Highest scored 'dg.differential-geometry+smooth-manifolds+complex-geometry' questions

11 votes

1 answer

593 views

Examples of 6-manifolds without an almost complex structure

Question: I am searching for examples for closed (hence orientable ), smooth $6$-manifolds without an almost complex structure. Finding such an example is equivelant to finding a manifold where the ...

Nick L

6,995

asked Apr 29, 2022 at 18:26

10 votes

1 answer

653 views

Algebraic atlas on smooth manifolds

A real/complex rational atlas on a smooth closed manifold $M$ is an atlas with charts homeomorphic to Euclidean open sets in $\Bbb{R}^n$/$\Bbb{C}^n$ covering $M$ and real/complex rational transition ...

Zerox

1,543

asked Dec 4, 2021 at 5:15

8 votes

0 answers

382 views

Universal property for complex blowup in smooth category

If $M$ is a smooth complex manifold and $N$ is a smooth complex surface, we may ask when a holomorphic map $f:M\rightarrow N$ lifts to a map $f:M\rightarrow [N:p]$, where $[N:p]$ denotes the blowup of ...

Marco Gualtieri

1,221

asked Apr 7, 2011 at 4:28

6 votes

2 answers

2k views

book on calabi yau manifolds

hi, does anybody know a good book on calabi yau manifolds (i am a beginner) ? thanks in advance lois

lois

63

asked Nov 6, 2011 at 11:55

6 votes

1 answer

605 views

Closed topological embedding of complex algebraic varieties into a smooth manifold

In the book "Representation Theory and Complex Geometry” by Ginzburg-Chriss page 93 they claim that (the analytification of) every complex algebraic variety admits a closed embedding in a smooth ...

user110443

141

asked May 27, 2017 at 5:39

6 votes

1 answer

1k views

Holonomy of a Kähler manifold

Hi, I have the following question: Let $(M,J, \omega)$ be a Kähler manifold (not necessary compact). We know that the holonomy group is a subgroup of $U_{n}$. Let $\Omega$ be a constant ($\nabla \...

Mina

93

asked Oct 25, 2012 at 6:37

6 votes

0 answers

324 views

Ricci-flat metrics on Cotangent bundles in adapted complex structure

greetings, Let $(M,g)$ be a compact Riemannian manifold. On some neighbourhood $X$ of the zero section in the cotangent bundle $T^{*}L$ we have a complex structure $J$ and a Kähler form $\omega$ s.t. ...

dominik

61

asked Sep 24, 2012 at 11:16

4 votes

2 answers

575 views

Do transvers foliations induce complex structure?

Hallo, I have the following question: Let $M$ smooth analytic manifold of dimension 4n. Assume furthermore that $M$ admits two foliations $A$, $B$, both with leaves of dimension 2n such that the ...

Marin

41

asked Nov 3, 2012 at 5:28

4 votes

2 answers

414 views

Is it impossible for the dimension of a topological space to increase under a smooth map?

First let me make a definition. Let $M$ be a smooth manifold and $S \subset M $ a topological subspace of $M$. We say that $S$ has "dimenion" at most $k$ if $S$ is a subset of $$ X_1 \cup X_2 \ldots ...

Ritwik

3,245

asked Jun 30, 2013 at 9:23

3 votes

3 answers

1k views

Rotation in Hyperkähler manifolds

Any Hyperkähler manifold has 3 complex structures $I_{1}, I_{2}, I_{3}$. Assume that there is an additional complex structure $J$. Can this be written as $J = aI_{1} + bI_{2} + cI_{3}$, where $(a,b,c) ...

hapchiu

339

asked Nov 6, 2012 at 5:10

3 votes

0 answers

426 views

Integration over a Surface without using Partition of Unity

Suppose we are given a compact Riemann surface $M$, an open cover $\mathscr{U}=\{U_1,U_2,\dots\}$ of $M$, charts $\{(U_1,\phi_1),(U_2,\phi_2),\dots\}$, holomorphic coordinates, $\phi_m:p\in U_m\mapsto ...

Wakabaloola

277

asked Mar 6, 2019 at 10:49

3 votes

0 answers

142 views

Non-snc locus relative to a smooth morphism

Let $f\colon X\rightarrow Y$ be a smooth morphism between smooth varieties and $B\subset Y$ a simple normal crossing divisor such that $f^*(B)$ is simple normal crossing as well. Consider a semiample ...

Joaquín Moraga

1,585

asked Nov 7, 2017 at 17:25

3 votes

0 answers

110 views

Thom form of holomorphic bundle over Kaehler manifolds/orbifolds

Consider a holomorphic vector bundle $\pi:E\rightarrow X$ of complex rank $m$ over a Kaehler manifold $X$. Can we find a Thom form $\Theta$ of $E$ such that as a form on the complex manifold $E$, it ...

Ryan Du

303

asked Jul 13, 2016 at 15:48

2 votes

1 answer

425 views

holomorphic extension of forms

hallo, I have the following question: Let $M$ be a $n-$dimensional complex manifold and $X \subset M$ be a compact $n-$dimensional totally real analytic Riemannian submanifold. Let furthermore $\...

bruno

29

asked Sep 5, 2012 at 6:41

2 votes

1 answer

392 views

Holonomy group of a non-compact Kaehler manifold

Hallo, I have the following question: Let $(M,I,\omega)$ be a not necessary compact Kaehler manifold of complex dimension $n$. Assume that there exists a nowhere vanishing holomorphic $(n,0)$-form $\...

Mina

93

asked Oct 24, 2012 at 15:33

2 votes

2 answers

513 views

Isometric embedding of a Kaehler manifold as a special Lagrangian in a Calabi-Yau manifold

Hallo, I am reading the paper "Hyperkaehler structures on the total space of holomorphic cotangent bundles" by D.Kaledin and I am asking if it is possible to embedd a real-analytic Kähler manifold, ...

Pavel

23

asked Oct 20, 2012 at 10:24

2 votes

0 answers

224 views

Intuition behind Nakano positivity

I am learning about Nakano positivity of hermitian vector bundles, which is the strongest notion of positivity we can ask. I don't understand what is the geometric meaning of it. Let me briefly ...

Dubious

1,237

asked Jan 24, 2021 at 4:42

2 votes

0 answers

446 views

$\partial \bar{\partial}$ on a complex manifold

Let $M$ be a complex $n$-dimensional manifold and $R \subset M$ be a totally real, compact, $n$-dimensional (real) manifold. Let $\alpha$ be a smooth nonnegative $(n,n)-$form on $M$. Does there exist ...

william

213

asked Feb 9, 2012 at 6:42

1 vote

1 answer

510 views

The space of generalized complex structures in sense of N.Hitchin is contractible?

Generalized complex structures were introduced by Nigel Hitchin in 2002. A generalized almost complex structure is an almost complex structure of the generalized tangent bundle which preserves the ...

user21574

asked Aug 13, 2013 at 14:55

1 vote

2 answers

622 views

Kähler manifold with Ricci-flat Kähler form

hallo, I have the following problem: Let $X$ be a $n-$dim Kähler manifold with Ricci-flat Kähler form $\omega$. There is a known fact that then there exists a holomorphic (n,0)-form $\Omega$ such ...

bruno

29

asked Sep 13, 2012 at 6:14

1 vote

1 answer

534 views

Unique symplectic form in an adapted complex structure

I have the following question: Due to Stenzel, Lempert, Szöke etc. we know that a Riemannian manifold $(M,g)$ admits a complex structure on a neighbourhood of the zero section of the cotangent bundle. ...

Joan

13

asked Oct 6, 2012 at 6:07

1 vote

1 answer

397 views

Einstein metrics on the tangent bundle

Let $M$ be a compact, real analytic, riemannian manifold with real analytic metric $g$. Does the tangent bundle always admit an Einstein metric ?

william

213

asked Feb 11, 2012 at 21:48

1 vote

1 answer

254 views

local kählerforms on complex manifold

hallo, Let $M$ be a complex manifold. Assume we have a covering of $M$ by complex charts $\{U_{i}\}$. Furthermore assume that we have on each $U_{i}$ a Kählerform $\omega_{i}$ (i.e. $d\omega_{i} = 0$)...

gary

221

asked Dec 7, 2011 at 6:57

1 vote

0 answers

648 views

Proof of Ehresmann's theorem

In Huybrechts' book Complex geometry: An introduction p.269, Proposition 6.2.2, the author gives a proof of the following theorem (Ehresmann) Let $\pi:\mathcal X\to B$ be a proper family of ...

Tom

471

asked May 9, 2022 at 8:01

1 vote

0 answers

88 views

Submersion function from a product space

Let $\Phi(x,y) \colon U_N \times U_M \to \mathbb{C}^n$ be a submersion, where $U_N \subset \mathbb{C}^N$ and $U_M \subset \mathbb{C}^M$. Under which condition on $\Phi$ can I find some $s \in \...

Serge the Toaster

919

asked Feb 25, 2022 at 23:09

1 vote

0 answers

346 views

HyperKaehler manifolds are Ricci-flat

Hi, I have the following question: Let $M$ be a Hyperkaehler manifold with complex structures $I,J,K$ and Hyperkaehler metric $g$. Let $\omega_{I} = g(I *, *), \omega_{J} = g(J *, *), \omega_{K} = g(...

Mina

93

asked Oct 24, 2012 at 15:54

1 vote

0 answers

169 views

monge ampere equation along totally real submanifolds

hi, are there some references when solving the complex monge ampere equation along totally real submanifolds of some compact (with boundary or without) complex manifold. i know that there are a lot ...

william

213

asked Feb 22, 2012 at 7:58

0 votes

2 answers

136 views

Boundary of the image of a projection

Let $U$ be a connected open subset in $\mathbb{R^n}$. Let $f: U \rightarrow U$ be a differentiable projection, i.e. $f\circ f = f$. It's well-known that $f(U)$ is a submanifold of $U$ (Henri Cartan, ...

Curiosity

293

asked May 16, 2017 at 9:29

0 votes

0 answers

80 views

Details of the proof of the inequality $ \int_{X}\left(2 r \mathrm{c}_{2}(E)-(r-1) \mathrm{c}_{1}^{2}(E)\right) \wedge \omega^{n-2} \geq 0.$

I'm trying to make sense of the following proof. Let $E$ be a holomorphic vector bundle of rank $r$ on a compact hermitian manifold $(X, g)$. If $E$ admits an Hermite-Einstein structure then $$ \int_{...

Nikolai

103

asked Jun 21 at 7:52

-1 votes

1 answer

237 views

Almost Complex Structure extending to Complex Structure, aka "Integrable"

Let $M$ be a smooth manifold of (real) dimension $2n$. An almost complex structure $J$ on $M$ is a linear vector bundle isomorphism $J \colon TM\to TM$ on the tangent bundle $TM$ such that $J^2 = − 1 \...

user267839

6,018

asked Dec 6 at 17:03

-2 votes

1 answer

946 views

Holonomy group of calabi yau manifold

Let $(M,J,\omega, \Omega)$ be a calabi-yau manifold (not necessary compact). Does it follow that the holonomy group of $M$ is $SU_{n}$, where $n$ is the complex dimension of $M$ ?

pascal

89

asked Mar 20, 2012 at 6:18

-3 votes

1 answer

1k views

Pull back a vector field [closed]

In Voisin's book Hodge theory and complex algebraic geometry, I Section 9.1.2, p.223, the author writes: Let $\phi:\mathcal X\to B$ be a family fo complex manifolds. The differential $\phi_*$ is a ...

Tom

471

asked May 4, 2022 at 13:54

Stack Exchange Network

All Questions

Examples of 6-manifolds without an almost complex structure

Algebraic atlas on smooth manifolds

Universal property for complex blowup in smooth category

book on calabi yau manifolds

Closed topological embedding of complex algebraic varieties into a smooth manifold

Holonomy of a Kähler manifold

Ricci-flat metrics on Cotangent bundles in adapted complex structure

Do transvers foliations induce complex structure?

Is it impossible for the dimension of a topological space to increase under a smooth map?

Rotation in Hyperkähler manifolds

Integration over a Surface without using Partition of Unity

Non-snc locus relative to a smooth morphism

Thom form of holomorphic bundle over Kaehler manifolds/orbifolds

holomorphic extension of forms

Holonomy group of a non-compact Kaehler manifold

Isometric embedding of a Kaehler manifold as a special Lagrangian in a Calabi-Yau manifold

Intuition behind Nakano positivity

$\partial \bar{\partial}$ on a complex manifold

The space of generalized complex structures in sense of N.Hitchin is contractible?

Kähler manifold with Ricci-flat Kähler form

Unique symplectic form in an adapted complex structure

Einstein metrics on the tangent bundle

local kählerforms on complex manifold

Proof of Ehresmann's theorem

Submersion function from a product space

HyperKaehler manifolds are Ricci-flat

monge ampere equation along totally real submanifolds

Boundary of the image of a projection

Details of the proof of the inequality $ \int_{X}\left(2 r \mathrm{c}_{2}(E)-(r-1) \mathrm{c}_{1}^{2}(E)\right) \wedge \omega^{n-2} \geq 0.$

Almost Complex Structure extending to Complex Structure, aka "Integrable"

Holonomy group of calabi yau manifold

Pull back a vector field [closed]

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