All Questions
102 questions
1
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39
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Currents with logarithmic poles compared with those with no poles
I am learning Deligne homology via U. Jannsen, "Deligne homology, Hodge-$\mathscr{D}$-conjecture, and motives." There, the currents with logarithmic poles are given in Definition 1.4 by
$$
'\...
- 161
0
votes
0
answers
76
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Constant mean curvature hypersurface
Assume that $f:\mathbb{B}^2\to \mathbb{C}$ is a holomorphic function defined in the unit ball in $\mathbb{C}^2$. Let $u(z)=|f(z)|(1-|z|^2)$ and consider $\Sigma =\{z: u(z)=c\}$. It seems to me that if ...
- 37
5
votes
1
answer
226
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Nontrivial extension of the action of complex hyperbolic group $H$ on $\mathbb{C}$
Inspired by this question about conjugation of reql analytic maps to a holomorphic function and with a group action view point we ask the following question.
The complex Lie group $H=\...
- 356
0
votes
0
answers
121
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Is any singularity a subgerm of $(\mathbb{C}^n, 0)$?
I am studying singularity theory. I have often come across, in the literature, the sentence which says "let $(X,0) \subset (\mathbb{C}^n,0)$ be a singularity". Here a singularity is a ...
- 369
4
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1
answer
209
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Is there a complete Kahler metric on a bounded domain？
Let $\Omega\subset\mathbb{C}^n$ be a bounded domain.
By the theorems of Cheng-Yau and Mok-Yau, $\Omega$ admits a complete Kahler-Einstein metric if and only if $\Omega$ is a pseudoconvex domain.
My ...
- 51
1
vote
0
answers
156
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Top cohomology of the canonical class of a compact non-Kähler manifold
Let $X$ be a complex compact manifold of complex dimension $n$. Let $K_X$ denote its canonical class. Is it true that the cohomology group
$$H^n(X,K_X)$$
is one dimensional?
Remark. If $X$ is Kähler ...
- 21.8k
5
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0
answers
225
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Energy bounds (or the lack thereof) for a functional between almost Hermitian manifolds
Suppose that $(M,g,J_M)$ and $(N,h,J_N)$ are two almost Hermitian manifolds. For a differentiable function $f:M\to N$ define its pseudoholomorphic energy to be
$E_+(f)=\frac{1}{4}\int_M |Df+J_N Df J_M|...
- 646
3
votes
0
answers
85
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Can a punctured ball $(B\setminus\{0\})\subset\mathbb{C}^n$ be foliated by complete leaves?
Recently Antonio Alarcón proved that in the case of the unit ball $B\subset\mathbb{C}^n$ for $n\geq 2$ every smooth closed complex submanifold of dimension $q\leq n$, $V\subset\mathbb{C}^n$ defines a ...
- 166
1
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1
answer
224
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Bott-Chern cohomology for singular complex spaces
I'm reading the book 'An Introduction to the Kahler-Ricci Flow' (Lecture Notes in Mathematics 2086). They discuss Bott-Chern cohomology on complex spaces:
Let $X$ be a complex space(i.e. analytic ...
- 361
2
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0
answers
108
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The dual of the Lefschetz operator under a perturbation
Let $(X, \omega)$ be a compact Kähler, or more generally, Hermitian manifold. Let $L_{\omega} : \Omega^k(X) \to \Omega^{k+2}(X)$ denote the Lefschetz operator given by $$L_{\omega}(\alpha) : = \omega \...
- 137
5
votes
1
answer
457
views
How to define a current on a complex analytic space
I'm reading a paper in which the author use $(p,q)$-forms and currents on a complex analytic space.
My question is how to define $(p,q)$-current on complex space? Does it have similar properties like ...
- 361
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 \...
2
votes
0
answers
231
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Does every non-compact hyperbolic manifold admit compact complex submanifolds?
Let $(X,\omega)$ be a complete Kähler manifold with a metric of negative holomorphic sectional curvature. Does $X$ admit a proper, positive-dimensional, compact complex submanifold?
In general, it is ...
- 1,379
1
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0
answers
70
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Classification of principal monodromy elements
Let $(X,0)$ be a germ of normal analytic space with an isolated singularity at $0$, and let $Y:=X\backslash\{0\}$. Suppose $Y$ has a complex-hyperbolic metric which is complete at $0$. Burns-Mazzeo ...
- 1,121
4
votes
1
answer
622
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Is there a compact complex manifold with $b_1(X)=b_2(X)=b_3(X)=b_4(X)=0$?
As the title suggests, I have the following question:
Is there a compact complex manifold with $b_1(X)=b_2(X)=b_3(X)=b_4(X)=0$?
Clarification:
Denote by $b_k$ the $k$th Betti number of a compact ...
- 1,379
2
votes
0
answers
358
views
Triangulating Riemann surfaces by using non-constant meromorphic functions
Let $X$ be a connected Riemann surface, i.e. $X$ is a one dimensional connected complex manifold (Hausdorff and second-countable as a topological space). The following is a classical result:
Theorem (...
- 395
6
votes
0
answers
144
views
What does it mean for the torsion to blow up?
Consider the following theorem which is the main result of the Hermitian Curvature Flow paper by Jeffrey Streets and Gang Tian:
Theorem 1.1. Let $(M^{2n}, g_0, J)$ be a complex manifold with Hermitian ...
- 651
1
vote
0
answers
55
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What are we to deduce from a structure theorem of this type concerning totally geodesic maps?
I apologise in advance for the vague nature of the question, but some insight would be greatly appreciated.
I'm reading a paper of Lei Ni concerning structure theorems for Kähler manifolds. Here is an ...
- 651
5
votes
1
answer
238
views
Volume of singular Kahler metric
Let $X$ be a compact complex manifold of complex dimension $n$ and let $\omega$ be a smooth Kahler form on it. Let $Y \subset X$ be a complex (possibly singular) hypersurface and let $u: X \setminus Y ...
- 135
10
votes
1
answer
662
views
Hartogs' theorem for real-analytic subvarieties
One version of Hartogs' extension theorem is the following (see, e.g. [1], Theorem 5B, p. 50).
Theorem. Let $U \subset \mathbb{C}^n$ be open and let $X \subset U$ be a complex-analytic subvariety of ...
- 1,383
3
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0
answers
191
views
Holomorphic sectional curvature and Kobayashi hyperbolicity
Let $(M,g)$ be a compact Hermitian manifold. Let $\text{HSC}(g)$ denote the holomorphic sectional curvature of $g$. The implication $$\text{HSC}(g) < 0 \implies M \ \text{is Kobayashi hyperbolic}$$ ...
- 651
4
votes
1
answer
847
views
Does $P_xP_y+Q_xQ_y=0 \implies$ "non-existence of limit cycle" for $P\partial_x+Q\partial_y$"? (Complex dilatation and limit cycle theory)
Let $X=P\partial_x+Q\partial_y$ be a vector field on the plane $\mathbb{R}^2$. Assume that we have :$$P_xP_y+Q_xQ_y=0$$ Does this imply that the vector field $X$ is a divergence-free vector field ...
- 356
3
votes
0
answers
55
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Infinitely many deformation equivalent Hodge diamonds II
Let $S$ be a connected smooth complex-analytic space. Let $\phi:X\to S$ be a proper holomorphic submersion with connected fibers. Can the fibers of $\phi$ have infinitely many distinct Hodge diamonds?
...
user164740
2
votes
1
answer
652
views
Complex manifold defined over $\mathbb{R}$
Let $M$ be a connected closed complex manifold with an antiholomorphic involution.
Must there be an atlas and a choice of a reference point in each chart such that the transition functions are ratios ...
user164740
2
votes
1
answer
128
views
Infinitely many deformation equivalent Hodge diamonds
Let $S$ be a connected open complex manifold. Let $\phi:X\to S$ be a proper holomorphic submersion with connected fibers. Can the fibers of $\phi$ have infinitely many distinct Hodge diamonds?
An ...
user164740
7
votes
1
answer
634
views
Hodge diamonds of complex threefolds
There is no closed complex curve or surface with $h^{1, 0}-h^{0, 1}=1$.
Now consider threefolds. Can this condition be satisfied?
Is Serre duality in fact the only restriction on the Hodge diamond?
user164740
6
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0
answers
228
views
All complex surfaces embed into a common complex manifold
Is there a closed complex manifold into which every closed complex surface embeds?
user164740
5
votes
0
answers
188
views
Proof of Tian's constant
Basically Tian's invariant relies fundamentally on a lemma of Homander which says that $e^{- \phi}$ is integrable for $\phi$ plurisubharmonic on a ball of radius $1$ under some extra assumption. I am ...
- 51
4
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0
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294
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Holomorphic covers pulling back the volume form to any integer multiple
Let $M$ be a closed connected complex manifold with $\mathrm{dim}\:M=n$. Can there exist holomorphic covering maps $\phi_k:M\to M$ for all integers $k\geq 1$ such that $\phi_k^*:H^n(M, \mathbb{Z})\to ...
user145520
8
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0
answers
315
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Singularities of a morphism from a smooth projective variety to an abelian variety
Let $f: X\to A$ be a (flat) morphism from a smooth complex projective variety $X$ to an abelian variety $A$. Consider the following natural diagram:
$$T^*X\overset{df}{\longleftarrow}X\times H^0(A, \...
- 1,081
5
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0
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245
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Dimension of highest discriminants of a morphism
Let $f: X\to Y$ be a flat morphism between smooth complex affine varieties. Let $Z$ be the closed set of most singular points of $f$ (in the sense: $p$ is a most singular point of $f$ if the tangent ...
- 1,081
6
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1
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219
views
Restriction of holomorphic functions on $G$-invariant subspace
Let $X$ be a complex manifold with a holomorphic action of a complex reductive group $G$. Let $Y \subset X$ be a $G$-invariant reduced complex analytic subspace. Is the restriction
$$
\mathcal{O}_X^G \...
- 1,383
2
votes
0
answers
158
views
Pull back of a Bounded form
Let $(X, \omega) $ be a complex manifold and let $\alpha $ be a $p$-form $\omega$-bounded on $X$.
Let $f:Y\to X$ be a holomorphic function.
Is $f^*\alpha$ $f^*(\omega)$-bounded on $Y$ ?
- 21
4
votes
1
answer
1k
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Norm of a differential form [closed]
How can we explicitly calculate the norm of a differential form?
For example let $(X, \omega) $ be a complex manifold such that locally
$$
\omega(z) =i\sum_{k,j} h_{k, j} (z) dz_k\wedge d\overline {...
- 41
3
votes
1
answer
304
views
Relationship between volume and area
Let $\mu(z) dV_n$ be a measure in $\mathbb{C} ^n$.
Let $B_n(r) := \{z \mid \|z\| < r\}$ be the ball of radius $r$ in $\mathbb C^n$, and $\partial B_n(r) $ be the corresponding sphere.
In $\mathbb{C}...
- 33
2
votes
0
answers
86
views
Estimates for tensors using local coordinates
Suppose that we have a Kähler manifold $(M, \omega_0)$ and another $(1,1)$ form $\eta$ on $M$. Let $\varphi$ be a smooth function such that $\omega_{\varphi} = \omega_0 + i \partial \bar \partial \...
- 121
2
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0
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117
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Equivariant resolution of singularities with equivariant centres
From what I understand, given a complex projective variety X inside a compact complex manifold Y, according to Hironaka, there is a sequence of $r$ blowups $Y_i$ of Y along complex submanifolds (...
- 3,383
3
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0
answers
259
views
Stokes's Theorem with singularities on projective line
Let $X$ be a complex manifold and $\omega\in \Omega(X\times \mathbb{P}^1)$ a form. I met the following identity:
$$\int_{\mathbb{P}^1}(\partial_z\bar{\partial}_z\omega) \log|z|^2=\int_{\mathbb{P}^1}\...
- 789
1
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0
answers
81
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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
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
9
votes
1
answer
321
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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
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0
answers
119
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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
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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
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