Questions tagged [complex-geometry]
Complex geometry is the study of complex manifolds and complex algebraic varieties, and, by extension, of almost complex structures. It is a part of both differential geometry and algebraic geometry.
2,239
questions
2
votes
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18 views
Complex Monge-Ampere equation with degenerate right hand side
Given a Kahler manifold $(X, \omega_0)$, and a smooth function $f$, suppose that I have a smooth solution to the following complex Monge-Ampere equation:
$(\omega_0 +i \partial \bar \partial \varphi)^...
3
votes
0answers
75 views
Analytic and algebraic definitions of intersection multiplicity of two complex algebraic curves coincide
There are two definitions of intersection multiplicity of two complex algebraic curves. One is given in An introduction to algebraic curves by Griffiths. Let $t \mapsto (t^k, y(t^k) )$ be the local ...
10
votes
1answer
165 views
Examples of Stokes data
I'm trying to learn Stokes data but can't find an example to get my teeth into it.
Background. It's well-known that on a complex manifold $X$, there is the Riemann Hilbert equivalence
$$\text{regular ...
4
votes
1answer
82 views
minimizing weighted length of closed curves
Let $\mathcal{A}$ be the family of closed smooth curves in the right half of the complex plane $\mathbb{C}$ such that any curve in the family must enlose the point $z=1$ and tangent to the $y$-axis at ...
2
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0answers
73 views
Equivariant resolution of singularities with equivariant centres
From what I understand, given a complex projective variety X inside a compact complex manifold Y, according to Hironaka, there is a sequence of $r$ blowups $Y_i$ of Y along complex submanifolds (...
3
votes
1answer
98 views
Stokes's Theorem with singularities on projective line
Let $X$ be a complex manifold and $\omega\in \Omega(X\times \mathbb{P}^1)$ a form. I met the following identity:
$$\int_{\mathbb{P}^1}(\partial_z\bar{\partial}_z\omega) \log|z|^2=\int_{\mathbb{P}^1}\...
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51 views
where is the condition class $C^{1}$ used in the definition of current of integration? [closed]
Here is the usual definition of current of integration.
Let $Z \subset M$ be a closed oriented submanifold of $M$ of dimension $p$ and class $C^{1} ; Z$ may have a boundary $\partial Z$. The ...
2
votes
1answer
138 views
Curves on a Kahler manifold
Let $(X,\omega )$ be a compact Kahler manifold. For any $d>0$ are there only finitely many families of curves $C_i$ such that $C_i\cdot \omega <d$? (More precisely, if $C$ is any curve such that ...
10
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0answers
149 views
Moments of Plücker coordinates on complex Grassmannian and log-concavity
Consider the Grassmannian $Gr(k,N)\simeq U(N)/(U(k)\times U(N-k))$ which parametrizes $k$-dimensional subspaces of $\mathbb{C}^N$. Let us put on it the $U(N)$-invariant probability measure. Let $\...
2
votes
1answer
88 views
Simple example of non-integrable holomorphic connection
Let $X$ be a complex manifold with complex dimension $d$ and structure sheaf $\mathcal{O}_X$. Let $E$ be a locally free sheaf on $X$. A $holomorphic$ connection on $E$ is a morphism of sheaves
$$\...
3
votes
1answer
73 views
Tangent bundle with Sasaki metric is Kähler iff $M$ is locally flat
I'm having a hard time proving the following:
If $M$ is an $n$-dimensional indefinite Riemannian manifold whose metric $g$ has index $s$, then the metric of Sasaki $g^{D}$ is an indefinite metric ...
4
votes
1answer
189 views
Critical points of polarized endomorphisms of algebraic varieties
My main question is the following:
Let $f: \mathbb{CP}^n \to \mathbb{CP}^n$ be a holomorphic endomorphism of degree $d \ge 2$ of $\mathbb{CP}^n$ .
1. Let $X \subset \mathbb{CP}^n$ be an ...
-1
votes
0answers
66 views
About the embedded resolution
Let $M$ be a Kähler manifold and $V$ a singular hypersurface of $M$. Assume we obtain an embedded resolution $M^{\prime}$ of $V$ in $M$ by finitely many blow-ups along smooth centers.
My question ...
7
votes
2answers
254 views
Vector field with constant divergence around embedded submanifold
Let $M$ be a smooth $n$-dimensional manifold and $N\subset M$ be a closed embedded submanifold of codimension at least $2$. Furthermore, let $\mu$ be a volume form on $M$.
Question: Does there ...
4
votes
1answer
207 views
Fun examples relating to Hopf surfaces
A Hopf surface is a compact complex surface whose universal cover is complex analytically isomorphic to $\mathbb{C}^2 \setminus \{ 0 \}$. I would like to know whether anyone has any of the following ...
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0answers
53 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 ...
2
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0answers
69 views
Is there Hodge isomorphism between Dolbeault and Harmonic on noncompact manifold
As is well known , Hodge theorem tells us
Let $(X, g)$ be a compact hermitian manifold. Then the canoni.
cal projection $\mathcal{H}_{\bar{\partial}}^{p, q}(X, g) \rightarrow H^{p, q}(X)$ is an ...
1
vote
0answers
78 views
Variation of the (Chern) curvature with respect to the metric
Let $E\rightarrow X$ be a holomorphic vector bundle, for any Hermitian metric $h$ on $E$ we denote by $F_h$ the curvature of the Chern connection associated to $h$. Fix a metric $h_0$ and consider a ...
5
votes
0answers
173 views
Derivative of the Bott-Chern forms
The Bott-Chern forms are constructed formally in Bismut's "Analytic Torsion and Holomorphic Determinant Bundle I" (page 74). This construction can be found as well in "Lectures on Arakelov Geometry" ...
1
vote
1answer
51 views
weak convergence of positive currents vs. $L^1$ convergence of normalized potentials
I have run into the following statement in the literature (e.g. here, p.5, after Theorem 1.1): that weak convergence of positive $(1,1)$-currents on a complex manifold is equivalent to $L^1$ (I ...
3
votes
1answer
152 views
Closed analytic subvariety of $\mathbb C^n$ not defined by global holomorphic functions
Here is the motivation of this question: $\mathbb C^n$ is already "local" in algebraic category. In other words, algebraic subvarieties of $\mathbb C^n$ are affine, so they are common zero locus of ...
2
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0answers
129 views
Relative Chow's theorem
Suppose you have a smooth quasiprojective complex algebraic variety that is not compact. Suppose you have a complex analytic fiber bundle over the algebraic variety where each fiber is a smooth ...
2
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0answers
61 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 ...
2
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0answers
49 views
About the current of finite mass
In Demailly's e-book Complex analytic and differential geometry,
chap3-(1.14) Proposition is stated as follows:
Every positive current $T=i^{(n-p)^{2}} \sum T_{I, J} d z_{I} \wedge d \bar{z}_{J}$ ...
0
votes
1answer
111 views
A differential geometric proof of the Riemann--Roch theorem for lines [duplicate]
I am looking for a differential geometric version of the proof of the Riemann--Roch theorem
for Riemann surfaces, that is, $1$-dimensional compact complex manifolds. The proofs one usually finds are ...
1
vote
1answer
95 views
Holonomy groups of Hermitian, and hyper-Hermitian, manifolds
An $n$-dimensional complex manifold $M$, endowed with an Hermitian metric $g$, is Kähler if and only if the holonomy group of $g$ is contained in $U(n)$. If $g$ is Hermitian but not Kähler do we ...
0
votes
0answers
25 views
Linear system corresponding to a holomorphic embedding from compact Riemann surface to projective space is complete
Le $X$ be a compact Riemann surface and $\phi$ be a holomorphic embedding of $X$ into projective space $\mathbb{C}\mathrm{P}^n$ which is induced by $(f_0,\dots , f_n)$. Then there is a linear system ...
2
votes
1answer
134 views
$S^3$ as a Sasakian Manifold
Reading about Sasakian manifolds one come across two slogans:
A) "A Sasakian manifold is an odd-dimensional analogue of a Kahler manifold."
B) "A Sasakian manifold sits between two Kahler manifolds -...
2
votes
0answers
149 views
cohomology classes of complex submanifolds
I was wondering if there were restrictions in what the cohomology classes corresponding to complex submanifolds of a complex manifold could be.
For example, say $T^4$ is regarded as a complex ...
7
votes
1answer
214 views
A relative version of Ehresmann's theorem
Edited: Phil Tosteson suggested Thom's first isotopy lemma, but it does not seem to be in the direction that I'm trying to generalize. Let me reformulate my question again.
Let $N\subset M$ be a pair ...
3
votes
1answer
99 views
Examples of constant scalar curvature kähler metric that is not kahler einstiein
It is well known that if the first Chern class is proportional to the kähler class given, then every cscK in that class has to be kähler Einstein. So there are two directions to generate examples as ...
2
votes
0answers
194 views
Kollar-Matsusaka's finiteness theorem from a topological perspective
I am looking to find a reference or proof for the following topological version of Kollar-Matsusaka's theorem, which does not seem to be stated explicitly in the literature.
First I recall the ...
6
votes
0answers
111 views
Sections of infinite order of elliptic surfaces
Let $X\to \mathbb{P}^1$ be a non-isotrivial elliptic surface over $\mathbb{C}$ with a section and with $X$ a smooth projective connected surface over $\mathbb{C}$. Let $\sigma:\mathbb{P}^1\to X$ be a ...
2
votes
1answer
153 views
Control the convex combination of two classes on the boundary of the kahler cone
Let $(X,w)$ be a compact kahler manifold, and $[\eta]$ be a class is on the boundary of the kahler cone. The claim is that one can find another class $[\beta]$ also on the boundary of the kahler cone ...
2
votes
0answers
73 views
Conformal welding of rectifiable curves
In classical conformal welding theory, we start with a homeomorphism $h$ of the unit circle and try to find a Jordan domain $D$ together with two conformal isomorphisms $f_1 \colon \mathbb D \to D$ ...
3
votes
1answer
124 views
Surfaces of general type with $q=1$
Let $X$ be a smooth projective connected surface of general type over $\mathbb{C}$ with $q(X) = 1$, where $q(X) = \mathrm{h}^1(X,\mathcal{O}_X)$.
Let $E$ be the Albanese variety of $X$, and let $X\to ...
3
votes
0answers
89 views
Organizing mirror pairs
At a maximally vague and naive level, mirror symmetry asks the following question: given a complex manifold $(X, I)$, is there a symplectic manifold $(M, \omega)$ and an equivalence between the ...
3
votes
1answer
264 views
Some questions about Hitchin's self-duality paper
I am reading this paper (The self-duality equations on a Riemann surface by N. Hitchin), and I don't understand a few things in page 67. In proof of Theorem 2.1 after Equation 2.4, he gives the ...
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vote
2answers
214 views
Vector bundle over compact complex manifold which is not holomorphic?
A vector bundle over a complex manifold is said to be holomorphic if its trivialization maps are biholomorphic maps. What is a "natural" example example of a vector bundle over compact complex ...
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0answers
30 views
How do I find the portion of a cell/voxel lying within a defined surface?
We have a 3-dimensional grid of voxels (or cells), with individual voxels being of volume $dx\,dy\,dz$ where $dx=dy=dz=1$.
A cone-like surface is defined by some function, $z = f(x, y)$, which in ...
1
vote
0answers
36 views
Is the $L^2$ or $L^p$ space of global sections of a holomorphic line bundle with respect to a singular metric a Hilbert space?
Let $(X,\omega)$ be a Kahler manifold and $L$ be a holomorphic line bundle over $X$ with a singular metric $e^{-\phi}$. We refer the definition of singular metric (weight) to Demailly 92' Singular ...
2
votes
0answers
110 views
Chow variety of 1-cycles on abelian surface
It is an easy exercise to show that on a K3 surface, a smooth genus $g$ curve moves in a $g$-dimensional linear system. Nearly the same exercise shows that on an abelian surface, the corresponding ...
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0answers
73 views
Equality of the derivative of the exponential map on Kähler manifolds
Let $M$ be a Kähler manifold, $\omega$ its Kähler form and $J$ the complex structure. Moreover, let $V$ be a smooth (or even analytic, I am not sure if this is important) vector field on $M$, $p \in M$...
2
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0answers
93 views
Structure of non-big divisors in an abelian variety
Let $A$ be an abelian variety over $\mathbb{C}$. If $A$ has an effective non-big divisor, then $A$ is not simple. (In a simple abelian variety, every non-zero effective divisor is ample.)
What can ...
5
votes
1answer
216 views
Vanishing theorems on a non-compact manifold
In complex geometry, various vanishing theorems for cohomology
groups of a hermitian line bundle E over a compact complex manifold X have been found.
My question is
Is there some vanishing ...
1
vote
1answer
84 views
Chart in $1$-parameter family of Lagrangians in a Kähler manifold
Let $(X,\omega,J)$ be a complex $n$-dimensional Kähler manifold ($\omega$ Kähler form, $J$ complex structure) and $L \subset X$ be a closed real-analytic Lagrangian submanifold. Furthermore, let $L_{t}...
1
vote
1answer
105 views
Can a real (1,1) form $\phi$ be represented by $u\sqrt{-1}\partial\bar{\partial}u$ on Kahler manifold?
Let M be a 2-dimension (complex dimension) K\"{a}ler maifold and $\phi$ be a real $(1,1)$ form. Is it possible that there exists a function $u$ such that $\phi=u\sqrt{-1}\partial\bar{\partial}u$
3
votes
0answers
90 views
singular support in the singular case
For any constructible sheaf (or D-module) $\mathcal{F}$ over a smooth variety $X$ over $\mathbb{C}$, there is a notion of singular support $SS(\mathcal{F})$ that lives in the cotangent bundle $T^{*}X$ ...
4
votes
1answer
301 views
The Yoneda pairing, hypercohomology, and cup product
Let $\mathcal{F}$ and $\mathcal{G}$ be coherent analytic sheaves on $\mathbb{P}^n$. Let $\mathcal{F}_\bullet$ be a locally free resolution of $\mathcal{F}$. In Principles of Algebraic Geometry by ...
4
votes
2answers
225 views
Holomorphic union of sets
Let $\mathbb{D}$ be the unit disk on the plane and let $U,V\subset \mathbb{D}$ be open and such that $U\cup V=\mathbb{D}$.
Is there a holomorphic map $\varphi:\mathbb{D}\times \mathbb{D}\to \mathbb{...
