All Questions
Tagged with sheaf-theory complex-geometry
11 questions with no upvoted or accepted answers
5
votes
0
answers
269
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Line bundle whose pushforward is a complex of vector bundles
If $E\to X$ is a holomorphic vector bundle, it is well known that the tautological line bundle $\mathcal{O}_E(1)$ over the projectivization $\pi:\mathbb{P}(E^*)\to X$ satisfies
$$\pi_*\mathcal{O}_E(1)=...
- 789
4
votes
0
answers
205
views
Sheaf-type property for Derived Categories?
Suppose $X$ is a finite dimensional complex space (I'm happy to restrict to $X$ being a scheme of finite type over $\mathbb C$ as well). I'm wondering if the following sheaf-like properties hold for ...
- 1,761
3
votes
0
answers
1k
views
Saturation of sheaves
Let $(X, \mathcal{O}_X)$ be a complex manifold, which we can take to be projective. A coherent subsheaf $\mathscr{F}$ of some sheaf $\mathscr{G}$ is said to be saturated in $\mathscr{G}$ if the ...
- 651
3
votes
0
answers
978
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How does one compute the Chern classes of the dual sheaf from the Chern class of the original sheaf?
Let $X$ be a smooth projective 4-fold (over $\mathbb{C}$). Let $Z \subset X$ be a codimension two subscheme. Let $I_{Z}$ denote the ideal sheaf of
$Z$. How does one compute the Chern classes of $I_Z^{...
- 3,245
3
votes
0
answers
160
views
Monodromy along strata of a pushforward
Work with complex varieties and constructible sheaves on the complex analytic site. All functors will be tacitly derived.
Let $X$ be a variety acted upon by a connected linear algebraic group. Let $X ...
- 1,595
2
votes
0
answers
168
views
Criteria for a sheaf to be locally free over subvariety
Let $X$ be a compact complex manifold and $\mathcal{F}\to X$ a sheaf.
Is there a regularity criteria (or a condition) for $\mathcal{F}$ that determines whether we there exists a closed subvariety $i:...
- 789
1
vote
0
answers
226
views
Resolution of the pushforward of a vector bundle
Let $i:Z\hookrightarrow X$ be a subvariety of a compact Kahler manifold. Assume that $Z$ can be realize as the zero locus of a section $s$ of a holomorphic vector bundle $E\to X$ of rank $r$. The ...
- 789
1
vote
0
answers
125
views
Explicit resolution of $\Omega^1_C$ for prestable curve $C$
Suppose $C$ is a complex projective curve (or a compact $1$-dimensional connected reduced complex space). If $C$ is smooth, then its module of differentials $\Omega^1_C$ is locally free. If $C$ is a ...
- 1,761
0
votes
0
answers
185
views
Recipe for resolving a coherent sheaf
Let $X$ be a complex manifold and let $V\subset X$ be a subvariety. Let $F\rightarrow V$ be a holomorphic vector bundle over $V$ and let $\mathcal{S}=\Gamma(F)$ be the sheaf of holomorphic section of $...
- 789
0
votes
0
answers
635
views
A coherent sheaf is a vector bundle over subvariety?
Over a complex manifold, can every coherent sheaf be seen as a holomorphic vector bundle over an analytic subset?
Thanks in advance.
- 789
0
votes
0
answers
265
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Explicit adjunction formula and local top form
I am reading the section 4.2 in Kollar-Mori, where they provide the explicit equations for Du Val Singularities. In the course of the proof, they reduce to studying the equation $x^2+f(x,y)=0$ in a ...
- 625