All Questions
26 questions
2
votes
1
answer
217
views
Dual of slope semistable vector bundle on higher dimensional variety
Recall the definition of slope semistability, taken from section 1.2 of Huybrechts and Lehn's "Geometry of Moduli Spaces of Sheaves" book. Let $X$ be a projective $\mathbb{C}$-scheme and $E \...
- 129
5
votes
1
answer
286
views
Origin of the name Trace resp Integral symbol for the trace map of Dualizing Sheaf
Let $X \subset \mathbb{P}^n_k$ be a normal projective subscheme over $k$ of dimension $n$. The dualizing sheaf is in context of Serre duality a pair $(\omega_X,t)$ (which exists in that case) ...
- 6,038
3
votes
1
answer
345
views
Normal bundle of a linear subspace
Let $X\subset\mathbb{P}^N$ be a smooth scheme theoretical complete intersection, and $H\subset X$ a linear subspace. Denote by $N_{H,X}$ the normal bundle of $H$ in $X$.
If $\dim(H) = 1$, that is $H$ ...
- 8,998
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
2
votes
1
answer
257
views
Understanding spaces is the same as understanding (sheaves of) functions on the space
I'm trying to understand Ravi Vakil's FOAG. In chapter 2 it is written:
[...] understanding spaces is the same as understanding (sheaves of) functions on the spaces, and
understanding vector bundles (...
- 61
5
votes
0
answers
269
views
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
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
4
votes
2
answers
340
views
Sheafification of presheaf of trivial vector bundles is the stack of vector bundles
This is a deliberately vague question, possibly obvious to experts. Let $F$ be a field. Over the (say, fpqc) site of $F$-schemes, we may define a presheaf $T^{\textrm{pre}}$ that takes a scheme $S$ ...
user333445
6
votes
2
answers
631
views
Geometric meaning of coherent sheaves $\mathcal{F} \otimes \mathcal{O}_{\mathbb{P}^n}(d)$ over $\mathbb{P}^n$
Maybe it sounds like a silly question but I'm not able to figure out in my head the geometric meaning of "twisting" vector bundles (or more generaly coherent sheaf) over $\mathbb{P}^n$.
I ...
- 1,343
1
vote
1
answer
186
views
Dual of stable vector bundle on a Fano threefold
Let $E$ be a rank $2$ stable vector bundle on a prime Fano threefold of genus $8$, with Chern numbers $c_1=1, c_2=6, c_3=0$.
Question. Is it true that $E(-1)=E^*$?
What I am able to show is that ...
- 565
1
vote
0
answers
131
views
Schur's lemma for sheaves with different reduced Hilbert polynomials
Recall Schur's Lemma for Gieseker-semistable sheaves, in particular the injectivity statement:
Let $\psi : F \to G$ be a morphism of Gieseker-semistable sheaves. If $p(F)=p(G)$ and $F$ is stable, ...
- 123
1
vote
1
answer
427
views
Flat familiy of coherent sheaves over a scheme
I'm studying the moduli problem of locally free sheaves over a connected smooth projective curve on an algebraically closed field, from the Lecture Notes of Victoria Hoskins, and I cannot fully ...
- 395
2
votes
1
answer
399
views
Locally free sheaves and vector bundles over smooth connected projective curve
Let $X$ be a connected smooth projective curve over an algebraically closed field $K$. Let $\mathcal{F}$ be a locally free sheaf on $X$ and $\mathcal{E}$ a subsheaf of $\mathcal{F}$, which is again ...
- 395
4
votes
1
answer
435
views
Push-out in the category of coherent sheaves over the complex projective plane
I'm trying to deal with an example of a rank two vector bundle over the complex projective plane which is non slope-stable (because the associated sheaf of sections has a coherent subsheaf of equal ...
- 395
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
4
votes
1
answer
216
views
$G$-torsor for topological space compared to that for sheaf of groups
I just read about the definitions about torsor of sheaf of groups and get a bit confused.
How does the notion of $G$-torsor for a topological space compared to that of a sheaf of groups? Is there a ...
- 365
2
votes
0
answers
315
views
Any vector bundle can be twisted to have sections
Let $X$ be an integral scheme proper over $\mathbb{C}$. Let $V$ be a locally free $\mathcal{O}_X$-module of positive finite rank. Does there necessarily exist a locally free $\mathcal{O}_X$-module of ...
user139965
1
vote
1
answer
187
views
Subbundle generated by linearly dependent sections
On $\mathbb{P}^1$ consider the trivial bundle $\mathcal{O}\oplus \mathcal{O}$, and the subbundle $\mathcal{L}_{a,b}\subset\mathcal{O}\oplus \mathcal{O}$ that on an open subset $U$ of $\mathbb{P}^1$ is ...
user68440
1
vote
0
answers
258
views
example of rank 2 torsion free sheaf with no global sections that is not stable
Ìn the book "Vector bundles on complex projective spaces" the authors prove in Chapter 2 Lemma 1.2.5 that, if $E$ is a rank $2$ reflexive sheaf on $\mathbb{P}^n$, then $E$ is stable if, and only if, $...
- 556
14
votes
1
answer
702
views
Donaldson and DT invariants
Let $X$ be a smooth projective surface. Then, using the compactified moduli space of anti self-dual connections or torsion free sheaves we can construct Donaldson invariants of $X$. Similarly, one can ...
- 587
0
votes
0
answers
182
views
Analytic-Local Germs of "General" Sections
Let $C$ be an algebraic curve over an algebraically closed field $k$ of characteristic $0$, and let $\mathcal{L}$ be a base-point-free line bundle on $C$. Furthermore, let $p \in C$ be a smooth point, ...
24
votes
1
answer
837
views
Is there a useful theory of D-modules on smooth (non-analytic) manifolds?
D-modules are related to flat connections on vector bundles, end hence local systems. The theory of D-modules (and related notions such as crystals etc.) seems to be very popular in complex analytic ...
- 6,025
2
votes
2
answers
607
views
Canonical (tautological) section of a family of sheaves
A couple of months ago, i saw a construction, that somehow looks like the construction of the tautological section of the pullback of a vector bundle to its total space, i am trying to piece it ...
- 1,025
11
votes
5
answers
8k
views
When is the push-forward of the structure sheaf locally free
Let $f:X\longrightarrow Y$ be a morphism of noetherian schemes. Under what conditions is $f_\ast \mathcal{O}_X$ a locally free $\mathcal{O}_Y$-module?
Example 1. Suppose that $f$ is affine. Then $f_\...
- 9,497
24
votes
4
answers
6k
views
What is a section?
This question comes out of the answers to Ho Chung Siu's question about vector bundles. Based on my reading, it seems that the definition of the term "section" went through several phases of ...
- 118k
4
votes
2
answers
2k
views
Modules, Sheaves and Vector bundles
Given a graded ring $S$ and a graded S-module $M$ we can carry out a construction in order to get $\tilde{M}$, which is a sheaf over the scheme $\mathrm{Proj}~ S$. With this in view, I have an ...
- 2,132