Tagged Questions
3
votes
0answers
29 views
Maps between total spaces of holomorphic vector bundles
I am wondering what is possible and what is not possible regarding maps between the total spaces of holomorphic vector bundles.
Let me outline a situation that is a bit more concrete, to help focus ...
1
vote
1answer
30 views
Positive curvature on holomorphic vector bundles
There must be a mistake in my understanding the definition of positivity for the curvature. Let me summarize:
Let $ (L,\nabla,h) \rightarrow M $ be a hermitian hol line bundle with Chern connection. ...
4
votes
0answers
47 views
Which is the correct universal line bundle: the tautological bundle or its dual?
With topological line bundles over $\mathbb{C}$, one learns that every line bundle is a pullback of the universal line bundle, which is the tautological line bundle over $\mathbb{C}P^\infty.$
In ...
1
vote
0answers
59 views
A funny condition for ampleness on a curve
Let $E$ be a locally free sheaf on a complete nonsingular curve $C$ over $\mathbb C$.
Suppose that, for all points $P$ in $C$, the locally free sheaf $$ E\otimes \mathcal{O}_C(P)$$ is an ample ...
8
votes
1answer
134 views
Are vector bundles on $\mathbb{P}_{\mathbb{C}}^n$ of any rank completely classified? (main interest $n=3$)
It is very well known that the group of line bundles on $\mathbb{P}_{\mathbb{C}}^n$ is exactly $\mathbb{Z}$. Are bundles of higher rank classified as well? If so, could anyone provide a nice ...
5
votes
4answers
138 views
Reference request: Chern classes in algebraic geometry
I have encountered Chern classes numerous times, but so far i have been able to work my way around them. However, the time has come to actually learn what they mean.
I am looking for a reference that ...
4
votes
1answer
102 views
Geometric meaning of Line-bundle product
I was wondering,
What could be a geometric/intuitive meaning of taking the (tensor) product of two line bundles on a smooth variety. Could you depict this to me with some example?
...
7
votes
2answers
85 views
Adjunction for varieties with higher codimension
For $X \subseteq \mathbb{P}^n$ a smooth hypersurface, the canonical divisor $K_X$ can be computed as
$$
K_X = (K_{\mathbb{P}^n} + X)|_X.
$$
Is there a similar formula where $X$ is of higher ...
6
votes
2answers
109 views
Conditions such that taking global sections of line bundles commutes with tensor product?
Let us work with projective algebraic varieties over $k= \mathbb{C}$. If necessary we can also assume smoothness of the varieties.
Of course it is not in general true that given two line bundles $L, ...
1
vote
0answers
49 views
Complex projective manifolds and holomorphic mappings
Let $X\subset\mathbb{P}^2$ be a complex manifold defined by a homogeneous polynomial of degree $d>3$. Let $$\phi:\mathbb{P}^1\rightarrow X$$ be a holomorphic map. Show that $\phi$ is constant.
...
7
votes
1answer
119 views
What is the line bundle $\mathcal{O}_{X}(k)$ intuitively?
I am always confused about how to understand the line bundle $\mathcal{O}_{X}(k)$ on a projective scheme $X=\mathrm{Proj}(\oplus_{n=0}^\infty A_{n})$. Of course this is by definition the ...
3
votes
2answers
236 views
When does a line bundle have a meromorphic section?
Let $X$ be a scheme and $D$ be a Cartier divisor on $X$. Then $D$ determines a line bundle $\mathcal{O}(D)$ on $X$. Under which condition, is the converse true? That is, when does a line bundle come ...
4
votes
1answer
127 views
Existence of Complex Structures on Complex Vector Bundles
Let $E$ be a real vector bundle on a smooth manifold $X$. Let $J : E \to E$ be a vector bundle morphism (i.e. $\pi \circ J = \pi$, where $\pi : E \to X$ is the projection map) with $J^2 = ...
1
vote
1answer
81 views
Definition of tautological section.
Reading Barth, Peters: Compact complex surfaces, i stumbled across the following:
Let $Y$ be an algebraic surface over $k =\mathbb{C}$, and $\mathcal{L}$ an invertible sheaf on $Y$. Denote by $p: L ...
1
vote
0answers
56 views
Complexification of a complex bundle.
Suppose that $E$ is a complex bundle already. This is to say that is can be viewed as a real bundle with an almost complex structure $j$. Then Taubes assert $E$ sits inside its complexification ...
5
votes
0answers
176 views
understanding this differential operator on a tensor product
I am currently trying to read the T. Kotake's paper "An Analytical Proof of the Classical Riemann Roch Theorem", in which he defines a differential operator which acts on smooth sections of a tensored ...
