# Questions tagged [cotangent-bundles]

The cotangent-bundles tag has no usage guidance.

42
questions

2
votes

1
answer

82
views

### Normal bundle of veronese as iteration extension of symmetric powers

In this post, an answer claims that the normal bundle of the Veronese decomposes into a filtration, such that the associated graded is $$\bigoplus_{i=2}^d S^i T,$$ where $T$ is the tangent bundle. But ...

1
vote

0
answers

105
views

### Affinization map of cotangent bundle is proper/projective?

Given a cotangent bundle of a projective variety $Y=T^*X,$ do we know that its affinization map,
$$Y \rightarrow \mathrm{Spec}(H^0(Y,\mathcal{O}_Y))$$
is proper or projective?

4
votes

1
answer

436
views

### Pairing of cotangent and tangent bundles

I am reading the survey paper: "The de-Rham Witt complex and Crystalline cohomology" by Luc Illusie.
In math line (2.1.12), Illusie considers the pairing $\langle-,-\rangle:\Omega_{X/S}^1\...

7
votes

1
answer

531
views

### Derivations on the continuous functions of a manifold

For a manifold $M$ a vector field is a derivation of the algebra $C^{\infty}(M)$ of smooth functions on $M$. What happens if look instead as derivations on the continuous functions of a manifold. I ...

2
votes

0
answers

135
views

### Is the cotangent sheaf of the symmetric product reflexive?

Let $X$ be a smooth projective surface and $X^{(n)}:= X^n/\mathfrak{S}_n$ be the nth symmetric product of $X$.
When is the cotangent sheaf of $X^{(n)}$ reflexive?

2
votes

1
answer

260
views

### tangent bundle of Hilbert schemes of points on a projective surface

Let $S$ be a smooth projective surface. We denote $S^{[n]}$ the Hilbert scheme of artinian subschemes in $S$ of length $n$, which is a smooth projective variety of dimension $2n$ by Fogarty. Let $I\...

0
votes

0
answers

143
views

### Projectivization of the normal bundle of $\mathbb P^4$ in the 10-spinor variety

Let $X=S_{10} \subset \mathbb P^{15}$ be the 10-dimensional spinor variety in its minimal embedding. Consider a $\mathbb P^4 \subset S_{10}$, hence we can define the normal bundle $N=N_{\mathbb P^4|S_{...

2
votes

1
answer

472
views

### Tangent bundle for orthogonal and isotropic Grassmannians

We will work over $\mathbb C$. Let us consider a $n$-dimensional vector space $V$, then we define the $k$-th Grassmannian as
$$
\mathbb G(k,V):=\{W \subset V : \dim W=k\}.
$$
Then consider a non-...

1
vote

0
answers

142
views

### Relating the Morse index with the Maslov index

In the following paper https://arxiv.org/pdf/math/0408280.pdf there is created an isomorphism between the Floer Homology of an hamiltonian functional $H$ in the cotangent bundle and the the Morse ...

3
votes

0
answers

88
views

### Displacing a conormal Lagrangian from the zero section

I was told that the conormal bundle $\nu^*K$ of a knot $K\subset S^3$ can be displaced from the zero section $0_{S^3}$ in $T^*S^3.$ Having no intuition about whether/how often this happens in general, ...

0
votes

1
answer

202
views

### Natural map from vector fields to cotangent variety

Let $X$ be a smooth variety, and let $\operatorname{Vec} (X)$ denote the $\mathcal{O}_X$-module of vector fields on $X$. It is stated in several books on D-modules, for example here in Corollary 6.6, ...

7
votes

0
answers

214
views

### Triviality, ampleness, nefness, and bigness of the tangent bundle

Let $X$ be a smooth projective connected variety over $\mathbb{C}$ and let $T_X$ be its tangent bundle.
If $T_X$ is ample, then $X$ is isomorphic to a projective space by Mori's theorem.
If $T_X$ is ...

2
votes

0
answers

59
views

### Schur Bundle of Smooth Manifold

I've seen hints at the following result:
Let $M$ be a 3-dimensional manifold and let $T := T^*M$ be the cotangent bundle. By Schur-Weyl Duality, the 3rd tensor product can be written as follows:
$$T^...

3
votes

0
answers

90
views

### On the degrees of the normal and tangent sheaves of Gorenstein curves in Pn

It is known that if an irreducible curve $C$ is a local complete intersection in $\mathbb{P}^n$, then
$\wedge^{n-1}\mathcal{N}\otimes\omega_{\mathbb{P}^n}$ is isomomorphic to the dualizing sheaf $\...

1
vote

0
answers

240
views

### Restriction of the sheaf of relative differentials

Let $f:X\rightarrow C$ be a morphism, where $C$ is a smooth curve, and let $\Omega_f$ be the sheaf of relative differentials.
For $t\in C$ let $i_t:X_t = f^{-1}(t)\rightarrow X$ be the inclusion of ...

15
votes

0
answers

324
views

### Beyond smoothness-the clear picture about the notion of a differential form

In this paper N.Teleman constructs the signature operator on an arbitrary (closed, oriented) Lipschitz manifold with coefficients in a vector bundle $\xi$. In particular the notion of a differential $...

2
votes

0
answers

193
views

### how to understand the manifold with boundary jet bundle and cotangent bundle with boundary

Suppose that $M\subset (W^{2n},\omega)$ is an $n$-dimensional manifold with smooth boundary $\partial M$, where $(W,\omega)$ is a $2n$-dimensional Kähler manifold and boundary with contact type ...

3
votes

0
answers

131
views

### Is there a precisely formulable obstruction for the tangent bundle being a Lie algebra bundle?

Although vector fields (which are sections of the tangent bundle) form Lie algebras, the bundle itself, as far as I know, almost never carries Lie algebra structure; that is, in general, I believe, ...

6
votes

0
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303
views

### Lagrangian up to Hamiltonian in cotangent bundle

I want to understand the folklore conjecture that, in a CY manifold, Lagrangians up to Hamiltonian isotopies are represented by special Lagrangians by examining cotangent bundle and Hodge theory.
...

4
votes

1
answer

311
views

### Natural extension homomorphism and wrong-way maps in K-theory

Let $X \subset Y$ be two smooth manifolds. To the inclusion $I:X \to Y$ corresponds the so called wrong-way map in $K-theory$ $i_!:K(X) \to K(Y)$. It is constructed as follows: to the inclusion $X \...

14
votes

3
answers

1k
views

### Splitting of tangent bundle

Is it possible to give an example of $n$ dimensional manifold with the property that the tangent bundle $TM$ cannot be expressed as Whitney sum of two subbundles? It is certain true for two sphere; it ...

3
votes

0
answers

327
views

### Reference for plumbing cotangent bundles as Liouville manifolds

A Liouville domain $(W, \omega,\alpha, X)$ is a compact manifold $W$ with boundary $\partial W$, and a exact symplectic structure $\omega = d\alpha, \iota_X \omega = \alpha$, such that $X$ points ...

6
votes

1
answer

655
views

### Are the Sasaki metrics on tangent and cotangent bundle isomorphic?

Let $(M,g)$ be a Riemannian manifold. Then there is the well-known
Sasaki metric that makes $(TM,\hat{g})$ a Riemannian manifold. In a
similar way, one can construct a Sasaki metric $\bar{g}$ on the
...

3
votes

1
answer

1k
views

### characteristic classes of homotopy equivalent manifolds

Let $M,N$ be two manifolds of different dimensions. Suppose $M\simeq N$, i.e. $M$ is homotopy equivalent to $N$. Do the Stiefel-Whitney classes of the tangent bundles of $M$ and $N$ equal
$$
w(TM)=w(...

2
votes

0
answers

221
views

### When is a symplectomorphism on a cotangent bundle the lift of a diffeomorphism on the base manifold [duplicate]

Let $X$ be a manifold and $T^\ast X$ the cotangent bundle. Let $\alpha$ denote the tautological $1$-form on $T^\ast X$ so that $(T^\ast X, \omega=-d\alpha)$ is a symplectic manifold. I want to know ...

7
votes

2
answers

1k
views

### Cohomology of the tangent sheaf of $\mathbb{P}(1,2,3)$

Using the exact sequence
$$0\mapsto\mathcal{O}_{\mathbb{P}^{2}}\rightarrow\mathcal{O}_{\mathbb{P}^{2}}(1)^{\oplus 3}\rightarrow T_{\mathbb{P}^{2}}\mapsto 0$$
it is easy to compute $H^{1}(\mathbb{P}^...

2
votes

0
answers

616
views

### A symplectic structure for cotangent bundle

Before that I mention my question explicitly, I start with my motivation:
Look at $\mathbb{D} \times \mathbb{C}=\{(x_{1},x_{2},y_{1},y_{2})\mid x_{1}^{2}+x_{2}^{2}< 1\}$.This can be identified ...

3
votes

1
answer

189
views

### A question on long line

Assume that $M$ is the long line. Is $TM$, the tangent bundle, isomorphic to $T^{*}(M)$, the cotangent bundle?

9
votes

2
answers

3k
views

### Cotangent bundle lift theorem

Let $M$ be a smooth manifold and $T^\ast M$ be its cotangent bundle. Consider the tautological 1-form $\theta$ on $T^\ast M$ ($\theta=\sum y_i dx^i$ in local canonical coordinate systems).
A ...

6
votes

2
answers

1k
views

### symplectic structure of tangent bundle of $\mathbb{S}^{n-1}$

It is well known that $T\mathbb{S}^{n-1}$ is diffeomorphic to $M= f^{-1}(1)$ where
$f:\mathbb{C}^n\rightarrow \mathbb{C}$ is $f(z):=\sum_{i=1}^{n} z_{i}^{2}$.
Two questions:
1) Is $M$ a ...

0
votes

1
answer

301
views

### Tangent bundles and birational morphisms

Let $f:X\rightarrow Y$ be a smooth morphism between smooth schemes. Then there is an exact sequence
$$0\mapsto T_{X/Y}\rightarrow T_{X}\rightarrow f^{*}T_{Y}\mapsto 0$$
Now let us assume $f$ to be a ...

1
vote

1
answer

461
views

### Is there a relationship between tensor (or form) bundles and iterated tangent/cotangent bundles on a manifold?

Let's say we denote by $T^{(n,m)}M$ the vector-bundle of rank $(n,m)$ tensors on a manifold $M$ and by $\Lambda^pM$ the vector-bundle of $p$-forms on $M$. Is there a relationship (perhaps a ...

5
votes

1
answer

220
views

### What is the role of $\sum (-1)^p[\wedge^pT^*M]$ in the K-theory $K(M)$

I apologize for the vague title. Let $M$ be a compact smooth manifold, then we have $T^*M$ and hence $\wedge^pT^*M$ as vector bundles on $M$. There for we have
$$
\sum (-1)^p[\wedge^pT^*M] \in K(M).
$...

-4
votes

1
answer

450
views

### Symplectic forms and 1-forms [closed]

Suppose we have a real symplectic manifold $(M,\omega)$. Under what conditions can we find a global 1-form $\alpha$ such that $\omega = \alpha \wedge\alpha$?
Obviously there are some simple ...

11
votes

2
answers

1k
views

### Do smooth ind schemes have Dualizing sheafs?

Say I have an ind scheme $X = \cup_i X_i$ over a field $k$. I have its tangent bundle $\hom_k(k[\epsilon], X)$ which I can think of as ind scheme via $\cup_i \hom_k(k[\epsilon],X_i)$. The problem is ...

1
vote

0
answers

217
views

### Co-normal bundle of orthogonal compliment

Is the following fact well known?
Let $X$ be a manifold and $V$ be a vector space. Let $E_1$ be a sub-bundle of the constant bundle $X \times V$. Let $E_2$ be its orthogonal compliment in $X \...

41
votes

5
answers

4k
views

### Can cotangent bundles see exotic smooth structures?

I have two questions that are inspired by a couple of questions here on MO (referenced below), as well as by a conversation with some other grad students at a summer school.
Caveat: I'm not a ...

1
vote

2
answers

2k
views

### tangent and cotangent bundle

Hi, I am reading "Introduction to symplectic topology" by McDuff and salamon. At some point I cant go further. My question is: Let $(M,g)$ be a Riemannian manifold and consider the cotangent bundle $T^...

20
votes

1
answer

2k
views

### Functoriality of the cotangent bundle

Recall that to any manifold $X$, I can assign in a canonical way a manifold $\mathrm T ^* X$, the total space of the cotangent bundle over $X$. Recall also that, unlike the tangent bundle ...

4
votes

1
answer

917
views

### Does the preimage of the Slodowy slice in $T^*G/P$ have a name?

Let $G$ be your favorite simple complex Lie group, and $P\subset G$ your favorite parabolic subgroup. We can identify $T^*G/P$ with the space of pairs $$\{(gP,x)\in G/P\times \mathfrak g | x\perp \...

3
votes

1
answer

618
views

### Can the class of the canonical bundle be recovered from the total space of the cotangent bundle if one forgets that it is a cotangent bundle?

This is a somewhat speculative question, so bear with that (or not, as is your preference).
Let $X$ be a smooth projective variety, and let $\omega_X$ be its canonical sheaf. The Euler class of ...

32
votes

6
answers

4k
views

### Kähler structure on cotangent bundle?

The total space of cotangent bundle of any manifold $M$ is a symplectic manifold.
Is it true/false/unknown that for any $M$, $T^*M$ has Kähler structure?
Please support your claim with reference or ...