All Questions
53 questions
29
votes
5
answers
6k
views
"The complex version of Nash's theorem is not true"
The title quote is from p.221 of the 2010 book,
The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions
by Shing-Tung Yau and Steve Nadis. "Nash's theorem" here ...
- 151k
27
votes
5
answers
7k
views
References for "modern" proof of Newlander-Nirenberg Theorem
Hi,
I'm starting to prepare a graduate topics course on Complex and Kahler manifolds for January 2011. I want to use this course as an excuse to teach the students some geometric analysis. In ...
24
votes
5
answers
4k
views
Weitzenböck Identities
I asked this question at Maths Stack Exchange, but I haven't received any replies yet (I'm not sure how long I should wait before it is acceptable to ask here, assuming there is such a period of time)....
- 19.3k
23
votes
3
answers
3k
views
Hsiung on the Complex Structure of $S^6$
In 1986 C. C. Hsiung published a paper "Nonexistence of a Complex Structure on the Six-Sphere" and in 1995 he even wrote a monograph "Almost Complex and Complex Structures" to further elaborate on his ...
- 329
16
votes
3
answers
3k
views
References for holomorphic foliations
I'm looking for an introduction to holomorphic foliations and foliations of complex manifolds.
Any little helps, but I'm particularily interested in problems of the type where we have a hermitian ...
- 4,343
10
votes
2
answers
526
views
Two smooth tangent almost complex curves in a $4$-manifold
I would like to know if following is correct.
Statement. Suppose we have a smooth (i.e., $C^\infty$) almost complex structure on $\mathbb R^4$ and $C_1, C_2$ are two $J$-holomorphic curves passing ...
- 14.3k
9
votes
1
answer
3k
views
Complex geometry text/research introduction for the analyst
To give some background, I am mainly an analyst trained in harmonic/functional and do work on geometric pde's and spectral multipliers. Of late, I am trying to learn more about (research level) ...
- 141
9
votes
1
answer
581
views
Smoothing of a Kähler orbifold metric on a complex surface
Let $S$ be a smooth complex projective surface and $D\subset S$ be a smooth complex curve. Fix an integer $m>1$ and consider $(S,D,m)$ as an orbifold with orbi-locus $D$ with stabilizer $\mathbb ...
- 14.3k
9
votes
0
answers
996
views
Complexification of smooth manifolds
Given a $C^\infty$- smooth manifold $M$, is there a natural way to complexify it?
By this I mean finding a complex manifold $N$ and a smooth function $f:M\to N$ such that $df:T_xM\otimes \mathbb{C}\to ...
- 435
8
votes
1
answer
1k
views
Calculating a curvature tensor by polarization
I'm reading some articles by Siu and Nannicini on the Weil-Petersson metric associated to families of compact Kahler-Einstein manifolds. In each article Siu and Nannicini construct a Weil-Petersson ...
- 4,343
7
votes
4
answers
2k
views
$E$ is a holomorphic vector bundle if and only if there is a Dolbeault operator $\bar{\partial}_E$
I am looking for a reference which shows that the following statements are equivalent for a complex vector bundle $E$:
$E$ is a holomorphic vector bundle.
There is a Dolbeault operator $\bar{\partial}...
- 19.3k
7
votes
2
answers
1k
views
References for the moduli space of complex structures
I am looking for references where the moduli space of complex structures on a complex manifold is well explained: in particular the infinitesimal deformations, the obstructions, the elliptic complex ...
- 2,816
7
votes
1
answer
428
views
A geometric characterization of smooth points of a complex algebraic variety
Let $X^m\subset \mathbb{C}^n$ be an irreducible $m$-dimensional complex algebraic subvariety. Let $\mathbb{C}^n$ be equipped with the standard Hermitian metric.
Fix an arbitrary point $p\in X$. Let $...
- 21.8k
7
votes
1
answer
257
views
Bimeromorphic equivalence of reduced spaces for Kähler $S^1$-actions
Let $(X,\omega)$ be a smooth Kähler manifold (not necessarily compact) with an isometric $S^1$-action with a Hamiltonian $H$. It is a well known fact that
1) The reduced spaces $X(c)=H^{-1}(c)/S^1$ ...
- 14.3k
6
votes
2
answers
706
views
Reference request: uniformization theorem
I would appreciate if someone could point me to some introductory literature/resources where I can learn about Poincaré's uniformization theorem at a basic level.
Any good powerpoint notes, short ...
- 63
6
votes
2
answers
2k
views
Line bundles over Kähler–Hodge manifolds
A Kähler–Hodge manifold $M$ can be defined as a Kähler manifold whose Kähler form $\omega$ is integral, namely $\omega\in H^{2}(M,\mathbb{Z})$. It is known then that there always exists a Hermitian ...
- 2,816
6
votes
0
answers
239
views
Complex submanifolds via Kähler reduction
Let $X$ be a Kähler manifold with an isometric $S^1$-action (which of course complexifies to a $\mathbb C^*$ action). Consider the corresponding Hamiltonian $H$ and let $X_0=H^{-1}(0)/S^1$ be the ...
- 14.3k
5
votes
1
answer
153
views
An estimate on deviation of two smooth tangent $J$-holomorphic curves
Take $\mathbb C^2$ with coordinates $(z,w)$. Suppose that $J$ is a $C^{\infty}$ almost complex structure on $\mathbb C^2$ such that the line $w=0$ is $J$-holomorphic and $J(0,0)$ is given by $(z,w)\to ...
- 14.3k
5
votes
1
answer
1k
views
On the complexification of a Riemannian manifold
Let $(M,g)$ be a Riemannian manifold and $TM$ be its tangent bundle. If we suppose $TM\otimes\mathbb{C}$ is the complexification of $TM$ then how can we define a natural metric on the complex bundle $...
- 601
5
votes
1
answer
234
views
Does $F_{A}^{0,2}=0$ for a connection $A$ on $TM$ almost complex give a complex structure?
First, some motivation. Let $X$ be a complex manifold, and $A$ a Hermitian connection on some complex vector bundle $E$ over $X$. It is known that the existence of $A$ such that the $(0,2)$-part of ...
- 1,763
5
votes
1
answer
270
views
Varying a Kahler metric in a neighborhood of a point
I would like to know if the following statement (or a more general version of it) is contained in some book or article:
Statement. Let $(U,g)$ be a complex manifold with a Kahler metric $g$ and let $...
- 14.3k
5
votes
1
answer
442
views
Chern-Weil theory for degenerated metric
If $\omega$ is a Kähler metric on a compact complex manifold $X$, the standard Chern-Weil theory says that the Chern classes $c_{i}(M)$ can be represented by forms involving the curvature of $\omega$....
- 51
4
votes
1
answer
150
views
Linearisation of complex $S^1$ actions at fixed points
Let $(U,x)$ be an open complex $n$-manifold (say an $n$-ball) with an action of $S^1$ by holomorphic transformations that fix $x$. How to prove that there is a neighbourhood $U_1\subset U$ of $x$ ...
- 14.3k
4
votes
2
answers
758
views
Riemannian metric of hyperbolic plane
I'm fishing for the origin of the idea to consider "trace scalar product" on the space of ($G$-)orthogonal projectors as means of defining a Riemannian metric on some subset of lines in a vector space....
- 8,597
4
votes
1
answer
434
views
Curvature and Symmetry on Kähler manifolds
Hi there,
Suppose $X$ is a Kähler manifold that has an analytic isometry $S$, with $S^k = \operatorname{Id}$ ($k \in \Bbb N$). In a situation like this (maybe with additional assumptions on $X$) can ...
- 1,211
4
votes
1
answer
710
views
Reference request for instantons
I've been researching instantons lately and I'd like to learn more about them but would like some help finding what to read. I have read about the ADHM equations and their noncommutative analogues. ...
- 161
4
votes
0
answers
114
views
Degeneration formula and Donaldson-Floer theory
Is there a relation between the degeneration formula of GW Invariants of Jun Li and the Donaldson-Floer theory? Is there an example / discussion anywhere of/on this relation?
- 153
4
votes
0
answers
310
views
PDE obtained while trying to construct a complex structure
Upon reading this answer to this question, the last paragraph mentions the following. "Requiring the [almost complex] structure to be integrable corresponds to a certain PDE for this map." ...
- 1,763
4
votes
0
answers
129
views
Geodesic distances and Grassmannians
Let $Gr(n, V)$ be the Grassmannian parametrising $n$-dimensional subspaces of a vector space $V$.
When $V$ is an inner product space, formulae exist for calculating the geodesic distance between ...
- 61
4
votes
0
answers
129
views
Coordinate-free B.Feix's construction of a hyperkähler metric
In the 2001's paper 'Hyperkähler metrics on cotangent bundles' B.Feix gives a construction of a hyperkähler metric on a neighbourhood of zero section in $T^*X$ where $X$ is a real analytic Kähler ...
- 2,305
4
votes
0
answers
157
views
Analytic maps $\varphi: \mathbb C^n\to \mathbb C^n$ with degenerate differentials
Let $B^n\subset \mathbb C^n$ be a unit ball with center $p$ . Let $\varphi: B^n\to \mathbb C^n$ be a complex analytic map such that $d\varphi$ has rank at most $n-1$ at $p$. I would like to know if ...
- 14.3k
4
votes
1
answer
234
views
A trivialization of an almost complex structure
Recently, I have been studying the Carleman Similiarity Principle, which is used to study the regularity and unique continuation of J-holomorphic curves.
Roughly, one takes a solution $ u $ of a ...
- 337
3
votes
1
answer
202
views
Non-positive metric on a product is a product metric
I have heard mentioned the following theorem:
If a Riemannian metric on a product of two Riemann surfaces has non-positive sectional curvature then the metric is a product metric.
And am trying to ...
- 198
3
votes
1
answer
466
views
Finding global sections of a sheaf of sets using (some kind of) sheaf cohomology?
Let $X$ be a compact manifold, say, and $G$ a Lie group, and $H$ a closed Lie subgroup such that $M \cong G/H$ is a homogeneous space. (For my purposes, $X$ and $M$ would be a smooth projective ...
- 1,763
3
votes
1
answer
524
views
Kahler-Einstein metrics on Toric manifolds are Torus-invariant?
let $(M,\omega)$ be a Kahler-Einstein toric manifold of complex dimension $m$. By toric manifold i mean a manifold that has an open dense subset $X$ biholomorphic to an algebraic torus $\mathbb{T}^{m}...
- 1,727
3
votes
1
answer
286
views
Kähler metric of constant scalar curvature with positive bisectional curvature is Kähler-Einstein
Suppose $\omega$ is a Kähler metric of constant scalar curvature with positive bisectional curvature, how to prove $\omega$ is Kähler-Einstein?
I was told that we can use the following method:
Step ...
- 79
3
votes
3
answers
908
views
Good exposition of "Calabi ansatz"
As far as I understand, Calabi ansatz is (in particular) a way to produce Kähler metrics on total spaces of line bundles (or their disk subbudles) over Kähler manifolds of the following form:
Calabi ...
- 14.3k
3
votes
1
answer
277
views
A (non-Kahler) metric on projectivised vector bundles
Given a hermitian holomorphic vector bundle (E, h) on a complex manifold-with-a-metric (X,g), then consider the following (natural) construction of a metric on the total space of $\mathbb{P}(E)$ : ...
- 3,383
3
votes
0
answers
336
views
Understanding Calabi's conjecture proof: What is it meant by the logarithm of a differential form?
I'm reading several books and articles concerning Yau's proof of the Calabi conjecture. I want to have a deep understading of how and why such proof actually works, but most articles are aimed at ...
- 153
3
votes
0
answers
637
views
English reference for Fischer-Grauert theorem and its generalization by Schuster
From this MSE question and its answer, and from this MO question I have learned of the following remarkable theorem of Wolfgang Fischer and Hans Grauert.
Theorem. A proper holomorphic submersion with ...
- 10.5k
3
votes
0
answers
447
views
Complex structures on Riemann surfaces
This is cross posted from math.SE: https://math.stackexchange.com/q/876432/9
Let $M$ be a Riemann surface and $[\alpha] \in H^{0,1}(M; T^{1,0} M) \simeq (H^0(M;K^2))^*$. Considering $\alpha$ as a ...
- 3,244
2
votes
2
answers
427
views
Analytic Lagrangian Submanifolds
Hallo,
I am looking for a preprint "Analytic Lagrangian Submanifolds" by Guillemin, Sternberg. I googled it but without any success. Does any one know how I could get this preprint. Or are there ...
- 339
2
votes
1
answer
290
views
On the stack of semistable curves
This is a question related to
Semistable curves of genus $g\geq 2$ form an Artin algebraic stack in the etale topology?
Let $\mathcal C\rightarrow \mathcal M^{ss}_g$ be the universal curve over the ...
- 494
2
votes
0
answers
152
views
Period map on non-Kähler manifold
Is there a theory of period map on non-Kähler manifolds that has Hodge decomposition? Any reference is helpful. Thank you.
- 263
2
votes
0
answers
172
views
Construction of Kahler Einstein Metric of Poincare Type
I am reading Kobayashi's Kahler-Einstein metric on an open algebraic manifold. In this paper he constructs a Kahler-Einstein of Poincare type on an open manifold X' = X\D, where X is projective and D ...
- 31
2
votes
0
answers
75
views
Notation and geometry facts in a paper on the Diederich-Fornæss index
I am reading this article by Bingyuan Liu on the Diederich-Fornæss index.
I am having some problems with both the notation and the geometrical side.
1)I don't know what kind of objects $N,L$ are ...
- 779
2
votes
0
answers
59
views
Group of real analytic isometries of $g$-fold product of the Poincare upper half plane
Let $\mathfrak{h}^g$ be the cartesian product of $g$ copies of the Poincare upper half plane. We endow $\mathfrak{h}^g$ with the usual Poincare metric given in local coordinates by $ds^2=\sum_{i=1}^g ...
- 7,609
2
votes
0
answers
282
views
Generalizing a result of Paul Andi Nagy
I am trying to generalizate a result of Paul Andi Nagy which says that in a almost Kähler manifold with parallel torsion we have $\langle \rho,\Phi-\Psi\rangle $ is a nonnegative number; in fact
$$
4\...
- 21
1
vote
2
answers
1k
views
Reference on Complex Geometry
For the preparation of a complex geometry lecture I am looking for a good literature. I already have standard literature like Huybrechts "Complex Geometry. An Introduction" and I am also using it. But ...
- 39
1
vote
1
answer
1k
views
Hermitian form, fundamental $2$-form of Kahler structure on $\mathbb{C}^n$
I've come across the following (it is an excerpt of Stolzenberg's lecture notes 19):
Wirtinger's Inequality.
Let $L$ be a complex linear space and let $M$ be a real
even-dimensional subspace....
- 103