All Questions
Tagged with complex-geometry coherent-sheaves
25 questions
3
votes
2
answers
145
views
Vector bundles over a Stein space are projective
It is a "well known" fact that
locally free sheaves over a Stein space $X$ are projective as $\mathcal{O}_X$-modules
(see e.g. just after Lemma 1.6 in O'Brian-Toledo-Tong's "The trace ...
- 1,109
3
votes
0
answers
130
views
Trace map on Ext group
Let $R$ be a (possibly non-commutative) unital ring and $M$ be a perfect left $R$-module. Then, we have the trace map
$$
\operatorname{Tr}\colon \mathrm{Hom}_R(M,M)\to R/[R,R]\,.
$$
According to the ...
- 985
7
votes
1
answer
607
views
Converses to Cartan's Theorem B
Here is a phrasing of some Cartan Theorem B statements:
Consider the following conditions:
$X$ is a {Stein manifold, affine scheme, coherent analytic subvariety of $\mathbb{R}^n$, contractible ...
- 1,109
1
vote
0
answers
109
views
Monomorphism/Isomorphism of $C_4$-tangent cones for complex varieties
Suppose that $(M,\mathcal{O}_M)$ is a reduced complex analytic space (or complex algebraic variety if you prefer). The tangent linear fiber space $TM$ associated to $M$ is defined as the analytic ...
- 513
2
votes
0
answers
53
views
Pullback of coherent sheaves on Stein manifolds
Let $i:X\to Y$ be a closed embedding of Stein spaces, $G$ be a coherent $O_Y$-module. Set $I=\ker(i^*:O(Y)\to O(X))$. Then $I$ is an ideal of the ring $O(Y)$. Is that true that $\Gamma(X,i^*G)=G(Y)/...
- 615
2
votes
1
answer
315
views
Resolving complexes of coherent analytic sheaves
Background
Throughout, let $X$ be a smooth complex manifold.
It is a classical fact that a coherent analytic sheaf admits a local resolution by locally free sheaves (also known as a local syzygy). ...
- 1,109
3
votes
0
answers
98
views
Analogous tensor product operation for reflexive sheaf
Suppose now $(X,\mathcal O_X)$ is a normal complex space, and $\mathcal F$ is a coherent analytic sheaf on it.
Product the reflexive sheaf $$\mathcal F^{[p]}:=(\mathcal F^{\otimes p})^{**},$$ where $\...
- 295
0
votes
1
answer
264
views
Stability of sheaves of non-constant rank
Let $E\to X$ be a coherent sheaf over a compact (projective) Kahler manifold. The definition of stability of sheaves as stated in Huybrechts-Lehn (Definition 1.2.12) says that $E$ is stable if for all ...
- 789
1
vote
0
answers
178
views
Projectivization in the derived category of coherent sheaves
Let $X$ be a compact Kahler manifold. There exists a notion of projectivization of holomorphic vector bundles and coherent sheaves over $X$. Does that concept extend to objects in the derived category ...
- 789
2
votes
0
answers
276
views
Projectivization of a coherent sheaf using resolution by vector bundles
Let $\mathcal{F}\to X$ be a coherent sheaf over a compact Kahler manifold and let $E^{\bullet}\to \mathcal{F}$ be a resolution of $\mathcal{F}$ by holomorphic vector bundles.
Is there a way to ...
- 789
3
votes
0
answers
241
views
Stability and simplicity of tangent sheaf of Grassmannian
Everything is over the complex numbers. Let $ X = \text{Gr}(k,n) $ be a Grassmannian variety and with tangent sheaf $ T_X $.
(1) Is $ T_X $ simple, i.e. is $\text{Hom} ( T_X, T_X) = \mathbb{C} $?
(2)...
- 936
4
votes
2
answers
350
views
When are two resolutions of a coherent sheaf homotopic
Let $\mathcal{F}$ be a coherent sheaf on a projective manifold $X$. It is well known that one can construct a resolution of $\mathcal{F}$ by holomorphic vector bundles (locally free sheaves).
Are two ...
- 789
1
vote
0
answers
99
views
Is the resolution of a sub-sheaf into complex of holomorphic vector bundles a "sub-resolution"?
Let $F\rightarrow X$ be a coherent sheaf on a projective Kahler manifold. We can resolve it into a complex of holomorphic vector bundles $E^{\bullet}\rightarrow X$. Let $G\subset F$ be a subsheaf of $...
- 789
1
vote
0
answers
305
views
Stability of vector bundles and corresponding coherent sheaf
Let $j:Y\hookrightarrow X$ be an embedding of projective complex manifolds. Let $E\rightarrow Y$ be a vector bundle and $S=j_*E$ the corresponding coherent sheaf on $X$ (see Push forward of a Vector ...
- 789
1
vote
0
answers
360
views
On definition of stable vector/Higgs bundle
Recall that the slope of a holomorphic vector bundle $\mathcal{E}$ over a smooth projective variety (or rather a compact Kähler manifold) $X$ is defined as
$\mu(\mathcal{E}) :=\frac{\operatorname{deg}(...
- 1,170
3
votes
0
answers
119
views
Organizing mirror pairs
At a maximally vague and naive level, mirror symmetry asks the following question: given a complex manifold $(X, I)$, is there a symplectic manifold $(M, \omega)$ and an equivalence between the ...
- 3,020
4
votes
0
answers
194
views
Do we have $D^b_{coh}(X)\simeq D^b(coh(X))$ for a compact complex manifold $X$?
Let $X$ be a compact complex manifold and $\mathcal{O}_X$ be the structure sheaf of holomorphic functions. We call a sheaf of $\mathcal{O}_X$-module $\mathcal{F}$ coherent if it satisfies the ...
- 9,019
5
votes
1
answer
386
views
Normal Cones for Complex Spaces
Suppose $U\subset\mathbb C^n$ is an open subset and $f_1,\ldots,f_k$ are analytic functions on it, generating the coherent ideal sheaf $\mathcal I$ which defines a closed complex subspace $Z\...
- 1,761
6
votes
0
answers
507
views
de Rham isomorphism with holomorphic forms
For a non-compact Riemann surface $X$ there is an isomorphism:
$$\Omega(X)/\mathrm d \mathcal O(X)\simeq H^1(X,\mathbb C)$$
where $\Omega$ is the sheaf of holomorphic forms on $X$. The group on the ...
- 81
4
votes
2
answers
594
views
H. Cartan's "Variétés analytiques complexes et cohomologie"?
Does anyone know where I might find an online version (for free or purchase, translated or in french) of this paper by Henri Cartan from 1953? I know it was published in Colloque sur les fonctions de ...
- 43
14
votes
1
answer
459
views
Obstruction to the existence of global resolution of coherent sheaf
It is well known that any coherent sheaf on a complex manifold (or more generally on some complex spaces) admits locally a resolution with locally free sheaves. It is also well known that for non-...
- 912
2
votes
0
answers
357
views
$G$-equivariant coherent sheaves on Bott$-$Samelson resolutions
Let $G$ be a Lie group and $B$ a Borel subgroup. $G/B$ is the corresponding flag variety.
Let $w$ be an element of the Weyl group $W$ with a reduced expression
$w = s_1 \cdots s_n$. Let $X_w$ be ...
- 1,719
8
votes
2
answers
2k
views
Coherent sheaves and holomorphic vector bundles
For a complex manifold $M$, I'm trying to understand (from a differential geometry point of view) what its category of coherent sheaves is. If I understand correctly, then the sheaf of holomorphic ...
- 1,462
3
votes
1
answer
3k
views
question about torsion sheaf
Hi, i'm stuck on the following, please can someone help?
Let $E$ be a complex holomorphic vector bundle of rank r over a compact kahler
manifold $M$, let me indicate $\mathcal{E}$ the associated ...
- 1,727
6
votes
2
answers
1k
views
Deformations of sheaves via automorphisms. How to express $Ext^1$?
Let $X$ be a complex manifold (for example $\mathbb CP^n$), let $v$ be a holomorphic vector field on $X$, and let $F$ be a coherent sheaf (for example a vector bundle or a structure sheaf of a point). ...
- 28.9k