All Questions
Tagged with complex-geometry ag.algebraic-geometry
1,708
questions
0
votes
0
answers
114
views
Hyperplane section argument on Kähler manifold
On a projective variety, there exist very ample line bundles and hyperplane sections. It is a useful trick to take a general hyperplane section to reduce a problem to a lower dimension. However, on ...
- 361
3
votes
1
answer
176
views
Negative definite of exceptional curve in higher dimension
One direction of the Grauert's contractibility theorem shows
Let $f:S\rightarrow T$ be a surjective holomophic map where $S$ is a compact holomorphic surface. If $C$ is a reduced connected effective ...
- 361
1
vote
0
answers
105
views
Enriques-Kodaira classification of minimal resolution of surface with quotient singularities
Let $X$ be a normal projective complex surface with at worst quotient singularities. Let $\bar{X}\to X$ be the minimal resolution. Further assume that $b_2(X)=1$ and $b_1(X)=b_3(X)=0$. Since quotient ...
- 189
3
votes
0
answers
175
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$H^2(X,T_X)=0$ implies the Frölicher spectral sequence degenerates at $E_1$?
Let $X$ be a compact complex manifold, if $X$ satisfies $H^2(X,T_X)=0$, then it is well-known that the Kuranishi space of $X$ is smooth by Kodaira and Spencer's deformation theory. On the other hand, ...
- 449
1
vote
0
answers
164
views
How to "eliminate" the log pole of a logarithmic $(p,q)$-form？
Let $X$ be a compact complex manifold, and $D=\sum_{i=1}^{r} D_i$ be a simple normal crossing divisor on $X$. Let $\alpha$ be a logarithmic $(p,q)$-form, namely, on an open subset $U$, we can write
$$\...
- 295
1
vote
0
answers
57
views
Seeking for bridges to connect K-stability and GIT-stability
We consider the variety $\Sigma_{m}$ := {($p$, $X$) : $X$ is a degree $n$ + 1 hypersurface over $\mathbb{C}$ with mult$_{p}(X) \geq m$} $\subseteq$ $\mathbb{P}$$^{n}$ $\times$ $\mathbb{P}$$^{N}$, ...
- 41
1
vote
0
answers
79
views
Birational deformations of holomorphic symplectic manifolds
Let $X$ and $X'$ be birational holomorphic symplectic manifolds. Then the birational morphism between them identifies $H^2(X)$ with $H^2(Y)$. The period space of $X$ is defined to be a subset of $\...
- 153
5
votes
0
answers
241
views
Does the Poincaré lemma (Dolbeault–Grothendieck lemma) still hold on singular complex space?
Let $X$ be a complex manifold, then we have the Poincaré lemma (or say, Dolbeault-Grothendieck lemma) (locally) on $X$, whose formulation is as follows:
( $\bar{\partial}$-Poincaré lemma) If $\...
- 317
4
votes
0
answers
190
views
Is there Riemann-Roch without denominators for complex manifolds?
Let $X \subset Y$ be an inclusion of compact complex (possibly Kähler) manifolds. I'm wondering if "Riemann-Roch without Denominators" [1, Thm 15.3] holds in that situation. The statement is
...
- 1,141
7
votes
0
answers
146
views
Example of closed non-exact torsion differential form on variety
I asked this question some time ago on MSE and received close to no interest. I feel it is appropriate for this site:
I am interested in finding a particular example. I would like to find a variety (...
- 513
2
votes
0
answers
217
views
Chern classes and rational equivalence
Let $X$ be a complex variety and let $l_1$ and $l_2$ be line bundles on $X$. Let $f_1$ and $f_2$ be sections of $l_1$ and $l_2$ respectively, and let $Z_1$ and $Z_2$ be their zero-sets.
I would like ...
- 89
1
vote
1
answer
254
views
Is the associated G/B fibration to a G-torsor projective?
Let $X$ be a smooth projective variety over $\mathbb{C}$, $G$ a reductive group, and $P \to X$ a $G$-torsor. Let $B \subset G$ be a Borel subgroup. Is the associated $G/B$ fibre bundle $$ Y=G/B \...
- 757
2
votes
0
answers
132
views
Hodge numbers of a complement
Let $Y\subset X$ be an analytic subvariety of codimension $d$ of a smooth compact complex variety $X$. Denote $U = X\setminus Y$. The relative cohomology exact sequence implies that
$$
H^i(X) \to H^i(...
- 2,305
16
votes
0
answers
502
views
Gabriel's theorem for complex analytic spaces
Let $X,Y$ be noetherian schemes over $\mathbb{C}$.
Then, it is known that
$$
\text{Coh}(X) \simeq \text{Coh}(Y) \Rightarrow X \simeq Y,
$$
by P. Gabriel(1962).
Are there some results in the case of ...
- 869
6
votes
1
answer
709
views
Understanding the Hodge filtration
Let $X$ be a smooth quasiprojective scheme defined over $\mathbb{C}$, and let $\Omega^{\bullet}_X$ denote its cotangent complex, explicitly, we have:
$\Omega^{\bullet}_X:=\mathcal{O}_X\longrightarrow \...
- 2,113