Extension of Ulrich bundles

Technical Summary: Extension of Ulrich Bundles

Problem Statement
This paper investigates the conditions under which an Ulrich bundle defined on a smooth, non-degenerate subvariety $X \subset \mathbb{P}^n$ can be extended to a vector bundle on the ambient projective space $\mathbb{P}^n$. While the existence of Ulrich bundles on smooth subvarieties is a major conjecture (proven for complete intersections), the question of their extendability to the ambient space remains largely open. The author seeks to determine when such an extension is possible and, conversely, to characterize the obstructions preventing it.

Methodology and Framework
The study is conducted over the field of complex numbers $\mathbb{C}$. The author utilizes the cohomological characterization of Ulrich bundles: a vector bundle $E$ on $X$ is $L$-Ulrich if $H^i(X, E \otimes L^{-p}) = 0$ for all $i$ and $1 \le p \le \dim X$.

Key methodological tools include:

1. Cohomological Vanishing and ACM Properties: The paper leverages the fact that Ulrich bundles are arithmetically Cohen-Macaulay (ACM). It employs induction on dimension using hyperplane sections and the short exact sequences associated with them to propagate ACM properties from subvarieties to the ambient space.
2. Chern Class Invariants: A central tool is the invariant $\tau(E) = \mu(E(k))$, where $\mu(E) = c_1(E)/\text{rank}(E)$ and $k$ is the smallest integer such that $E(k)$ is globally generated. The paper analyzes the behavior of $\tau$ under exterior powers, symmetric powers, and tensor products, establishing inequalities that constrain the possible geometry of $X$ if an extension exists.
3. Inductive Arguments: The proofs for arbitrary subvarieties rely on a minimality argument on the dimension of $X$, reducing higher-dimensional cases to curves or surfaces via general hyperplane sections.
4. Explicit Constructions: For the existence results on curves, the author constructs specific extensions using multiplication maps of polynomial rings and properties of the evaluation map for vector bundles on $\mathbb{P}^n$.

Key Results

• Non-Extendability for Complete Intersections (Theorem A):
  The paper proves that if $X$ is a smooth, non-degenerate complete intersection of dimension $\ge 2$, an Ulrich bundle on $X$ cannot extend to a vector bundle on $\mathbb{P}^n$ unless the extension is trivial and $X = \mathbb{P}^n$. The proof relies on showing that if an extension $E$ exists, it must be a split bundle (by Horrocks' criterion), and the restriction of a split bundle to a complete intersection of dimension $\ge 2$ cannot satisfy the Ulrich condition unless the bundle is trivial and the variety is the whole space.

• Characterization of Extendable Bundles (Theorem B):
  For an arbitrary smooth non-degenerate subvariety $X$ of degree $l$, the paper characterizes when a rank $r$ bundle $E$ on $\mathbb{P}^n$ restricts to an $\mathcal{O}_X(a)$-Ulrich bundle, assuming the condition $\tau(E) < 1$. Under this assumption, $E(1-a)$ is globally generated, and the pair $(X, E)$ must fall into one of five specific geometric configurations:

  1. $X = \mathbb{P}^2$, $a=2$, $r$ is even, and $E \cong T_{\mathbb{P}^2}^{\oplus r/2}$.
  2. $X \cong \mathbb{P}^1$ and $l$ divides $r$.
  3. $X = \mathbb{P}^n$, $a=1$, and $E$ is trivial.
  4. $X \cong \mathbb{P}^2$, $a=1$, $l=4$, and $n \in \{4, 5\}$.
  5. $X \cong \mathbb{P}(\mathcal{E})$ for an ample bundle $\mathcal{E}$ on $\mathbb{P}^1$, with specific line bundle identifications.
     In all cases except (1), $\tau(E) = (l-1)/l$.

• Applications to Tangent Bundles (Theorem C):
  Applying Theorem B, the author determines when twists of exterior powers ($\wedge^p T_{\mathbb{P}^n}$), symmetric powers ($S^p T_{\mathbb{P}^n}$), and tensor powers ($T_{\mathbb{P}^n}^{\otimes p}$) restrict to Ulrich bundles on subvarieties. The results show that such restrictions are only possible if $X$ is a rational normal curve (with specific twists) or if $X = \mathbb{P}^2$ with specific parameters. This generalizes previous results by López and Zamora regarding the tangent and cotangent bundles.

• Existence of Extending Bundles on Curves (Theorem D):
  Contrasting with the non-existence result for higher-dimensional complete intersections, the paper proves that non-trivial Ulrich bundles on curves do extend to $\mathbb{P}^n$.

  1. If $X$ is a complete intersection curve, there exists a non-trivial Ulrich bundle on $X$ that extends to $\mathbb{P}^n$.
  2. For a smooth rational curve of degree $l$ in $\mathbb{P}^n$, any $\mathcal{O}_X(a)$-Ulrich bundle of rank $r$ (where $l|r$) extends to a globally generated vector bundle on $\mathbb{P}^n$ (specifically, the extension $E$ satisfies that $E(1-a)$ is globally generated). The author notes this global generation property is optimal given the vanishing of $H^0(X, F(-a))$.

Significance and Claims
The paper claims to provide a definitive obstruction to the extension of Ulrich bundles in the case of complete intersections of dimension $\ge 2$, showing that such extensions are essentially impossible. Conversely, it provides a complete classification of the rare cases where extensions are possible for arbitrary subvarieties, subject to a bound on the invariant $\tau(E)$.

The work generalizes existing characterizations of when restrictions of tangent bundles are Ulrich, unifying these under the framework of Theorem B. Furthermore, it establishes the existence of extending Ulrich bundles on curves, offering explicit constructions that achieve optimal positivity (global generation) for the extended bundle. The author explicitly states that the global generation result in Theorem D(2) is a new contribution, as previous literature (specifically [20]) did not guarantee this strong property without the global generation hypothesis.