools for the minimal resolution conjecture, Boij–Söderberg theory, and the study of syzygies, as Ulrich bundles often manifest as test objects in these domains. Further avenues for research include a detailed classification of extension behaviors for other homogeneous spaces, explicit construction of Ulrich bundles in new cases, and exploration of the moduli-theoretic consequences for families of varieties.

Conclusion

This paper delivers a comprehensive analysis of the extension problem for Ulrich bundles on subvarieties of projective space, proving stark rigidity in higher-dimensional complete intersection cases and offering sharp extension criteria for arbitrary subvarieties under Chern class restrictions. The theorems unify and broaden previous work, illuminate the geometric subtleties of Ulrich bundles, and provide a toolkit for further explorations in algebraic vector bundle theory and projective geometry.