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Let $\pi \colon E \rightarrow \mathbb{CP}^1$ be a complex vector bundle. It is a well-known fact that a Dolbeault operator on $\pi\colon E \rightarrow \mathbb{CP}^1$ gives a holomorphic structure on $E$.

My questions are derived from this fact:

  1. Let $\{D_t\}_{t\in [0,1]}$ be a smooth family of Dolbeault operators on $\pi \colon E \rightarrow \mathbb{CP}^1$. This family induces a family of holomorphic vector bundles whose underlying complex vector bundle is the given one. Are these holomorphic vector bundles isomorphic as holomorphic vector bundles?
  2. Suppose that the answer to the first question is no in general. What condition on $\{D_t\}_{t\in [0,1]}$ except for a constant family gives us a family of isomorphic holomorphic vector bundles?

Any comments and references are welcome. Thank you in advance.

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The answer to the first question is in fact no: consider the family of Dolbeaut operators on the complex vectorbundle of degree 0 and rank 2 underlying $$V=\mathcal O(-1)\oplus\mathcal O(1).$$ Consider the family of operators $$\bar\partial^t=\begin{pmatrix}\bar\partial^{\mathcal O(-1)} & t\, \gamma \\0& \bar\partial^{\mathcal O(1)} \end{pmatrix}$$ parametrised by $t\in\mathbb C$, where $\gamma\in\Gamma(CP^1,\bar K K)=\Omega^2(CP^1,\mathbb C)$ is a 2-form with non-vanishing integral. By Serre-duality the bundle $\mathcal O(1)$ is a holomorphic subbundle w.r.t. the holomorphic bundle defined by the Dolbeaut operator $\bar\partial^t$ if and only if $t=0$.

Concerning 2: You should compute the dimensions $D(d)$ of $H^0(CP^1, (V,\bar\partial^t)\otimes \mathcal O(d))$ for some $d\in\mathbb Z.$ If the dimension is constant for all $d\in\mathbb Z$ the bundles are isomorphic by Grothendieck splitting. But it is enough to restrict to some $d$'s, depending on the actual situation. For example, if the holomorphic bundle is trivial at one point in the (connected) family, then all bundles are isomorphic if the dimension $D(-1)=0$ stays trivial.

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  • $\begingroup$ Thank you for answering the questions. Let me ask some questions about Concerning 2. I cannot understand why the condition that $H^0(CP^1, (V,\bar\partial^t)\otimes \mathcal{O}(d))$ is constant implies the answer to the second question. Could you please explain more details? Also, I'd like to ask whether this cohomology group is computable in practice. Thank you. $\endgroup$
    – Math1016
    Commented Aug 10, 2021 at 0:22
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    $\begingroup$ This follows from the Grothendieck splitting theorem, see en.wikipedia.org/wiki/Birkhoff–Grothendieck_theorem. In fact, you order the degrees of the line subbundles $a_1\geq a_2\geq\dots\geq a_k$. Then the largest degree occurs $l$-times, where $l$ is the dimension $D(-a_1)$. Note that $D(-a_1-1)=0.$ Similarly, you can detect the other degrees, and their multiplicities. $\endgroup$
    – Sebastian
    Commented Aug 10, 2021 at 9:37
  • $\begingroup$ It depends on the specific situation in which you want to calculate the bundle type. For example, in the theory of integrable systems, where these families might show up as push forward sheafs of families of line bundles over the spectral curve, the above criteria is quite useful. See for example Hitchin's article 'Riemann surfaces and integrable systems'. In some different situations it is very hard to obtain specific informations about the bundle types in a family. $\endgroup$
    – Sebastian
    Commented Aug 10, 2021 at 10:00
  • $\begingroup$ Thanks! The first comment makes sense to me. That is clear. As for the second comment, I am not so sure about whether the method is applicable to my situation. In any case, thank you so much for explaining the details kindly. $\endgroup$
    – Math1016
    Commented Aug 11, 2021 at 7:00

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