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15
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0answers
460 views

Characteristic Classes for $E_8$ Bundles

Given a principal $E_8$ bundle $P\rightarrow X$ one can take the adjoint representation $\rho :E_8\rightarrow SU(\mathbb C^{248})$ and form the associated vector bundle $V=P\times_{\rho}\mathbb ...
10
votes
0answers
525 views

conformal blocks for beginners

I have given now a couple of talks that involve conformal blocks bundles on the moduli stack $\overline{\mathcal{M}}_{g,n}$, in front of a public of algebraic geometers but not specialists of the ...
10
votes
0answers
298 views

Atiyah-Bott from Beauville-Laszlo

This is a question about the cohomology groups of the stack of vector bundles (with fixed discrete invariants) on an algebraic curve. Explicit formulas for these cohomology groups are known, and they ...
10
votes
0answers
369 views

Exotic smoothness and Parallelizability

Regarding the parallelizability of the Milnor's seven dimensional exotic spheres: Parallelizability of the Milnor's exotic spheres in dimension 7 The following question naturally arises: Suppose ...
9
votes
0answers
427 views

Deformations of some simple quotient stacks.

I am interested in stacks of vector bundles on varieties and how deformations of the variety (including non-commutative ones) reflect themselves in deformations of the stack of vector bundles. I will ...
7
votes
0answers
328 views

Big tangent bundle

Let $X$ be a nonsingular complex algebraic variety whose tangent bundle is $T_X$. I use Lazarsfeld's book for the definition of a big vector bundle. A line bundle $L$ is big if it has Itaka dimension ...
5
votes
0answers
140 views

Does the suspension isomorphism $K_1(A) \to K_0(SA)$ descend from a more refined invariant?

If $A$ is a C*-algebra, denote its minimal unitization by $\tilde A$ and its suspension by $SA$, thought of as all continuous $a:[0,1] \to A$ with $a(0)=a(1)=0$. The unitized suspension ...
5
votes
0answers
442 views

Ample vector bundles, $H^1=0$ and global generation in characteristic $p$

This is a follow up from this question, in characteristic $p>0$. First, some background. It is known that in characteristic $0$, a vector bundle $E$ on say a projective curve $X$ is ample if and ...
4
votes
0answers
292 views

Vector bundles of schemes and their topological realizations

Hi, there is a realization functor $R_\mathbb{R}$ from schemes of finite type over $\mathbb{R}$ to topological spaces and there is also a functor $R_\mathbb{C}$. Does $R_\mathbb{R}$ send an ...
4
votes
0answers
287 views

Harder-Narasimhan filtration of rank 4 vector bundles on $P^2$

Given a non-semistable vector bundle on $P^2$ of rank 4, are explicit conditions known for when the ranks in its Harder-Narasimhan filtration are (3,1), (2,2) and (1,3) respectively? I would be very ...
3
votes
0answers
143 views

Vector bundle connection over complex manifold vs. over underlying real manifold

Let $(X,g)$ be an Hermitian manifold, and $(E,h)$ be an Hermitian vector bundle over $X$ of rank $r$. Denote by $(X^{\mathbb{R}},g^{\mathbb{R}})$ the underlying Riemannian manifold of $(X,g)$. ...
3
votes
0answers
110 views

Orthonormal frame bundle orthogonal to a curve

This is a duplicate of this question on math.stackexchange, since I got there not a single answer. Let $M$ be a $n$-dimensional smooth riemannian manifold and ...
3
votes
0answers
170 views

Stack of vector bundles (on a curve) over a strictly semi-stable point of the moduli space

Consider the stack $Bun_{r,d}^{ss}$ of rank $r$ semi-stable vector bundle of degree $d$ over a fixed curve. There exists also a coarse moduli space $M$ built via GIT. Over the stable locus of $M$ it ...
3
votes
0answers
174 views

Extending intersection bundles

Let $X$ be the product $Gr_i(V)\times Gr_j(V)$ of two Grassmannians where $V$ is a complex vector space of dimension $d$. There is an open $U\subset X$ formed by all those $(V',V'')\in X$ such that ...
3
votes
0answers
198 views

Ricci flat metrics on holomorphic vector bundles over Riemann surfaces

I am interested in the local geometry of holomorphic curves in Calabi-Yau threefolds. The setup and question are then the following: Consider a $\mathbb{C}^2$ bundle over a compact Riemann surface ...
3
votes
0answers
388 views

Topological obstructions to extending algebraic vector bundles

Ariyan and Kevin Lin have asked about the problem of extending vector bundles defined on an open subvariety across the rest of the variety. There can be subtle commutative algebra obstructions, as in ...
2
votes
0answers
77 views

Subbundles of conformal blocks bundles over $\overline{\mathcal{M}}_{g,n}$

Let $\mathfrak{g}$ be a semisimple split Lie algebra over a field $k$ of characteristic zero. Let $l$ be a non-negative integer and let $P_{l}^{n}$ be the set of $n$-tuples of dominant integral ...
2
votes
0answers
48 views

Formal adjoint of covariant derivative on endomorphism bundle

Let $(X,g)$ be an Hermitian manifold, and $(E,h)$ be an Hermitian vector bundle over $X$, equipped with an integrable, unitary connection $D=D'+D''$. Let $\beta\in\Lambda^{p,q}(\mathrm{End}\,E)$ be ...
2
votes
0answers
123 views

A natural tower of bundles over Grassmannian manifolds

There is a well-known exact sequence of vector bundles $$ 0\to U\to T\to N\to 0 $$ over a Grassmannian manifold $Gr(V,n)$: the fibers over $L\in Gr(V,n)$ are, respectively, $L$ itself, the whole $V$ ...
2
votes
0answers
66 views

Properties of special Cayley-Bacharach bundles on a K3-surface

Assume we have a $K3$-surface $X$ over $\mathbb{C}$ and two rational curves $C_1$ and $C_2$ on $X$ with $C_1.C_2=1$ and $C_i^2=-2$. Let $x$ be a closed point on the reducible curve $C_1\cup C_2$. We ...
2
votes
0answers
104 views

Bundles of (co)chain complexes

A stupid question: does anybody know a good book or something in which invariants of a vector bundle of finite-dimensional acyclic (co)chain complexes (which, I believe is equivalent to a acyclic ...
2
votes
0answers
653 views

Vector bundles on some non-projective surfaces

Let $X$ be a smooth projective curve over a field $k$ and let $L$ be a line bundle on $X$. I will denote by $S$ the total space of $L$ -- this is a smooth surface over $k$ containing $X$ (as the zero ...
1
vote
0answers
38 views

Construction of a Hermitian metric on a locally free subsheaf which is not a subbundle

Assume $X$ is a smooth projective curve over $\mathbb{C}$ and let $\mathcal{E}$ be a locally free sheaf of rank 2 on $X$. We pick a closed point $x\in X$ and a surjection $f: \mathcal{E}\rightarrow ...
1
vote
0answers
79 views

Degeneracy divisor of the “trace” morphism

Let $f\colon X\to Y$ be a finite morphism of smooth curves over an alg. closed field of characteristic zero. I recently asked how methods reminiscent of basic algebraic number theory can be used to ...
1
vote
0answers
139 views

extending a vector bundle

given a vector bundle $V \rightarrow N$ over a manifold $N$ and let's assume $N \hookrightarrow M$ is embedded into a manifold $M$ is there a way to extend $V$ to a bundle over $M$, i.e. is there a ...
1
vote
0answers
116 views

Can one always extend a smooth section defined on a non compact submanifold to the whole manifold, provided it extends continuously to the closure?

Let $V \rightarrow M$ be a smooth vector bundle over a smooth compact manifold $M$ (without boundary) and $X \subset M$ a smooth submanifold of $M$, that is not necessarily closed. Suppose $s: X ...
1
vote
0answers
467 views

Cech Cohomology of Sections of Holomorphic Bundle over Contractible Space

Is there anything that can be said in general about the Cech cohomology of the sheaf $\mathcal{F}$ of sections of a holomorphic bundle over a contractible space $X$? I know that such a bundle must ...
1
vote
0answers
100 views

Topological index and Dirac operator with a non compact group

A spinor which belogs to a representation of a group $G=SO(p,q)$ is a section of a product bundle $S(M)\otimes E$, where $S(M)$ is a spin bundle over a four dimensional orientable and compact manifold ...
1
vote
0answers
268 views

Splitting of vector bundles on a complex torus

Let $X$ be a complex torus (a finite dimensional complex vector space modulo a lattice) and let $E$ be a smooth (not necessarily holomorphic) complex vector bundle over $X$. Is it true $E$ is ...
1
vote
0answers
139 views

Partial ordering of vector bundles on projective spaces

I would like to know if there are some interesting partial orders defined on the isomorphism classes of vector bundles on $\mathbb P^n_k$ (you can assume $k$ is $\mathbb C$ if that helps). ...
0
votes
0answers
68 views

Parabolic bundles on elliptic curves

as a warm up for his thesis I would like a student of mine to read something on parabolic bundles. He is reading the famous Atiyah paper on vector bundles on elliptic curves, so I think it would be ...
0
votes
0answers
38 views

a section invariant under Reeb flow

Let $M$ be a compact contact manifold with $R$ the Reeb vector field. Let $E$ be some vector bundle of gauge group $G$, say, $G = SU(2)$ and $E$ is the adjoint vector bundle. So my question is: If ...
0
votes
0answers
36 views

Double vector bundle vs vector bundle over supermanifolds

Double vector bundle is roughly a vector bundle over (horizontal) vector bundle. Vector bundle is a supermanifold in a nature way (non-nature on the other way around). My question: is a double ...
0
votes
0answers
56 views

Definition of complex smooth vector bundle

Let $\xi:E \overset{\pi}{\rightarrow} B$ be a $2n$-dim. real vector bundle. Are the following two definitions for $\xi$ to be smooth and complex equivalent? 1) For every point $x \in B$ there is a ...
0
votes
0answers
102 views

Mumford's vector bundle stability equivalent the notion orbit stability for a G-space?

Everyone seems to use the slope definition of stability for vector bundles without making any mention to the fact that this should be the correct definition describing that a stable equivalence class ...
0
votes
0answers
289 views

Does extending a section by the exponential map make it transverse to the zero set?

Let $V_1, V_2 \rightarrow M$ be two smooth vector bundles over a smooth riemannian manifold $M$ and $s_1:M\rightarrow V_1$ a section transverse to the zero set and $s_2: s_1^{-1}(0) \rightarrow V_2 ...
0
votes
0answers
53 views

Is it obvious that the defining conditions to obtain a particular singularity are well defined on the quotient space?

Let $~f:\mathbb{C}^2 \rightarrow \mathbb{C}$ be a holomorphic function vanishing at the origin, with the following properties: $$ f_{00}, ~f_{10}, ~f_{01}, ~f_{20}, ~f_{11} =0,~~f_{20} \neq 0 ...
0
votes
0answers
151 views

Does every real vector bundle admit a metric?

Let $ E \longrightarrow B $ be a vector bundle. I know that if B is paracompact, the bundle admits a metric. My question is the following: is this true for any B? I also know that a reduction of ...
0
votes
0answers
81 views

Chern classes on compact manifold with boundary

Dear all, I am studying topological classes and I would like to know how Chern classes are defined on smooth compact manifolds with (non-empty) boundary. I am wondering in particular if there exists ...
0
votes
0answers
66 views

sections of vector bundles transversal to a divisor

Let $X$ a smooth projective curve over $\mathbb{C}$, $S$ a finite subscheme of $X$. $E$ a vector bundle over $X$ with a divisor $D$. We look at the sections $A:=H^{0}(X,E)$ with $\deg E$ big enough. ...
0
votes
0answers
164 views

sections of vector bundles

Let $X$ a smooth projective connected curve over $\mathbb{C}$. Let $E$ a vector bundle and $E'$ a subbundle of $E$. Let $(x_{1},\dots,x_{n})$ $n$ closed points on the curve $X$ with $n>2g$ and $z$ ...
0
votes
0answers
71 views

on trivialisation of T-torsors

Let $X$ a smooth connected projective curve over an algebraically closed field $k$ and $F$ its function field. $T$ a $X$-torus. Let $R$ be any ring. Let $E$ a $T$-torsor on $(X-x)\times_{k}R$. Does ...
0
votes
0answers
190 views

Variation of the Chern connection according to the variation of hermitian metric

Whats is the relation between the Chern connections of tow Hermitian metrics in a holomorphic vector bundle?
0
votes
0answers
606 views

First order differential equation of vector fields

Given vector fields $Y$, $Z$ on a (possibly compact) manifold $M$, I would like to know about the existence of solutions $X$ to the differential equation $$ \nabla_Y X + a \cdot \mathrm{div}(Y)\cdot X ...