Let $X$ be a smooth projective variety. Given two vector bundles $V_1$ and $V_2$ such that $[V_1]=[V_2]\in K^0(X)$, can one expect that $V_1$ and $V_2$ can be connected by a family of vector bundles? Or are there any counterexamples?

For example, if we have a short exact sequence of vector bundles

$$0\rightarrow E_0\rightarrow E_1\rightarrow E_2\rightarrow 0 $$,

then $E_1$ can be deformed to $E_0\oplus E_2$ by finding a curve between corresponding elements in $Ext^1(E_2,E_0)$. Since the relations in $K^0$ are generated by short exact sequences of vector bundles, I was attempting to generalize this kind of argument but failed.

Another motivation for this question is that I want to know the possibility of identifying the Gromov-Witten theories of $\mathbb P(V_1)$ and $\mathbb P(V_2)$ for such $V_1$ and $V_2$ by deforming one to another. Therefore I think I would like to see, for example, families of vector bundles over a reducible variety.

  • $\begingroup$ The answer by abx below completely settles your question (in the negative). I recommend that you accept abx's answer. $\endgroup$ – Jason Starr Sep 2 '15 at 19:32
  • $\begingroup$ I enjoyed the two examples. Thank you both for the time. $\endgroup$ – Honglu Sep 3 '15 at 4:31

Maybe the simplest example of the phenomenon mentioned by Jason is the case of rank 2 vector bundles on $\mathbb{P}^3$ with $c_1$ even, studied in this paper of Atiyah and Rees. Besides $c_1$ and $c_2$ which encode the $K$-theory class, there is another topological invariant $\alpha $ with values in $\mathbb{Z}/2$, which can be $0$ or $1$ for holomorphic bundles with the same Chern classes.

  • $\begingroup$ That is wonderful! $\endgroup$ – Jason Starr Sep 2 '15 at 14:31
  • $\begingroup$ Possibly meaningless: the parity issue in Atiyah and Rees seems also to be related to the parity issue from this MO question:mathoverflow.net/questions/214655/… $\endgroup$ – Jason Starr Sep 2 '15 at 14:38
  • $\begingroup$ Wow thanks! That's interesting. Sorry for accepting it late. $\endgroup$ – Honglu Sep 3 '15 at 4:29

As always, you should specify that $X$ is projective (or at least proper). Even so, this really depends on what you mean by a "connected family". Certainly this is false if you want the base of your family to be irreducible. Here is the simplest counterexample I see. On $X=\mathbb{P}^2$, let $V_1$ be $\mathcal{O}(-7)\oplus \mathcal{O}(-7)\oplus \mathcal{O}(14)$ and let $V_2$ be $\mathcal{O}(-11)\oplus \mathcal{O}(-2)\oplus \mathcal{O}(13)$.

Since both rank $3$ locally free $V_i$ have zero first Chern class, $[V_i] - 3[\mathcal{O}]$ is in the second filtered subspace of the gamma filtration. Thus, the class in K-theory is uniquely determined by the second Chern class, and this is $-147c_1(O(1))^2$ for both $V_1$ and $V_2$. Thus $[V_1]$ equals $[V_2]$.

On the other hand, $\text{Ext}^1_{\mathcal{O}_{\mathbb{P}^2}}(V_i,V_i)$ equals $\{0\}$ for both $i=1,2$. Thus both locally free sheaves are infinitesimally rigid. Thus, there is no family of vector bundles on $\mathbb{P}^2$ over an irreducible base that parameterizes both $V_1$ and $V_2$.

  • $\begingroup$ Thanks! Let me add $X$ being projective to the problem. I like your example a lot. But I think I would like to allow reducible bases as well. My ultimate goal is to see if one can identify the Gromov-Witten theories of $\mathbb P(V_1)$ and $\mathbb P(V_2)$ solely by deforming one to another. $\endgroup$ – Honglu Sep 2 '15 at 6:48
  • $\begingroup$ Even so, almost certainly there will be counterexamples. Even as topological vector bundles, there is more information in the homotopy class of the map $u:X \to BGL_n$ than just the Chern classes, i.e., the pullback map $H^*(u;\mathbb{Z})$ on cohomology (John Morgan gave me a reference for this some time ago -- I will try to dig it up again). $\endgroup$ – Jason Starr Sep 2 '15 at 7:20

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