Sign up ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $X$ and $Y$ be connected quasi-projective varieties over $\mathbf{C}$. Let $\mathcal{L}$ be an algebraic vector bundle over $X\times Y$. Let $p_2:X\times Y\rightarrow Y$ be the projection.

($\star$) Assume that for all $y_0\in Y$ one has that $\mathcal{L}|_{X\times \{y_0\}}$ is trivial.

In this case, the first (naive) intuition is that $\mathcal{L}\simeq p_2^*\mathcal{F}$ for some algebraic vector bundle $\mathcal{F}$ over $Y$.

Q1: Give an example where $(\star)$ is satified but where $\mathcal{L}$ is not the pullback of any algebraic vector bundle $\mathcal{F}$ over $Y$.

Q2: Under what (interesting) additional assumptions on $X$, $Y$ and $\mathcal{L}$ is it possible to conclude that $\mathcal{L}$ is the pullback of a vector bundle over $Y$?

For example, if $X$ is complete then the square theorem says that $\mathcal{L}$ is the pullback of a vector bundle over $Y$.

share|cite|improve this question

1 Answer 1

There are many examples. Take $X := \mathbb A^1$ and $Y := \mathbb A^2 \smallsetminus \{(0,0)\}$; since all vector bundles on $X$ and $Y$ are trivial, it is sufficient to give an example of a vector bundle on $U := X \times Y$ that is not trivial.

Consider the embedding $j \colon U \subseteq \mathbb A^3$. The are reflexive sheaves $F$ on $\mathbb A^3$ that are locally free on $\mathbb A^3 \smallsetminus \{(0,0, 0)\}$, but not at the origin. For example, you can take the sheaf associated with a second syzygy of the $\mathbb C[x,y,z]$-module $\mathbb C[x,y,z]/(x,y,z)$; this is not projective, because the projective dimension of $\mathbb C[x,y,z]/(x,y,z)$ over $\mathbb C[x,y,z]$ is 3, but it is reflexive, because every second syzygy is. The restriction $G$ of $F$ to $U$ is a non-trivial locally free sheaf on $U$. In fact, since $F$ is reflexive we have $F = j_*G$, so if $G$ were trivial $F$ would also be a free sheaf, which is not the case.

share|cite|improve this answer
Hi @Angelo, thanks for the example. I never played with reflexive $\mathcal{O}_X$-modules. They seem to enjoy nice properties like $j_*G=F$. Do you have a reference for this last property? –  Hugo Chapdelaine Mar 22 '13 at 15:54
A good introduction to reflexive sheaves is "Stable reflexive sheaves", by Hartshorne, Mathematische Annalen (1980). –  Angelo Mar 22 '13 at 19:39
thanks for the reference! –  Hugo Chapdelaine Mar 23 '13 at 17:49

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.