Questions tagged [complex-geometry]
Complex geometry is the study of complex manifolds, complex algebraic varieties, complex analytic spaces, and, by extension, of almost complex structures. It is a part of differential geometry, algebraic geometry and analytic geometry.
3,142
questions
6
votes
1
answer
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Puiseux series expansion for space curves?
This result is apparently well known and used by many people.
I am, however, quite frustrated that I cannot seem to find a proof that I can understand.
For plane algebraic curves, this is not too hard....
- 1,101
2
votes
1
answer
254
views
What does non-levi flat point mean geometrically
Hello,
$CR$ manifold for example $S^1\times C^{n-1}$ is every where levi flat. Can I have example of $CR$ manifold which has at least one non levi flat point.
I can't see what the happening in Non-...
- 541
2
votes
2
answers
587
views
calabi conjecture on compact manifolds
hi,
is the calabi conjencture formulated for compact manifolds with boundary ? or only for those without boundary ? excuse me if the question is too trivial but in my literature it isn't mentioned ...
- 213
10
votes
1
answer
1k
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On compact Kähler manifold diffeomorphic to complex projective space
In the paper On the complex projective spaces, Hirzebruch and Kodaira prove the following:
If $X$ is compact Kähler manifold diffeomorphic to $\mathbb{CP}^n$, then $X$ is biholomorphic to $\mathbb{...
- 503
7
votes
3
answers
768
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Strong Kodaira vanishing
Let $X$ be a smooth projective variety (say, over a field of characteristic zero).
Let us say that strong Kodaira vanishing holds for $X$ if
$$
H^q(X,\Omega^p\otimes L)=0
$$
for every $p\geq 0$, $q&...
- 6,957
6
votes
1
answer
1k
views
Blowing-up an ordinary double point, then contracting the exceptional locus to a curve
Let $X\subset\mathbb P^4$ a projective hypersurface with an ordinary double point at $o\in X$.
Blow-up $\mathbb P^4$ at $o$ and let $E\simeq\mathbb P^3$ the exceptional divisor of this blow-up. ...
- 7,852
1
vote
0
answers
300
views
einstein metrics on the tangent bundle
hi,
i have the following question. let $M$ be a compact, real analytic, riemannian manifold with real analytic metric $g$. does the tangent bundle admit always a einstein metric ?
marco
- 213
2
votes
0
answers
445
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
3
votes
1
answer
1k
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A corollary to Stone-Weierstrass theorem
Can i get the answer to the following problem. I am having a proof, i feel there is something wrong here..Can you please point out!
Let $D\subset \mathbb C$ be a simply connected domain, and $\gamma:...
- 541
32
votes
2
answers
2k
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Example of a compact Kähler manifold with non-finitely generated canonical ring?
A celebrated recent theorem of Birkar-Cascini-Hacon-McKernan and Siu says that the canonical ring $R(X)=\oplus_{m\geq 0}H^0(X,mK_X)$ of any smooth algebraic variety $X$ over $\mathbb{C}$ is a finitely ...
- 6,656
3
votes
3
answers
599
views
$\partial \bar{\partial}$ on a riemann surface
hallo,
i have the following question. let $M$ be a Riemann surface and $R \subset M$ be a compact totally real manifold (1 dimensional real). Furthermore assume there is a holomorphic 2-form $\alpha$...
- 213
1
vote
0
answers
200
views
Question about specifying complex 1-motives
A 1-motive over a field $k$ is an algebraic torus $T$, an abelian variety $A$, a group scheme $G$ that's an extension of $A$ by $T$, a finitely generated free abelian group $L$, and a group ...
- 183
4
votes
0
answers
215
views
Kähler Cones in $\mathbb{C}^4$ and a foliation of $\mathbb{P}^3$
Take the 3-dimensional complex projective space $\mathbb{P}^3$. Consider the action of the group $SU(2)\times SU(2)$. I have read in physics related articles that these group gives a singular ...
- 101
4
votes
1
answer
604
views
Orbits of the action of $A_6$ on $\mathbb{P}_2$
By a paper of Scott Crass http://xxx.lanl.gov/pdf/math/9903111v1.pdf
we know that $A_6$ (Permutation on 6 elements) is an automorphism group of $\mathbb{P}_2$ which fix a sextic. What is the ...
- 645
5
votes
2
answers
726
views
spin structures on full flag manifolds
It is known that any full flag manifold $G/T$ is a spin manifold.
For example, we can prove it using that $G/T$ is a complex manifold,
by computing its 1st Chern class as follows:
For full flag ...
- 1,209
5
votes
1
answer
626
views
A simple question about the degree of some vector bundle over rational curve.
Let $E$ be a holomorphic vector bundle (infact complex vector bundle is enough) over $\mathbb{P}^1$. Let $c: \mathbb{P^1} \rightarrow \mathbb{P^1}$ be the anti-holomorphic involution, $c(z)=\frac{-1}{...
4
votes
1
answer
4k
views
How many points determine a line?
Consider the affine space $\mathbb C^n$ and then, because of reasons, compactify it to obtain the projective space $\mathbb P^n$. One of the most basic axioms or propositions of geometry is that ...
- 4,201
3
votes
1
answer
594
views
The Levi form of the distance squared function in a non-positively curved Kaehler manifold
Suppose that if $X$ is a complete, simply connected Kaehler manifold with non-positive sectional curvatures. Let $P \in X$ and $h : X \to \mathbb{R}$ be the function defined by $h(x) = dist(P,X)^2$. ...
- 75
5
votes
1
answer
1k
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first chern class and spin structures
Let M be a compact complex manifold. Then is it true that if the first Chern class of M is even, then M admits a spin structure?
- 1,209
10
votes
2
answers
572
views
Non-bimeromorphic compactifications
Let $X$ be a (smooth, non-compact) complex space. By a compactification of $X$ we mean a compact complex space $\overline X$ which contains a dense open subset biholomorphic to $X$ (we shall identify ...
- 7,852
2
votes
1
answer
1k
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Donaldson's proof of Narasimhan Seshadri theorem [closed]
I would like to know what in the Donaldson's proof make it work only for Riemann surfaces.
- 31
1
vote
0
answers
228
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every where levi flat
"Suppose $N$ is $2n-1; n\geq 2$ dimensional $CR$ manifold and everywhere Levi flat, then it will be locally $CR$ equivalent to $S^1\times \mathbb C^{n-1}.$"
Above statement can be found in Loop ...
- 541
8
votes
0
answers
1k
views
Can you make the cotangent bundle to a complex manifold?
The cotangent bundle of a manifold has a canonical symplectic form and if we choose a riemannian metric on $M$, we can give it an almost complex structure.
Is this structure integrable, and if it ...
- 9,591
1
vote
0
answers
303
views
Strong minimum principle for maximal plurisubharmonic functions
Suppose $u$ is a bounded maximal plurisubharmonic function in a bounded domain $D \in \Bbb C^n$. If $u$ is $C^2$ one can see that $u$ cannot have a local strict minimum inside $D$. Is there an analog ...
- 1,171
1
vote
1
answer
420
views
skoda el-mir theorem
now i'm studying the skoda el-mir theorem about the extension of a positive closed current $T$.
But if $T$ ed $S$ are two positive closed currents on a manifold $X$ such that are equal on $X\setminus ...
- 75
7
votes
2
answers
981
views
Explicit way to construct simple complex tori/abelian varieties of dimension at least 2
The following question was motivated by one of the earliest exercises of Complex Abelian Variaties by Birkenhake and Lange during my presentation last year.
It can be shown that any complex torus $X$...
- 1,651
5
votes
1
answer
1k
views
Dual Lefschetz Operator and Contraction with the Fundamental Form
Let $M$ be a Kahler manifold, with metric $g$, fundamental form $\omega$, and dual Lefschetz operator $\Lambda$. Now $\Lambda$, and contraction with $\omega$, both map the two forms $\Omega^2(M)$ to $...
- 389
8
votes
4
answers
1k
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Monge Ampere equations
I am a graduate student trying to understand complex Monge-Ampere equations(mostly on complex manifolds with or without boundary, but also in C^n), but I can't put my hand on any monograph/textbook ...
- 1,171
3
votes
1
answer
1k
views
Global sections of tensor product of pull-back of two vector bundles
Suppose X,Y are two complex manifolds and E,F are vector bundles over them respective, what can I say about their global sections?
Does this formula $\Gamma (X\times Y,p_1^{*}E\otimes p_2^{*}F)=\Gamma ...
- 843
8
votes
2
answers
1k
views
Hyper-complex and quaternionic Kähler Geometry
What is the exact relationship hyper-complex and quaternionic Kahler manifolds? From Wikipedia we get that hyper-Kahler manifolds are both hyper-complex and quaternionic Kahler. Thus, the two families ...
- 3,359
1
vote
2
answers
1k
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Composition of circle inversions
I would like to understand the map of $\mathbb{C}$ to $\mathbb{C}$
that results by iterating inversion in a unit circle.
Let $f(z)$ for $z \in \mathbb{C}$ invert $z$ in a unit circle
centered on $q_1$,...
- 149k
1
vote
2
answers
1k
views
direct image of currents
I'm studing currents from Demailly's Complex Geometry, and the author defines the direct image of a current by a $C^{\infty}$ map and also for the case of a submersion. My question is about the ...
- 75
3
votes
2
answers
226
views
Uniformity of injectivity for maps associated to linear systems
Let $X$ be a compact complex manifold and $L\to X$ a holomorphic line bundle (without any a priori assumption on its positivity).
Suppose that for each $x,y\in X$, with $x\ne y$, there exists a $k_0\...
- 7,852
1
vote
2
answers
329
views
Abelian subgroups of ball quotient
Let $X$ be a compact complex surface of general type which a ball quotient. Is it true that $\pi_{1}(X)$ can not contain ${\mathbb{Z}}^{2}$ as a subgroup? What kind of infinite abelian groups can ...
- 13
1
vote
1
answer
251
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
2
votes
1
answer
259
views
Little Picard for (open) complex manifolds?
"Little Picard" states that if a complex function $f(z)$ is entire and non-constant, then the set of values that f(z) assumes is either the whole complex plane or the plane minus a single point. The ...
- 3,758
3
votes
1
answer
345
views
extended forms from foliations [closed]
hi,
i have the following question: Let $M$ be a n-dimensional manifold (or riemannian or everything thats nice ...) and let $\mathcal{F}$ be a foliation of $M$ by surfaces. Assume, furthermore, that ...
- 221
-1
votes
1
answer
412
views
isolated points
can $E_c(T)=\{x\in X~:~\nu(T,x)\geq c\}$ have isolated point?
where T is a positive current of bidegree 1, c is a positive real number, $X$ is a complex variety, and $\nu(T,x)$ is the Lelong number of ...
- 1
3
votes
1
answer
445
views
If Lie(G) is semi simple then the moment map exists!
Let's assume $M$ is a symplectic manifold with the group action $G$. If $Lie(G)$ is semi simple then why the Hamiltonian condition, which requires the existence of linear map $Lie(G)\to C^{\infty}(M,...
- 645
18
votes
1
answer
3k
views
Theorem of Bryant in higher dimensions
I have the following question. I read about Bryant's theorem which says that: any real-analytic 3-dimensional Riemannian manifold $(Y,g)$ with real-analytic metric $g$ can be isometrically embedded as ...
- 221
4
votes
3
answers
1k
views
Non-integrable almost-complex structures for homogeneous spaces
Let $M$ be a smooth homogeneous $G$-space for a Lie group $G$, and let $J$ be a $G$-invariant almost-complex structure for $M$. Do there exist succinct sufficient (and necessary) conditions for $J$ to ...
- 389
12
votes
4
answers
1k
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Wanted: an example of a natural non-K\"ahler metric on a Kahler manifold
Let $X$ be a Kahler manifold. Associated to any hermitian metric $h$ on $X$ is a smooth real $(1,1)$-form $\omega = -\text{Im } h$, called the Kahler form of $h$. One of several equivalent conditions ...
- 4,201
3
votes
1
answer
461
views
Software for calculating products and sums of Kronecker deltas
I am looking at a Kahler metric $g$ on a certain manifold $M$, which has the good taste to be invariant under a transitive group of isometries, and I want to say something about its holomorphic ...
- 4,201
1
vote
0
answers
579
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 ...
- 3,975
5
votes
1
answer
861
views
sections of morphisms of complex spaces
A smooth morphism of schemes $f: X \to Y$ admits an étale-local section through any point $x \in X$.
One might wonder if this fact is true in the more general context of complex spaces (i.e. things ...
- 4,060
14
votes
1
answer
824
views
$\mathbb{Z}/2$ is to $\mathbb{Z}/3$ as K3 is to what?
I'd like to know "what" (say, in the classification of complex surfaces) the following complex manifold $X$ is:
Construction: Let $\Lambda$ be the hexagonal lattice in $\mathbb{C}$; that is, the ...
- 3,192
2
votes
1
answer
148
views
Degree of Zariski closure of curve parametrized by hypocycloids
I have a curve $(x(\theta),y(\theta))$ in $\mathbb{C}^2$, where $x(\theta)$
is described as
$$x(\theta) = (k-1)\cos(\theta) + \cos((k-1)\theta) + i[(k-1)\sin(\theta)- \sin((k-1)\theta)]$$
and $y(\...
- 15.6k
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
2
votes
2
answers
312
views
Condition on the canonical divisor for Yau Inequality - effective or ample?
Let $X$ be a complex, projective, nonsingular variety. We also understand it as a Kähler Manifold. My question now is, when people say $c_1(X) < 0$, what exactly do they mean? Let me elaborate. In ...
- 4,822
4
votes
2
answers
418
views
Proper morphisms
Suppose that $f:X\to S$ is a holomorphic morphism of Hausdorff complex manifolds and that $s\in S$ such that $f^{-1}(s)$ is compact (and maybe singular). Then is it true that there is an open ...
- 3,052