All Questions
32 questions
-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 \...
- 6,018
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_{...
- 103
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 ...
- 471
-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 ...
- 471
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 ...
- 6,995
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 \...
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 ...
- 1,543
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 ...
- 1,237
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 ...
- 277
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 ...
- 1,585
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 ...
- 141
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, ...
- 293
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 ...
- 303
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
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 ...
- 3,245
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) ...
- 339
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 ...
- 41
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 \...
- 93
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(...
- 93
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 $\...
- 93
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, ...
- 23
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. ...
- 13
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. ...
- 61
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 ...
- 29
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 $\...
- 29
-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 $M$ ?
- 89
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 ...
- 213
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 ?
- 213
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 ...
- 213
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. $d\omega_{i} = 0$)...
- 221
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
- 63
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 ...
- 1,221