All Questions
8 questions with no upvoted or accepted answers
4
votes
0
answers
179
views
How the existence of holomorphic sections depends on the choice of complex structure
In this Mathoverflow question it is asked how many invariant complex structures exist on the full flag manifold of $SU(m)$. In this question it is asked when a line bundle over a flag manifold has ...
- 443
2
votes
0
answers
128
views
Nonabelian Hodge correspondence for $\mathbb{G}_m$
Please excuse me if this question is too naive. I know very little about the nonabelian Hodge correspondence but I am trying to understand how the correspondence works in the simplest case of the ...
- 3,446
1
vote
0
answers
146
views
Compact complex manifolds with nef canonical bundle have nonnegative Kodaira dimension
Let $X$ be a compact Kähler manifold with nef canonical bundle. The (Kähler extension of the) abundance conjecture asserts that $K_X$ is semi-ample, and thus $K_X^{\otimes m}$ admits a section for ...
- 275
1
vote
0
answers
255
views
Proof of uniqueness in the universal property of Poincaré line bundles
My question concerns the proof of a part of Lemma IV.2.2 (pag. 168) of the book Geometry of Algebraic Curves. vol. I by Arbarello, Cornalba, Griffiths and Harris. In order to state my problem, let me ...
- 55
1
vote
0
answers
726
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Questions on Néron–Severi group
$\DeclareMathOperator\NS{NS}\DeclareMathOperator\Pic{Pic}$I have two questions on a comment from Daniel Hyubrechts's Complex Geometry on pages 133/134.
Let $X$ be a compact Kähler manifold. Consider ...
- 6,016
1
vote
0
answers
52
views
Connection on line bundle over general simplicial toric variety
In https://arxiv.org/pdf/hep-th/0005247.pdf, on page 60 and 61, it is mentioned that the connection of $\mathcal{O}(-n)$ over a (simplicial) toric variety of the form
$$
(\mathbb{C}^N \backslash U)/(\...
- 1,155
1
vote
0
answers
137
views
Holomorphic line bundles associated to multiple U(1) groups, defined over toric manifolds
The sections of the holomorphic line bundle $\mathcal{O}(n)$ are acted on by the covariant derivative
$$
d+nA,
$$
where $A$ is the connection on the $U(1)$ bundle to which $\mathcal{O}(n)$ is ...
- 1,155
0
votes
0
answers
183
views
When can one find holomorphic sections vanishing at a point to a certain order?
Let $X$ be a compact complex manifold (say of dimension $2$) and $L \rightarrow X $ a holomorphic line bundle. Consider the following statements:
Statement $A_0$: Given any point $p\in X$, there ...
- 3,245