All Questions
Tagged with divisors complex-geometry
21 questions
2
votes
0
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232
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Chern classes and rational equivalence
Let $X$ be a complex variety and let $l_1$ and $l_2$ be line bundles on $X$. Let $f_1$ and $f_2$ be sections of $l_1$ and $l_2$ respectively, and let $Z_1$ and $Z_2$ be their zero-sets.
I would like ...
- 6,377
1
vote
0
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47
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Positivity of self-intersection of dicisor associated to meromorphic function
In the book "Holomorphic Vector Bundles over Compact Complex Surfaces" by Vasile Brînzănescu, in the proof of theorem 2.13 there is the following claim
Let $X$ be a compact non-algebraic ...
- 111
3
votes
1
answer
216
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How to determine the type of a divisor on a product of elliptic curves?
I already asked this on Math.SE, but didn't receive an answer yet.
Say $E_1, \dotsc, E_n$ are elliptic curves (everything over $\mathbb C$), and $D \subset E_1 \times \dotsc \times E_n$ is an ...
- 1,286
1
vote
0
answers
184
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Divisor cohomology through spectral sequences
I don't know if it belongs here but anyway, I need to compute arithmetic genus of divisors pulled back from a Fano base space to a bundle (which may or mayn't be trivial) defined through the ...
- 11
1
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0
answers
190
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How to define a principal divisor on general complex spaces?
[I am not a native English speaker, so my sentences may sound strange. ]
I'm studying about complex analytic spaces. For meromorphic functions, I don't know how to define their principal divisors ...
- 11
1
vote
0
answers
168
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Rational classes of $(-2)$-curves in a minimal surface of general type
Let $X$ be a minimal surface of general type over $\mathbb{C}$. One can show that if for any set of $(-2)$-curves $C_1,\cdots,C_l$ on $X$, there exists $k$, $1\le k\le l$ such that $$\sum_{i=1}^k\...
- 41
3
votes
1
answer
529
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Examples of complex manifolds with trivial Néron–Severi group?
$\DeclareMathOperator\NS{NS}\DeclareMathOperator\Pic{Pic}$Let $X$ be a compact complex manifold, assume projective if you'd like. Define the Néron–Severi group to be the quotient $$\NS(X) = \Pic(X) / \...
- 498
2
votes
0
answers
129
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principal divisor on complex surfaces
Let $X$ be a non compact complex surface non projective and non algebraic, and let $S$ be compact Riemann surface embedded in $X$ ( i mean that $S$ is a compact complex sub variety of $X$ of ...
- 25
6
votes
2
answers
2k
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Generalisations of Riemann-Roch for surfaces
Let $X$ be a smooth projective algebraic surface (over $\mathbb{C}$ ). For all $L\in \mathrm{Pic}(X)$, we have
$$\chi(L)=\chi(\mathcal{O}_X)+\frac{1}{2}(L^2-L\cdot \omega_X).$$
This is the famous ...
- 2,215
0
votes
1
answer
281
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Question about Correspondences from Mumford’s Complex Projective Varieties
I study David Mumford's Algebraic Geometry I - Complex Projective Varieties
and have some problems to understand a step in the proof of Lemma 6.7 (b).
Firstly, the general setting & preparations ...
- 7,320
4
votes
1
answer
371
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Kähler metric on compact complex manifolds with simple normal crossing divisor
Let $X$ be a reduced compact complex analytic space of $\dim_{\mathbb{C}}X\ge2$; by [KJ] definition 3.29, remark 3.44 and theorem 3.45, it admits a strong resolution $R(X)$ which is smooth, $E=\pi_X^{-...
- 828
3
votes
1
answer
295
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Infinitely small intersections with nef $\mathbb R$-Cartier divisors
Suppose $X/\mathbb C$ is a projective $\mathbb Q$-factorial variety with wild singularities. Let $N$ be a nef $\mathbb R$-Cartier divisor. Then is it possible that there are infinitely many curves $...
- 28.8k
1
vote
1
answer
299
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A necessary condition for existence of Ricci flat metric on pair (X,D)
Let $X$ be a complex compact manifold with simple normal crossing divisor $D$. Is the condition $K_X +D = 0$ necessary for the existence of Ricci-flat metric?
CommunityBot
- 1
4
votes
2
answers
2k
views
Ample divisors on blown-up projective space
Let $\mathbb{P}=\mathrm{Proj}(\mathbb{C}[x_0,\ldots,x_n])$ be complex projective $n$-space. Assume I have linear subvarieties $L_1,\ldots,L_k\in\mathbb{P}$ of codimension $r_i\ge 2$, respectively. Let ...
- 625
6
votes
1
answer
354
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Fundamental groups of complements of divisors in $\mathbb P^2$
Let $D$ be a divisor in $\mathbb P^2_{\mathbb C}$ and let $X= \mathbb P^2_{\mathbb C} - D$.
Under what condition on $D$ is the fundamental group of $X$ infinite?
- 666
10
votes
1
answer
314
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Is there a divisor in $\mathbb P^2$ such that all analytic maps into its complement algebraize?
Is there a closed subscheme $D$ in $\mathbb P^2_{\mathbb C}$ pure of codimension one such that, for all algebraic varieties $X$ over $\mathbb C$, any analytic map
$$ \phi: X(\mathbb C) \to \mathbb P^...
- 103
2
votes
0
answers
167
views
Ricci flat metric on pair (X,D)
Let $(X,\omega)$ be a Calabi-Yau variety and $D$ be a simple normal crossing divisor on $X$ with conic singularities with cone angle $2\pi\theta$, $0<\theta<1$ such that $K_X+D>0$, then is ...
- 75
1
vote
0
answers
217
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Family $(X_y,D_y)$ with trivial canonical bundles
Let $i:D\hookrightarrow X$ and $f : X \to Y$ be holomorphic mappings of complex manifolds
such that $i$ is a closed embedding and $f$ as well as$ f \circ i$ are proper and smooth and $D$ is a divisor. ...
CommunityBot
- 1
2
votes
1
answer
300
views
a question on the space of divisors on a curve
Let $X$ a complex curve and $x\in X$ a point.
We consider the space of effective divisors $D$ with fixed degree $d$, whic we know is isomorphic to $X^{d}/S_{d}$ where $S_{d}$ is the symmetric group.
...
- 663
5
votes
2
answers
3k
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Embedded resolution of singularities
I'd like to check with my colleagues whether I have correctly understood "embedded resolution of singularities".
Let $X$ be a nonsingular projective variety over $\mathbf C$ and let $D$ be a "nice" ...
- 42.9k
9
votes
3
answers
3k
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Cone of movable curves
Let $X$ be a smooth complex projective variety of dimension $n$.
Under the duality between $N_1(X)$ and $N^1(X)$ we know that closure of cone of effective curves $\overline{NE}(X)$ is dual to closure ...
CommunityBot
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