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Kähler metric on the projective space

"Is there a Kähler metric on the complex projective space $\mathbb {P} ^n(\mathbb {C} ) $ different from the Fubini-Study metric?
Samir's user avatar
  • 43
1 vote
1 answer
92 views

Derivative of the symplectomorphism evaluated at a point of the zero section of the cotangent bundle

It might be an easy question, possibly not worth posting here. In the proof of the Lagrangian neighborhood theorem, the authors have written the expression for the derivative of a symplectomorphism at ...
Random's user avatar
  • 1,087
2 votes
1 answer
101 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 ...
maxo's user avatar
  • 33
1 vote
0 answers
143 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, $$Y \rightarrow \mathrm{Spec}(H^0(Y,\mathscr{O}_Y))$$ is proper or projective?
Filip's user avatar
  • 1,647
4 votes
1 answer
492 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\...
kindasorta's user avatar
  • 2,015
8 votes
1 answer
724 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 ...
Dave Shulman's user avatar
2 votes
0 answers
152 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?
Sourav's user avatar
  • 21
2 votes
1 answer
343 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\...
Pène Papin's user avatar
0 votes
0 answers
151 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_{...
Bobech's user avatar
  • 381
2 votes
1 answer
656 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-...
Bobech's user avatar
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0 answers
165 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 ...
Someone's user avatar
  • 791
3 votes
0 answers
95 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, ...
Filip's user avatar
  • 1,647
0 votes
1 answer
209 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, ...
Markus Zetto's user avatar
7 votes
0 answers
238 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 ...
Pat's user avatar
  • 513
2 votes
0 answers
68 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^...
Rdrr's user avatar
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3 votes
0 answers
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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 $\...
Andre Contiero's user avatar
1 vote
0 answers
292 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 ...
user avatar
15 votes
0 answers
332 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 $...
truebaran's user avatar
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2 votes
0 answers
212 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 ...
John Sung's user avatar
  • 111
3 votes
0 answers
136 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, ...
მამუკა ჯიბლაძე's user avatar
6 votes
0 answers
333 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. ...
Mimi's user avatar
  • 61
4 votes
1 answer
326 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 \...
truebaran's user avatar
  • 9,200
14 votes
3 answers
2k 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 ...
truebaran's user avatar
  • 9,200
3 votes
0 answers
362 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 ...
Peng's user avatar
  • 316
6 votes
1 answer
706 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 ...
Jaap Eldering's user avatar
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(...
QSR's user avatar
  • 2,213
2 votes
0 answers
224 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 ...
JonHerman's user avatar
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7 votes
3 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}^...
Puzzled's user avatar
  • 8,892
2 votes
0 answers
625 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 ...
Ali Taghavi's user avatar
3 votes
1 answer
212 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?
Ali Taghavi's user avatar
11 votes
2 answers
4k 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 ...
Pengfei's user avatar
  • 2,204
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 ...
Ali Taghavi's user avatar
0 votes
1 answer
310 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 ...
Puzzled's user avatar
  • 8,892
1 vote
1 answer
484 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 ...
Kensmosis's user avatar
  • 285
5 votes
1 answer
221 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). $...
Zhaoting Wei's user avatar
  • 8,737
-4 votes
1 answer
463 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 ...
Blake's user avatar
  • 1,025
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 ...
solbap's user avatar
  • 3,958
1 vote
0 answers
221 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 \...
Rami's user avatar
  • 2,591
41 votes
5 answers
5k 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 ...
Marco Golla's user avatar
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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^...
marco's user avatar
  • 11
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 ...
Theo Johnson-Freyd's user avatar
5 votes
1 answer
967 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 \...
Ben Webster's user avatar
  • 44.2k
3 votes
1 answer
631 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 ...
Ben Webster's user avatar
  • 44.2k
34 votes
6 answers
5k 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 ...
Mohammad Farajzadeh-Tehrani's user avatar