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If you have a complex manifold $Z$, and if you have a $\mathbb{C}P^1$ embedded in $Z$, what are the available techniques/methods for computing the normal bundle of such a $\mathbb{C}P^1$? I am aware that any holomorphic vector bundle (of finite rank) over $\mathbb{C}P^1$ splits as a direct sum of holomorphic line bundles (is that result due to Grothendieck by the way?). So if $Z$ for instance has low dimension, computing Chern classes can come in handy, as well as using adjunction formulas. But what other methods are available in the general case, where the dimension of $Z$ could be high, say.

As another question, suppose you were able to describe the normal bundle $N$ using a short exact sequence

$0 \to E \to N \to F \to 0$

where both $E$ and $F$ are known, and their direct sum decomposition is known. I would not expect $E \oplus F$ to be the only possibility for $N$. However, it is clear that you get several restrictions, using Chern classes say. Can one describe all possibilities for $N$ using maybe cohomology? I remember stumbling on an old paper (perhaps by Sir Michael Atiyah) on this, but I don't remember the details. Perhaps I should split this into two posts, but they are related sets of questions.

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    What inputs are you allowed? For example, one of the easiest cases is when you have a projective line sitting inside a projective space. Then, the tangent bundle of the projective space maps onto the normal bundle and thus it is ample. But, not much more can be said without extra information. For your exact sequence, in the case I describe, you always have one such with $E$ a trivial bundle and $F$ a line bundle, but $N$ will have many possibilities. – 2017-01-09
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    @Mohan thank you. I understand that my questions are a bit vague here. Let us say that $Z$ is the intersection of two hypersurfaces of degree $d_1$ and $d_2$ in some $\mathbb{C}P^{n+2}$ and that our $\mathbb{C}P^1$ is a submanifold of $Z$ which is embedded linearly in tbe ambient projective space. In the generic case in this setting, does it make sense to ask about the precise decomposition of the normal bundle of a generic such $\mathbb{C}P^1$, say? – 2017-01-09
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    @Mohan or you can interpret my question algorithmically. Is there any algorithm that can determine the decomposition of the normal bundle, if $Z$ is described as the intersection of two specific known hypersurfaces in a complex projective space and the the line is generic, sits in $Z$ and is embedded linearly in the ambient projective space? – 2017-01-09
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    I guess my questions were more philosophical and about the existence of some general enough method to do such calculations and compute the decomposition of the normal bundle. – 2017-01-09
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    Before asking philosophical questions, a few computations should always be made, I think. For example, in the case you suggest, you will have an exact sequence, $0\to N\to \mathcal{O}_X(1)^{n+1}\to \mathcal{O}_X(d_1)\oplus\mathcal{O}_X(d_2)\to 0$, where $X=\mathbb{P}^1$. You can see that this give many possibilities for the normal bundle $N$ and many of them can occur. – 2017-01-09
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    Perhaps you can formulate your comments as an answer please? Next time I will be more specific with my questions (and apparently it is bad if you delete posts here). But your comments were really helpful to me. – 2017-01-09

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Regarding your first question, there are fairly obvious facts such as the adjunction formula plus the fact that if $C$ is general in a family filling up a $k$-dimensional subvariety then at least $k - 1$ line bundle summands in the normal bundle are nonnegative. Beyond that, I'm not aware of any really general results, just specific ones in specific situations (e.g. a general rational curve of given degree in $\mathbb{P}^n$).

The other question is much easier, and is certainly answerable "in principle" though I don't know of a neat formulation. $F$ is endowed with an essentially canonical filtration with quots $\mathcal{O}(a_i)$, where $a_i \ge a_2$$\ldots$ so working inductively you can assume $F$ has rank $1$. Then, dualizing, the question becomes: when is a general map between direct sums of line bundles$$\mathcal{O}(a_1, \ldots, a_{r + 1}) \to \mathcal{O}(b_1, \ldots, b_r)$$surjective? This means exactly that $\mathcal{O}(a.)$ occurs as an extension of $\mathcal{O}(b.)$ by $\mathcal{O}\left(\sum a.- \sum b.\right)$. But that question has an easy answer: consider the matrix of coefficients $[b_i-a_j]$ and let $A$ be the matrix with zeros for each $b_i-a_j<0$ and general entries for each $b_i-a_j \ge0$. This is an echelon-type matrix with zeros on the left bottom. Then $A$ should be have rank $r$ everywhere, meaning there should be at most one of

  • zero first column (i.e. $b_1
  • consecutive columns of the same shape (i.e. number of zeros).

Moreover for each of the $r$ many columns not followed by a column of the same shape, the degree at the corner should be zero. Call such a corner "essential". The polynomials at the essential corners should be constant, else they would vanish somewhere and lower the rank. There should be a neat way of phrasing this, maybe written down somewhere. But I don't have a reference.

Actually, come to think of it, there is a simple condition that the matrix $A$ above has rank $r$ everywhere: for all $i=1, \ldots, r$, we have $b_i \ge a_{i+1}$, and if $b_i

The first condition ensures that all the corners lie on or below the "diagonal" line $(1,2)$, $\ldots$ , $(r, r+1)$; the second ensures that the corners on the diagonal correspond to constants.

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    Thank you. I have two questions to help me understand your answer. First, is it WLOG that one can assume that F has rank 1? Second, what is the "positivity" result you are using? You seem to be using that $F$, before dualizing, must satisfy some positivity condition? Can you state the proposition/theorem that you are using please? – 2017-01-10
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    I mean, it is true that the map from $N$ onto $F$ I had in mind is obtained by projecting the normal bundle $N$ of the rational curve onto the restriction of the normal bundle of that same curve inside a submanifold of the ambient manifold containing that curve. So it would satisfy a positivity condition. But I wonder how general is the positivity result you are referring to, and what it states exactly (maybe I should buy those famous books on positivity by Lazarsfeld!). – 2017-01-10