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Holomorphic tangent bundle can be defined by cocycle of holomorphic Jacobians of transition maps. But this method will give different bundles which only agree up to isomorphism. I see in some text that show the holomorphic tangent bundle of a complex submanifold is naturally a subbundle of the holomorphic tangent bundle of the ambient complex manifold. But the argument confuse me like other categorical constructions. The author "choose" a local holomorphic chart such that the submanifold embedded flatly. Why he doesnt check the effect of choose? I know this case can be save by two methods:

1 Check another choose give an essential same map;

2 Show the holomorphic tangent bundle has an invariant description by almost complex structure.

I cant understand the attitude of completely ignoring the checking. Furthermore if you can choose arbitrary cocycle, then you get different bundles and different maps, and in fact you define nothing. I wonder if we should fix a choice in a definition.

More generally, any categorical constructions will face similar case. E.g, when you define the kernels of maps between universal objects and consider their products, even compute some more complex constructions, why people never consider the agreement problem via different choose?

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Regarding universal objects: the problem of choice is an illusion, because these are always unique up to unique isomorphism. (If they weren't, then they don't deserve to be called universal!) –  Zhen Lin Dec 24 '11 at 16:19
    
But in concrte constructions, one often changes between distinct categories. The universal property is not clear at all. –  MiGang Dec 26 '11 at 6:43

1 Answer 1

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One way to define the holomorphic tangent bundle invariantly is the following.

Let $TM$ be the real tangent bundle and consider de complexification $T_{\mathbb{C}}M = TM \otimes \mathbb{C}$. The almost complex structure $J:TM \to TM$ extends $\mathbb{C}$-linearly to $J:T_{\mathbb{C}}M \to T_{\mathbb{C}}M$ and because $J^2=-Id$, it is diagonalizable with eigenvalues $\pm i$. Therefore the complexified tangent bundle splits as $T_{\mathbb{C}}M = T^{1,0}M \oplus T^{0,1}M$ where $T^{1,0}M$ and $T^{0,1}M$ are, respectively, the $+i$ and $-i$ eigenspaces.

Define the holomorphic tangent bundle of $M$ to be $T^{1,0}M$. It is not hard to see that this is a holomorphic vector bundle and that its cocycles are the holomorphic Jacobians.

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My problem is not how to define it invariantly, but why one doesn't consider the consistency in the cocycle definition. E.g. using cocycle one can show the existence of exact sequence of holomorphic conormal bundle and then define the normal bundle as the quotient, but no one show that the sequence is independent with the choice of the cocycles, and this will cause the validity of definition of normal bundle. –  MiGang Jan 3 '12 at 11:37

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