**10**

votes

**3**answers

802 views

### Is there a notion of “flat vector bundle over a topological space”?

I am reading this paper and at the top of page 5 the author makes reference to categories consisting of flat complex vector bundles over $X$ where $X$ is an arbitrary topological space. However, the ...

**8**

votes

**3**answers

335 views

### Trivialisation of vector bundles on Stein spaces

Does every vector bundle on a Stein space have a finite local trivialisation?
Definitions:
Stein space means either a complex analytic Stein space or a nonarchimedean Stein space in the sense of ...

**1**

vote

**0**answers

92 views

### Bilinear form and double cover

Let $\pi:X\rightarrow Y$ be a double cover of curves (smooth and projective), and $E$ a vector bundle on $X$ with a non-degenerate symmetric bilinear form $\phi:E\otimes E\rightarrow \mathcal O_X$. ...

**5**

votes

**1**answer

202 views

### Stiefel-Whitney class of unordered configuration space

Let $S^m$ be the $m$-sphere and $$F(S^m,2)/\mathbb{Z}_2=\{(a,b)\mid a,b\in S^m, a\neq b\}/(a,b)\sim (b,a)$$ be the $2$-nd unordered configuration space on $S^m$. How to compute the total Stiefel-...

**4**

votes

**1**answer

216 views

### characteristic classes of tangent bundle of 2-nd unordered configuration space

Given a (real or almost complex) manifold $M$, Let the 2-nd unordered configuration space be the quotient space
$$
B(M,2)=(M\times M\setminus\ \Delta)/\ \mathbb{Z}_2
$$
where
$$
\Delta=\{(m,m)\mid m\...

**0**

votes

**0**answers

123 views

### Chern classes of ideal sheaf of locally complete intersection

Let $C\varsubsetneq X$ be a reducible locally complete intersection closed pure $1$-dimensional subscheme in a smooth projective variety of dimension $3$ over a algebraically closed field $k$ of ...

**3**

votes

**2**answers

225 views

### Connectedness of moduli of vector bundles

Let $X$ be a smooth projective variety. Given two vector bundles $V_1$ and $V_2$ such that $[V_1]=[V_2]\in K^0(X)$, can one expect that $V_1$ and $V_2$ can be connected by a family of vector bundles? ...

**5**

votes

**0**answers

144 views

### Non trivial rank 2 holomorphic vector bundles in complex dimensions greater than or equal 2

Does every compact complex manifold of complex dimension greater than or equal two
possess a nontrivial rank 2 holomorphic vector bundle?

**4**

votes

**0**answers

131 views

### No irreducible parallelizable manifold of given dimension

What is an example of a closed 4-manifold $M$ such that $M$ is parallelizable and $M$ is topologically (or at least smoothly) irreducible?
Topological irreducible: it is not homemorphic to ...

**1**

vote

**1**answer

214 views

### realization map for K-theory of spheres

Let $\overset{\sim}{K}(X)$ and $\overset{\sim}{KO}$ denote the reduced stable isomorphic classes of complex and real bundles over X and $\rho$ be the realization map. We know that $\overset{\sim}{K}(\...

**6**

votes

**0**answers

166 views

### Simplicity of a rank 2 vector bundle over a principally polarized abelian surface

Let $A := \textrm{Jac}(C)$ be the Jacobian of a genus $2$ curve $C$, with principal polarization $\Theta$.
Studying some branched covers of $A$, I was led to consider some rank $2$ holomorphic ...

**1**

vote

**1**answer

165 views

### Orientability of Surfaces and the Fundamental Group [closed]

Let $(M,g)$ be a compact riemannian 3-manifold and $\Sigma \subset M$ an embedded compact surface homeomorphic to the projective plane. Consider the application $i_\#:\pi_1(\Sigma)\to \pi_1(M)$ given ...

**1**

vote

**0**answers

119 views

### addition on an affine scheme [closed]

At the Brenner's introduction to the geometric view of the tight closure the author states that an affine scheme has a natural addition (this addition will be extended to the vector bundles). I wonder ...

**7**

votes

**2**answers

486 views

### Generalising the Penrose Twistor Fibration

As is well known, there exists a fibration $\mathbb{CP}^3 \to S^4$, of the four sphere by complex projective $3$-space, called the Penrose twistor fibration. Does this fibration admit a "canonical" ...

**6**

votes

**0**answers

166 views

### Bundles over Grassmanian with given top Chern class

So, I have been working on Chern classes for my master thesis and apparently (My proofs could be wrong and a few things are still vague) I was able to give a construction method and exhibit, via ...

**3**

votes

**0**answers

48 views

### Determinant twist and $Pin _{\pm}$ structure on $4k$-dimensional bundles [Reference request]

Consider the automorphism $\varphi$ of $O(2n)$ given by $\varphi (A)=det(A)\cdot A$. The induced map in cohomology $H^*(BO(2n))$ sends $w_2$ to $w_2+(n+1)w_1^2$ (a proof is given at the end of the ...

**0**

votes

**0**answers

113 views

### Chern classes, vanishing of smooth sections or vanishing of holomorphic?

I have seen both definitions and this is getting me more and more confused.
Are Chern classes dual to the degeneracy cycles of smooth sections or holomorphic?
They can't be the same thing, can they?

**1**

vote

**1**answer

209 views

### Geometric interpretation of Chern classes over flag manifolds

I would like to find that Chern classes of the tautological bundles over a flag manifold are dual to some cells in homology, analogously to what happens for the Grassmanian case. I have not been able ...

**3**

votes

**1**answer

97 views

### Involution on the set of isomorphism classes of 2n-dimensional vector bundles induced by an involution of $O(2n)$

Denote by $\varphi$ the automorphism of $O(2n)$ given by $\varphi (A)=det(A)\cdot A$.This induces a self-map $B\varphi$ of $BO(n)$, so it induces a self-map (actually an involution)
$B\varphi ^*$ on $[...

**0**

votes

**1**answer

49 views

### s-equivalence and transition function

Two vector bundles $E$ and $F$, are said two be S-equivalent if they have isomorphic gradients. My question is: Is it possible to caracterise this properity using transition functions?
Thanks

**0**

votes

**1**answer

99 views

### Trying to relate the fundamental groupoid to vector bundles

Fix a topological space $X$. Now consider a functor from the fundamental groupoid of $X$ to the category $Vect$. In other words, we assign a vector space to each point of $X$, we allow ourselves to ...

**2**

votes

**0**answers

122 views

### Connection and reduction of the structure group

I apologize for this question which is not really research-level, but I don't get any answer on mathStackExchange, and I asked a professor in my university... which couldn't find the solution.
I am ...

**2**

votes

**0**answers

176 views

### Fiber Bundle with a perfect bilinear map

Let $\pi:X\rightarrow Y$ a double cover of Riemann surfaces, and consider a $SL_n-$bundle $E$ with a perfect $\mathcal O_Y-$bilinear pairing $$\psi:\pi_*E\times\pi_*E\rightarrow\pi_*\mathcal O_X$$
...

**3**

votes

**2**answers

90 views

### Name for construction on two vector bundles

I have a construction on two vector bundles and I would like to give it a name and a symbol but I can't find anything.
For two vector bundles $A=\{(x,A_x) : x \in X\}$ and $B=\{(y,B_y):y\in Y\}$ ...

**4**

votes

**2**answers

173 views

### What is the space for the coefficients of the connection 1-form of a connection in a vector bundle?

Let $E\to X$ is a a (smooth real) vector bundle with structure group some Lie group $G$. Suppose we have a (linear) connection $\nabla$ on $E$.
Is it true that if $A$ is the connection 1-form of ...

**1**

vote

**1**answer

155 views

### Tensoring by Line Bundles to Produce Holomorphic Sections

Inspired by the line bundle case, I have the following question:
Given an equivariant holomorphic vector bundle over complex projective space, is it true that tensoring it by line bundles often ...

**1**

vote

**2**answers

180 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 ...

**5**

votes

**2**answers

201 views

### Lindel's theorem for semisimple simply connected G

Let $k$ be a field.
$G/k$ be a simply connected semisimple algebraic group.
Let $X/k$ be a smooth affine $k$-scheme.
Question: Is every principal $G$ bundle on $X\times {\mathbb A}^1$ a pull back ...

**0**

votes

**1**answer

69 views

### Constructing vector bundles of specific stiefel whitney classes?

Is it possible to construct a vector bundle over a given base $X$ such that the $n$th stiefel whitney class vanishes for a given $n?$ What about for some set of integers? Can we make vector bundles ...

**1**

vote

**0**answers

72 views

### A holomorphic vector bundle structure for $\Omega^{(0,1)}(M)$

For a complex manifold $M$, the complexified tangent space $\Omega^1(M)$ splits into a direct sum $\Omega^1(M) = \Omega^{(1,0)}(M) \oplus \Omega^{(0,1)}(M)$. As is well-known $\Omega^{(1,0)}(M)$ is ...

**1**

vote

**0**answers

107 views

### Is it possible to find an explicit definition of the “universal” (co)tangent bundle?

Let $H_{0,1}(\mathbb{P}^2, d)$ be the space of holomorphic degree $d$
maps (that are not multiply covered) from $\mathbb{P}^1$ to $\mathbb{P}^2$ with one marked point
$y \in \mathbb{P^1} $ $\textit{...

**1**

vote

**1**answer

109 views

### Sobolev multiplication $\otimes$ of $H^1=W^{1,2}$ in vector bundles

Let $E\to X$ be a vector bundle with an inner product and fix a reference connection $A_0$ on $E$. Then for $1\leq p < \infty$ and $k\geq 0$ we can define the Sobolev space $W^{k,p}(E)$ as the ...

**1**

vote

**0**answers

41 views

### the associated action on the transition functions

Let $X$ be a curve with an involution $\sigma$ generically unramified, given a $G-$bundle $E$ of rank $r$, than we ca take its pull-back, I want to describe the action of $\sigma$ on $G$. Fix a ...

**1**

vote

**0**answers

124 views

### Contraction with a vector field and pullback bundle

While trying to understand the proof of "smooth" version of Kostant-Hochschild-Rosenberg theorem (which is due to Connes for compact smooth manifolds) I found the following argumentation: one is ...

**3**

votes

**0**answers

154 views

### Deformation of vector bundle on projective space with same Hilbert polynomial as multiple of structure sheaf

Let $E$ be a vector bundle on projective space ${\bf P}^n$ whose Hilbert polynomial is the same as $\mathcal{O}^{{\rm rank}(E)}$.
Does there exist a vector bundle over ${\bf P}^n \times {\rm Spec}(R)$...

**1**

vote

**0**answers

103 views

### Morphism of modules of sections and pullback bundles

I'v asked this question on StackExchange but unfortunately nobody answered. I thought that maybe it would be more apropriate to post it here:
so suppose that we have a morphism $\theta: \Gamma(B,E_1) ...

**0**

votes

**1**answer

148 views

### Why does this vector bundle on the surface sit in this exact sequence?

Let $X$ be a K3 surface. Let $E$ be a semistable rank 3 vector bundle. Now suppose $0 = E_0\subset E_1\cdots\subset E_s=E$ be the Harder-Narasimhan filtration. Suppose $E_1$ is $\mu$-stable and rank $...

**1**

vote

**0**answers

183 views

### Vector bundle is semistable if only if it's pull back is semistable?

If $X$ is a smooth projective variety and $D$ is a divisor on $X$, and let $i:D\longrightarrow X$ be the closed immersion. Let $E$ be a vector bundle on $X$. Are there any theorems which say that $E$ ...

**2**

votes

**1**answer

91 views

### Space of invariant sections

I have a principal bundle $\pi: M \to B$ with structure group $G$ and a vector bundle $p: V\to M$. I need to work with $\Gamma_G(V,M)$, the space of invariant (under the fiber action on $M$) sections ...

**1**

vote

**0**answers

167 views

### What is the co-kernel of the morphism of vector bundles?

Let $X$ be a surface, and $i:C\subset X$ be a smooth curve. Let $A$ be a line bundle on $C$, and $E$ be a vector bundle of rank $r$ on $X$.
Suppose there is a surjection: $E\longrightarrow i_*A\...

**1**

vote

**1**answer

152 views

### Pushforward of locally free sheaves under open immersion

Let $X$ be a connected projective noetherian scheme over $\mathbb{C}$, with every irreducible component of the same dimension. Let $\dim X=n \ge 2$ and $p$ be a closed point on $X$. Denote by $U$ the ...

**0**

votes

**0**answers

106 views

### conjugate operation on vector bundle

Is the conjugate operation on $\overset{\sim}{K}(\mathbb{C}\mathbb{P}^n)$ known? If so, can I get the full formula at least in terms of the basis $\eta^i$? Here $\overset{\sim}{K}(X)$ denotes the ...

**1**

vote

**0**answers

53 views

### How to prove Butler's inequality for the maximal slope of the kernel bundle?

In Butler' paper "Normal generation of vector bundles over a curve" (J.d.g.,1994). Proposition 1.4 said that
$$prop^+(M_E)\leq \max\left\{-2,\frac{-prop^+(E)}{prop^+(E)-g}\right\}$$
where $E$ is a ...

**2**

votes

**1**answer

155 views

### Exterior product

I have asked this question in the Mathematics forum but I received no answer.
Let $E$ be an algebraic vector bundle of rank $r$ and degree $d$,
Then $\Lambda^2 E$ is of rank $r'=r(r-1)/2$, but is of ...

**7**

votes

**1**answer

279 views

### Are all quotients of a weakly contractible space via a free group action classifying spaces of the group?

I asked this question on math.stackexchange a week ago, but did not get an answer.
First of all, I don't want to restrict to any kind of "nice spaces" since I am interested in the most general ...

**0**

votes

**1**answer

136 views

### Vector Bundles of small rank

I recently started the study of vector bundles on $\mathbb{P}^n$, and started to read Rao's article 'A family of vector bundles on $\mathbb{P}^3$'. There, there is a notion of spectrum of a vector ...

**4**

votes

**1**answer

210 views

### Symmetric product of a vector bundle

Let $E$ be a vector bundle of rank $r$ and degree $d$ over a smooth curve $X$. Is there any canonical exact sequence for $Sym^k(E)$? in particular what is the degree of $Sym^k(E)$?
Suppose $E$ is ...

**0**

votes

**0**answers

167 views

### Wedge product of Endomorphism-Valued Forms

To define characteristic classes in smooth vector bundles $E\longrightarrow M$ there is a more or less standard procedure: to choose a connection $\nabla$ and to derive the curvature $\Omega$, which ...

**2**

votes

**0**answers

196 views

### Geometric proof of Borel-Weil theorem

I am curious if there is any geometric proof of Borel-Weil theorem.
Borel-Weil is a geometric realization of irreducible unitary representation. The proofs I found, however, all use Weyl unitarian ...

**2**

votes

**1**answer

333 views

### Yang-Mills Functional and Energy

I have a question about the meaning of Yang-Mills Functional.
It is stated everywhere that the Yang-Mills Functional is a measure of energy. But the formal definition of the Yang-Mills Functional is:
...