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Results tagged with vector-bundles
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user 9871
A continuously varying family of vector spaces of the same dimension over a topological space. If the vector spaces are one-dimensional, the term line bundle is used and has the associated tag line-bundles.
6
votes
Positive vector bundles
The positivity you are talking about is named Nakano positivity. It implies Griffiths positivity (you only require positivity of the hermitian form on $T_M\otimes E$ on rank-one tensors) which implies …
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0
votes
Accepted
Normal bundle of $CP^1$ in $CP^2$
With your notations, the normal bundle is spanned over $\{Z_1\ne 0\}$ by $\partial/\partial v$. Now, over $\{Z_0\ne 0\}$, take affine coordinates $x=Z_1/Z_0$ and $y=Z_2/Z_0$, so that, where defined, y …
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1
vote
Accepted
Is there a "simple commutation" relation between $D^{''}$ and $\delta^{'}$, with $D^{''}$ ...
Yes, the relation is that they anti-commute.
Let's see this very briefly (you can find it in almost all books on Kähler geometry and Hodge theory).
We want to compute $[D''_E,\delta'_E]$, where $[\ …
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3
votes
Accepted
Semi-stability of $S^n\Omega_S$ with respect to $K_S$
Ciao Francesco!
The answer to your question is yes, and the work of Bogomolov you are looking for about semistability of the tangent space (for minimal surfaces of general type indeed, no need of ampl …
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9
votes
Direct sum of two stable bundles of same slope
One possible answer could be: by opening any book on vector bundles, and looking at the proposition right after the definition of stable vector bundle. :)
Another possible answer is as follows.
What …
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4
votes
Chern classes via degeneracy loci
The construction you want via degeneracy loci definitely goes through in the general setting of a complex vector bundles over a smooth compact oriented manifold.
It is often referred to as Gauss-Bonn …
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5
votes
Top self-intersection of the tautological line bundle
As Francesco showed you this is quite straightforward, therefore I think is a good opportunity to give a more general answer here.
If you have a rank $r$ holomorphic vector bundle $E$ over a compact …
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6
votes
2
answers
704
views
Ample vector bundles on complex tori
Let $X$ be a $n$-dimensional complex torus and $\omega$ a Kähler form on $X$. Then, it is well known that a real $(1,1)$-class $[\alpha]\in H^{1,1}(X,\mathbb R)$ is a Kähler class if and only if for a …
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12
votes
3
answers
3k
views
Relationship between monodromy representations and isomorphism of flat vector bundles
This question is somehow related to this one.
Let $M$ be a smooth (compact, if you wish) connected manifold.
Then, it is well known that there is an equivalence between the isomorphism classes of p …
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5
votes
Accepted
Nakano semipositivity
I would say more properly that nowadays it is not known any satisfactory algebraic description or characterization of the concept of Nakano's positivity for a hermitian vector bundle.
I would like al …
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5
votes
Accepted
Existence of a connection $A$ on a holomorphic line bundle $L$, s.t $F(A)=(\deg L)\omega$
Let me expand a bit Henri's (hi Henri!) answer, even if this is completely standard. In general, given a compact Kähler manifold $X$ of any dimension, given a holomorphic line bundle $L\to X$, and giv …
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