Lifting vector bundles along thickenings

Question

Let $X_0 \to X$ be a nilpotent closed immersion of schemes. Is every vector bundle on $X_0$ the pullback of a vector bundle on $X$?

The answer is yes when $X$ is affine. In general, there may be counterexamples?

Answer 1

In the case of line bundles, your question is related to the behaviour of the Picard variety in families, and counterexamples can be found by looking for examples in which the Neron--Severi rank jumps.

E.g. let $E_t$ be a one-parameter family of elliptic curves, chosen so that $E_0$ is CM, but $E_t$ is not CM for $t \neq 0$. More precisely, choose the family so that the CM locus is precisely $t = 0$ (rather than some non-trivial nilp. thickening of $t = 0$). (Concretely, this just means that we're choosing the family so that the $j$-invariant is of degree one in $t$; you could even choose a family in which $t = j$.)

Now let $X_0 = E_0 \times E_0$, and let $X$ be the thickening of $X_0$ given by taking $E_t\times E_t$, and setting $t^2 = 0$.

On $X_0$ you can take a divisor $D_0$ attached to the graph of a non-trivial complex multiplication. This divisor does not deform to a Cartier divisor on $X$, and the associated line bundle does not extend to a line bundle on $X$.

---

This question is related to the variational form of the Lefschetz $(1,1)$-theorem. If the thickening is a thickening over $k[\epsilon]$ (the dual numbers), as in my example, then the obstruction that Alex Youcis discusses is an element of $H^2(X_0,\mathcal O_{X_0}),$ and it is precisely the image in $F^0/F^1$ (here $F^{{\bullet}}$ denotes the Hodge filtration) of the transport via the Gauss--Manin connection of the Chern class of the given line bundle on $X_0$. The variational form of the $(1,1)$-theorem says that this image should vanish in order for the line bundle to deform (and one proof is via the obstruction computation that Alex discusses).