All Questions
Tagged with dg.differential-geometry cv.complex-variables
191 questions
3
votes
1
answer
304
views
Relationship between volume and area
Let $\mu(z) dV_n$ be a measure in $\mathbb{C} ^n$.
Let $B_n(r) := \{z \mid \|z\| < r\}$ be the ball of radius $r$ in $\mathbb C^n$, and $\partial B_n(r) $ be the corresponding sphere.
In $\mathbb{C}...
- 33
2
votes
0
answers
86
views
Estimates for tensors using local coordinates
Suppose that we have a Kähler manifold $(M, \omega_0)$ and another $(1,1)$ form $\eta$ on $M$. Let $\varphi$ be a smooth function such that $\omega_{\varphi} = \omega_0 + i \partial \bar \partial \...
- 121
2
votes
0
answers
117
views
Equivariant resolution of singularities with equivariant centres
From what I understand, given a complex projective variety X inside a compact complex manifold Y, according to Hironaka, there is a sequence of $r$ blowups $Y_i$ of Y along complex submanifolds (...
- 3,383
3
votes
0
answers
259
views
Stokes's Theorem with singularities on projective line
Let $X$ be a complex manifold and $\omega\in \Omega(X\times \mathbb{P}^1)$ a form. I met the following identity:
$$\int_{\mathbb{P}^1}(\partial_z\bar{\partial}_z\omega) \log|z|^2=\int_{\mathbb{P}^1}\...
- 789
1
vote
0
answers
81
views
zero extension of positive currents are always positive
In Demailly's Complex Analytic and Differential Geometry page 139:
He said the trivial (zero) extension of the positive current $T$ (on $X\setminus E$), which denoted by $\tilde T$ is always positive ...
- 295
6
votes
1
answer
261
views
The state of art of the singular Levi problem -- and hyperkähler quotients
One of the versions of the classical Levi problem asks the following:
Let $X$ be a complex manifold. Is it true that $X$ is Stein iff
$X$ admits a smooth exhaustion strictly plurisubharmonic ...
- 2,305
2
votes
1
answer
164
views
Conformal isomorphism uniquely determined by boundary identification?
Let $\Gamma$ be a smooth Jordan arc, and let $\Phi \colon \hat{\mathbb C} \backslash \overline{\mathbb D} \to \hat{\mathbb C} \backslash \Gamma$ be a conformal isomorphism that fixes the point at $\...
- 239
5
votes
1
answer
236
views
Which plane curves can be harmonically parametrized?
In this question, a “(closed oriented plane) curve” $\Gamma$ will mean a continuous map $f \colon \mathbb{U} \to \mathbb{C}$ where $\mathbb{U} := \{z\in\mathbb{C} : |z|=1\}$ is the unit circle, modulo ...
- 32.5k
14
votes
1
answer
395
views
Regularity of conformal maps
In order to define what it means for a map $f \colon \Omega \subseteq \mathbb R^n \to \mathbb R^n$ to be conformal, it is sufficient to require that $f$ is everywhere differentiable. Does conformality ...
- 1,347
1
vote
0
answers
153
views
A linear map on $\chi^{\infty}(\mathbb{R}^2)$ arising from the Cauchy integral formula
The space of smooth vector fields on $\mathbb{R}^2$ and open unit disc $\mathbb{D}$ are denoted by $\chi^{\infty} (\mathbb{R}^2)$ and $\chi^{\infty}(\mathbb{D})$, respectively. A vector field on $\...
- 356
23
votes
2
answers
1k
views
Theta functions on an elliptic curve and Serre duality
Given an elliptic curve $E$ (over $\mathbb{C}$) and line bundle $L$, one can identify $H^0(E,L)$ with a particular space of theta functions.
Serre duality gives a perfect pairing between $H^0(E,L)$ ...
- 411
4
votes
2
answers
559
views
A question about complex Laplacian on compact Hermitian manifolds
Let $(M,g)$ be a complex $n$-dim compact connected Hermitian manifold, and $\Delta_c(\cdot):=g^{i\bar{j}}\partial_i\partial_{\bar{j}}(\cdot)$ the complex Laplacian acting on smooth functions on $M$. ...
- 593
9
votes
1
answer
321
views
Notational question about quadratic differentials in Strebel's book "Quadratic differentials"
In Kurt Strebel's book "Quadratic Differentials", in Chapter 2, $\S4$, he begins by saying:
"Every analytic function $\varphi$ is a domain $G$ of the $z$-plane defines, in a natural way, a field of ...
- 7,527
2
votes
0
answers
119
views
Covariant derivative of the Monge-Ampere equation on Kähler manifolds
I am reading D. Joyce book "Compact manifolds with special holonomy" and I have some problems of understanding some computation on page 111, the first line in the proof of Proposition 5.4.6. More ...
- 21
0
votes
1
answer
101
views
Compatible solution of PDE
Let $c=c(z, \bar z)$ be a complex function satisfying $\partial_{z} \bar c=\partial_{\bar z} c$. It follows that there exists a real function $f$ such that $\partial_{\bar z} f=-c$. Would it be ...
- 99
5
votes
1
answer
613
views
On limits of manifolds
This question should be fairly elementary. I’d just like to check I’m not missing anything.
Let $\{M_n\}_{n\ge 0}$ be an inverse system of smooth manifolds with transition maps $f_{t,s} : M_t\to M_s$,...
user131191
3
votes
0
answers
148
views
Analytic Aspects of Rational Maps
I would like some help finding references for a analytic treatment of rational maps between compact complex manifolds (that is holomorphic maps defined away from a codimension at least 2 subvariety). ...
- 121
3
votes
0
answers
135
views
Asymptotic Expansion of Seiberg-Witten Differential?
Nekrasov & Okounkov proved (https://arxiv.org/pdf/hep-th/0306238.pdf) that the Seiberg-Witten prepotential can be given by
\begin{equation}
\mathcal{F}(\mathbf{a},\Lambda) = \lim_{\hbar\rightarrow ...
- 425
6
votes
0
answers
163
views
Reference request: normal form of k-differentials and flat surfaces at a puncture
Let $f(z)$ be a holomorphic function defined on a punctured neighborhood of $z=0$ with non-essential singularity of degree $d$ at $0$ (namely, $f(z)=z^dh(z)$, where $h(z)$ is a holomorphic function ...
- 1,804
3
votes
0
answers
98
views
Euler characteristic of an exhaustion of compacts of a surface
Let $X$ be an open (connected) Riemann surface of finite Euler characteristic. And $K_1 \subset \cdots K_n \subset$ be an sequence of closures of bounded open subsets with smooth boundary of $X.$
...
- 750
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
17
votes
2
answers
2k
views
Why only $\bar\partial$ but not $\partial$ in Dolbeault cohomology
While I learn about $\partial$ and $\bar{\partial}$ operators, I had some questions about the reason why people prefer $\bar\partial$ over $\partial$. Specifically,
When defining Dolbeault ...
- 419
7
votes
1
answer
535
views
Diffeomorphisms on a real manifold whose derivative are holomorphic maps on the tangent bundle
Edit: According to the answers to the linked MSE question and the comment of Holonomia, I understand that the answer to the second question is that " Every tangent bundles is a complex ...
- 356
0
votes
1
answer
703
views
flow of holomorphic vector field [closed]
Let $(M,J)$ be a complex manifold, where $J$ is the integrable complex structure. Let $X$ be a holomorphic vector field on $M$ and let $\varphi_{t} : M\rightarrow M $ be its flow. Question: Is $\...
- 21
5
votes
1
answer
395
views
Holomorphic Sard's theorem 2
My previous question on this topic had a negative answer, but Tom Goodwillie in the comments suggested a statement, which may be true, and even a strategy of how to prove it. I haven't been able to ...
- 5,529
3
votes
1
answer
177
views
Real solution of a complex equation with complex solution
Assume that $(M, [\lambda, \mu])$ defines an embeddable 3 dimensional CR structure where $\lambda$ is a real form and $\mu$ is a complex 1-form.
Because $M$ is embeddable, $\mu=dz$ for some ...
- 99
1
vote
0
answers
307
views
Fefferman metric and Einstein metric
From Lee's paper The Fefferman Metric and Pseudo hermitian Invariants, corresponding to any 3 dimensional strictly pseudo convex CR structure, there is a conformal class of Lorentzian metrics which ...
- 99
9
votes
1
answer
662
views
Holomorphic Sard's theorem?
I have originally posted this question on math.SE, but it received little attention, so I repost it here.
Let $U\subset \mathbb{C}^{n}$ and $V\subset \mathbb{C}^{m}$ be open and connected. Let $\Phi:...
- 5,529
3
votes
0
answers
84
views
Discrete set of critical points of a holomorphic map
I have originally posted this question on math.SE, but it received no attention, so I repost it here.
Let $U$ be an open domain in $\mathbb{C}^{n}$. Let $m\ge n$ and let $F:U\to C^{m}$ be a ...
- 5,529
13
votes
3
answers
1k
views
Do contact and CR structures have corresponding $G$-structures?
For an $n$-dimensional manifold $M$, almost complex and almost symplectic structures on $M$ correspond to reductions on the structure group of the tangent bundle, introducing a $\operatorname{GL}(n/2,\...
- 131
1
vote
1
answer
94
views
a question about complex Hessians on complex tori
Suppose we have a real-valued smooth function on a complex torus:
$$f: \mathbb{C}^n/(\mathbb{Z}+\sqrt{-1}\mathbb{Z})^n\longrightarrow\mathbb{R},$$
i.e., this $f$ is a real-valued smooth function on $\...
- 593
5
votes
1
answer
243
views
Deformation of the Plücker coordinates
Let $M_{2,4}(\mathbb{R})$ be the set of real $2\times4$-matrices of rank $2$. For any $A\in M_{2,4}(\mathbb{R})$ and $1\leq i<j\leq 4$, let $p_{ij}$ be the corresponding $2\times 2$-minors of $A$. ...
- 93
3
votes
0
answers
165
views
Is a non vanishing holomorphic vector field necessarily a geodesible vector field?
Motivated by the "The obvious Fact" part of this answer,, we ask the following question:
First we recall a definition, which is used in the above link:
Definition: A non vanishing vector ...
- 356
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
5
votes
0
answers
104
views
On the embedding of manifolds into infinite-dimensional spaces
Let $X$ be a (connected, finitely dimensional) topological/smooth/complex manifold and let $i$ be a weakly continuous/continuous/smooth/holomorphic map from $X$ into the dual $F^{*}$ of a real or ...
- 5,529
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
1
vote
0
answers
85
views
What does the space of holomorphic maps look like inside the space of equivariant maps
Consider a map from a closed topological surface $S$ into, for example, a fixed compact complex hyperbolic manifold.
For each point in the Teichmuller space of $S$ this map is homotopic to a unique ...
- 257
4
votes
1
answer
107
views
When do quotients of bounded domains contain closed Riemann surfaces?
Let $D$ be a bounded domain in $\mathbb{C}^n$, and let $\gamma$ be a biholomorphism of $D$ such that for all $\epsilon>0$ there is a point $z\in D$ such that the Kobayashi distance from $z$ to $\...
- 257
5
votes
1
answer
752
views
Gaussian integral over a ball
How to compute the following integral?
$$\int_{\|x\|^2\leq R} \exp(-x^\ast G x+2\mathcal{Re}(x^\ast a)) \,dx,$$
where $x$ is an $M \times 1$ vector ($M\gg 1$), $G$ is a positive definite matrix, and $...
- 129
3
votes
1
answer
303
views
Intersection multiplicity in the non-algebraic case
I know the definition of intersection multiplicity in algebraic geometry. However, I think it is possible to define it for some sort of non-algebraic functions such as $y=\sin x$.
How to define ...
user80247
6
votes
0
answers
147
views
What is the meaning of complex values/multiplicities in dimension spectrum?
If we have a manifold $M$ (say smooth, closed) it can be equipped with the Laplace operator $\Delta$. One can consider the function $\textrm{trace}(\Delta^{-s})$ where $s$ is complex parameter and $\...
- 9,330
6
votes
2
answers
755
views
Plurisubharmonic function and complete Kähler metric on certain Kähler manifold
Given a compact Kähler manifold $M$, let $D$ be an effective divisor on $M$.
Is $M\setminus D$ pseudoconvex? That is, can we find a smooth plurisubharmonic function that exhausts $M\setminus D$ ?
...
user40184
13
votes
5
answers
3k
views
A geometric proof of the Gauss-Lucas theorem
Motivated by a geometric proof of the Fundamental Theorem of Algebra we ask:
Is there a geometric proof for the Gauss-Lucas theorem? Since we are working on a half plane, can one imagine a possible ...
- 356
1
vote
1
answer
475
views
Gravitational instantons metric (change variables)
I'm trying to understand the paper by Hitchin called: ''Polygons and gravitons". I'm stuck at page 471. At this point, he does some computations and obtains a metric:
$$
\gamma dz d\bar{z}+\gamma^{-...
- 61
6
votes
0
answers
286
views
Is the space of holomorphic maps a manifold
To be more specific:
Let $Q\subset\mathbb{C}$ be a Lipschitz bounded domain, and $V$ is a compact complex manifold without boundary. Consider the set of holomorphic maps $f:Q\rightarrow V$, and $f\in ...
- 151
7
votes
0
answers
202
views
Biholomorphic neighborhoods of the boundary of Stein domains
Let $(X_1,J_1)$ and $(X_2,J_2)$ be Stein domains with the same contact boundary $(Y,\xi)$. Under what conditions does there exist a biholomorphism between a neighborhood of their respective boundaries ...
- 773
7
votes
2
answers
813
views
Criterion for deciding the conformal class of a metric on a complete surface
For orientable closed Riemannian surfaces $(S,g)$, there is a constant curvature metric $\overline{g}$ on $S$ that is conformal to $g$ in the sense that $\overline{g} = e^ug$ for some smooth function $...
- 173
6
votes
1
answer
409
views
Can the potential of a complete Kahler metric be bounded?
Let $X$ be a complex manifold and $\omega$ a Kahler form on $X$. A smooth function $\rho$ is called a potential of $\omega$ if $i\partial\bar\partial\rho=\omega$. By intuition, it seems that $\rho$ ...
- 285
1
vote
1
answer
190
views
What is the Fano index for Hermitian symmetric spaces of compact type?
As we know Hermitian Symmetric spaces of compact type are all Fano picard number one, we can talk about his Fano index. Suppose $X$ is one of those Hermitian symmetric spaces, $L$ is the generator of ...
- 1,121
5
votes
1
answer
342
views
harmonic extension of a curve by different parametrization
Let us consider a curve $\gamma :S^1 \rightarrow \mathbb{R}^3$ (or even a planar convex one if it simplifies). Then I look to the harmonic extension to the disc $h:\mathbb{D}\rightarrow \mathbb{R}^3$ (...
- 914