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.
3,129
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Criteria for when Gauss-Manin sheaves are vector bundles
Let $(X,D_X)$ and $(S, D_S)$ be smooth normal crossings pairs over $\mathbb C$; i.e. smooth schemes of finite type over $\mathbb C$ with a normal crossings divisor. If $f:X \to S$ is a proper, flat ...
3
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1
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Request for non-Einstein positive constant scalar curvature Kähler surfaces
I am curious about concrete examples of compact cscK manifolds in complex dimension two, in particular cscK surfaces with positive scalar curvature.
There are of course the Fano (del Pezzo) Kähler-...
4
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1
answer
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Equivariant projective embeddings with optimal dimension
Let $X$ be a complex projective manifold, and $f\in Aut(X)$ an automorphism, which is linearizable, that is, can be extended to an ambient projective space ${\mathbb P}^m$. I am interested to find ...
11
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1
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Cohomological bounds for scalar curvature of an extremal Kähler metric
There is an interesting trick used in Chen-LeBrun-Weber's paper on the extremal Kähler metrics of $\mathbb{CP}^2\#2\overline{\mathbb{CP}^2}$, and I would like to know whether it can be (has been?) ...
0
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0
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Some calculation about Chern connection
The Chern connection is the unique connection satisfying $\nabla^{0,1}=\bar{\partial}$ and
$$
\partial_k\langle u, v\rangle=\left\langle\nabla_k u, v\right\rangle+\left\langle u, \nabla_{\bar{k}} v\...
1
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1
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Proving that $H^1(X,\mathcal{Hom}(\mathcal{G},\mathcal{E})) \cong \text{Ext}^1(\mathcal{G},\mathcal{E})$ holds for locally free sheaves
The following passage is from a thesis I'm reading:
Suppose we have a short exact sequence of vector bundles $$0 \to \mathcal{E} \to \mathcal{F}\to \mathcal{G} \to 0.$$ Since these sheaves are ...
5
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0
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Theories of manifolds w/ extra structure and singularities
Many different objects in mathematics can be described as manifolds with extra structure. Among the most famous examples of these are smooth manifolds, Riemannian manifolds, complex manifolds, and ...
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44
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To study the elliptic PDE on complex manifold, when can we treat it as the real case?
I wonder when studying the elliptic PDE on complex manifold, especially studying the existence of solutions, when can we directly study the real case, for example, when studying
$$\Delta_c u = f(x,u),$...
1
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1
answer
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Monge–Ampère operator
I'm studying the article of Bedford–Taylor "Fine topology, Šilov boundary…" but I don't
understand the proof of the following proposition.
Let $u$, $v$ be plurisubharmonic functions defined ...
0
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1
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73
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Number of regions created by r hyper-planes in n-dimensional space [closed]
I found this formula for calculating maximum number of regions created by r hyper-planes in n-dimensional space (n<=r)
...
1
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0
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Dolbeault class of the curvature of the Chern connection equaling the Atiyah class
The following is a proposition from Complex Geometry by Huybrechts.
Proposition $4.3.10$. For the curvature $F_\nabla$ of the Chern connection on an hermitian holomorphic vector bundle $(E,h)$ one ...
2
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0
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A question about considering the solution of elliptic PDE with complex Laplacian as the critical point of a functional
I'm considering the elliptic PDE with complex Laplacian, for example, write $$
\Delta_c(\cdot):=-g^{i \bar{j}} \partial_i \partial_{\bar{j}}(\cdot),
$$
and $$\Delta_c(u)=f,$$
by [P.Gauduchon, Math.Ann,...
4
votes
1
answer
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A complex version of the Cahiers topos
Has anyone tried defining a complex version of the Cahiers topos?
If we take the definition of $C^\infty$-rings, replace "smooth" with "holomorphic" (of course, one has to take ...
2
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1
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Gluing local holomorphic connections
On page $180$ of Complex Geometry by Daniel Huybrecths, he defines the so called Atiyah class of a holomorphic vector bundle by the Čech cocycle $$A(E)=\{U_{ij}, \psi^{-1}_j \circ (\psi^{-1}_{ij}d\...
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Looking for counterexamples: Are maximal tori in the automorphism groups of smooth complex quasiprojective varieties conjugate?
Let $X$ be a smooth quasiprojective variety over $\mathbb{C}$. It has a group of (algebraic) automorphisms $
\DeclareMathOperator{\Aut}{Aut}
\Aut(X)$.
Define a torus in $\Aut(X)$ to be a faithful ...
4
votes
2
answers
411
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Nearby cycles for stacks
Let $X$ be a variety over $\mathbb C$. Let $f\colon X \to \mathbb{A}^1$ be a regular function. I understand that there is an analytic nearby cycles functor, defined in terms of the exponential map. I ...
1
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0
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Looking at a frequency reassignment rule as a Möbius transform
Suppose we have some Schwartz function $h$. Denote its Fourier transform $\widehat{h}$. Let $\xi_0$, $a$, $\Delta$ be positive and fixed.
I have a function $\Omega: \mathbb{R}\times \mathbb{R}^+ \to \...
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0
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44
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How do I find an algebraic expression for the function $F(ξ, \bar{ξ})$ from this paper?
I am working on understanding the paper "On $C^2$-smooth Surfaces of Constant Width" by Brendan Guilfoyle and Wilhelm Klingenberg. As part of their definition of equations for a 3D surface, ...
12
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4
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Interesting results for open Riemann surfaces
As far as I know, interesting results for open Riemann surfaces are quite rare. One of them is the theorem of Gunning and Narasimhan, which asserts that every connected open Riemann surface admits a ...
4
votes
1
answer
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Spin$^c$ structures induced by an almost complex structure
Let $M$ be a closed spin$^c$ $4$-manifold with determinant line bundle $L$.
If $c_1^2(L)=2\chi(M)+3\tau(M)$, where $\chi$ and $\tau$ denote the Euler characteristic and signature of $M$ respectively, ...
6
votes
0
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Reference request: Automorphisms of $\mathbb C\{x,y\}$ which preserve the equation of the cusp, $x^3 - y^2$
In my research I encountered automorphisms of the ring of convergent power series
$$\varphi: \mathbb C\{x,y\} \to \mathbb C\{x,y\},$$
which preserve $f = x^3 - y^2$, i.e. $\varphi(f) = f$. I'm ...
8
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2
answers
790
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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 ...
2
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0
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44
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Geometric explanation of Fueter-Sce-Qian Theorem and similar situations
In Clifford analysis there is a fundamental theorem due to Fueter and extended by Sce and Qian that says (in modern terminology) that the given a slice regular function $f:\mathbb{R}^{m+1}\to\mathbb{R}...
1
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0
answers
45
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Elliptic surfaces with monodromy in Borel subgroup
Are there restrictions on the invariants of an elliptic surface $M\overset{\pi}{\longrightarrow} C$ for the monodromy of its homological invariant to be contained in the upper triangular subgroup of $\...
4
votes
0
answers
140
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$\pm 1$-equivariant perverse sheaves on the affine line
Let $G=\mathbb{Z}/\mathbb{2Z}$ act by the map $z\mapsto -z$ on a complex line $\mathbb{C}$. The category $\mathcal{Perv}(\mathbb{C})$ of perverse sheaves smooth along the stratification by the origin ...
1
vote
1
answer
104
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Nonequidimensional birational Mori contractions
I have been looking for an excplicit example of a birational, divisorial Mori contraction such that the exceptional locus is not equidimensional onto its image.
To agree with the setup I like, the ...
0
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0
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139
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“Holomorphic” bump function
I was wondering in what sense can I construct a holomorphic “bump function”? Now, of course we cannot really construct a holomorphic bump function in the usual sense, but I have a much rougher idea in ...
1
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0
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187
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Constructing curves with large tangent space in complex variety
Suppose $M$ is a (singular) complex analytic/algebraic variety. Then for every $p\in M$ there exists a (possibly reducible) curve $C \subset U\subseteq M$ containing $p$ such that $T_pC=T_pM$, where $...
3
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0
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125
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Splitting of normal bundle exact sequence and Holomorphic neighbourhood retract
Let $X$ be a compact complex manifold and $Y\subset X$ a complex submanifold of $X$.
Consider the two following conditions:
The exact sequence $0\to TY\to TX|_{Y}\to N_Y\to 0$, where $TX$, $TY$ ...
7
votes
0
answers
226
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Is a smooth projective variety over $\mathbb{C}$ dominated by a Ball?
Suppose that $X$ is a smooth projective variety of dimension $d$ over the complex numbers.
Is it true that there is a ball $\Delta_d=\{ z\in \mathbb{C}^d / \lvert z\rvert<1\}$ and a surjective ...
1
vote
1
answer
423
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Vector bundles on $\mathbb{P}^1$
I am considering an alternative proof of Grothendieck's classification of vector bundles on $\mathbb{P}^1$. Given a vector bundle $E$ on $\mathbb{P}^1$ one can associate a graded module $\Gamma(E)$ ...
2
votes
0
answers
162
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Splitting of de Rham cohomology for singular spaces
I am currently trying to wrap my head around the following splitting result by Bloom & Herrera (here is a link to the ResearchGate publication) for the de Rham cohomology of (in particular) a ...
13
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2
answers
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Is the Gödel universe Wick rotatable?
Take Wick rotatability being as the way defined in the following article by Helleland and Hervik:
Christer Helleland, Sigbjørn Hervik, Wick rotations and real GIT, Journal of Geometry and Physics 123 ...
1
vote
1
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Solution to $a=e^t (t-r_1)(t-r_2)$ with Lambert $W$ function, where $r_1, r_2 $ are complex
Lambert $W$ works when $r_1$, and $r_2$ are real. However, I am trying to solve the equation when $r_1$, and $r_2$ are complex numbers.
2
votes
0
answers
120
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Hodge coniveaux of Calabi-Yau manifolds
Let $X$ be a strict compact Calabi-Yau manifold of dimension $n$. By this, I mean that $X$ is a simply connected projective manifold whose holomorphic forms are generated by a nowhere zero top degree ...
11
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1
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Hodge conjecture as the equality of arithmetic and algebraic weights of motivic L-functions
Recently I became aware of the following statement given on page 13 of this paper. First, let us recall the following definitions:
Definition 4.1. Suppose $L(s)$ is an analytic $L$-function with ...
8
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3
answers
1k
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How many non-orthogonal vectors fit into a complex vector space?
I am sitting on a problem, where I have a complex vector space of dimension $D$ and a set of normalized vectors $\{v_k\}$, $k\in\{1,2,\dots,N\}$ that are supposed to satisfy
$$\lvert\langle v_j\vert ...
2
votes
0
answers
194
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Smooth compactification of complex varieties and uniqueness
Since I'm working in differential geometry, for the following I'm strictly interested in the smooth setting over $\mathbb{C}$ and its relation to the setting over $\mathbb{R}$.
Here are a few useful ...
4
votes
0
answers
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Classification of affine varieties over the affine line whose central fiber is $\mathbb{C}$ and general fiber is $\mathbb{C}^*$
Consider an affine variety $Y$ equipped with a morphism $\pi: Y \rightarrow \mathbb{C}$. The conditions we have are that $\pi^{-1}(0)=\mathbb{C}$, and for any $x$ not equal to zero, $\pi^{-1}(x)=\...
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Let $f: X \to D$ be a proper holomorphic submersion. Does a holomorphic form on $X$, closed on each fiber of $f$, have holomorphic coefficients?
Let $D$ be the unit disc in $\mathbb{C}^n$ and let $f: X \to D$ be a proper surjective holomorphic submersion, which is trivial as a smooth fiber bundle, with connected fiber $F$. We get an induced ...
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0
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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 ...
5
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0
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273
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Formal neighborhood of stable curves
For a smooth projective curve $X/\mathbb{C}$, every (infinitesimal) deformation is trivial when restricted to $X \setminus x$ for any $x \in X$. In particular, all deformations can be obtained by “...
2
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0
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Is a Wick rotatable spacetime necessarily strongly causal?
There are a few viable ways to formulate Wick rotatability that preserve distinct features.
One is mentioned in the post:
Obtain Lorentzian manifolds from Riemannian ones by Wick rotation
There's also ...
1
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0
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Complex geodesic coordinate, local ramified map, and the conic metric
Remark: I have asked this question in MSE, however, I got no responses. This is the reason I come to ask here. I am looking forward to some advices. Thanks in advance
Let $X$ be a compact Kaehler ...
5
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2
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Some questions about Hitchin's self-duality paper
I am reading this paper (The self-duality equations on a Riemann surface by N. Hitchin; DOI: 10.1112/plms/s3-55.1.59), and I don't understand a few things in page 67. In proof of Theorem 2.1 after ...
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0
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Kähler potential for ALEs from resolving $\mathbb{C}^2/\mathbb{Z}_2$:
I am reading the famous paper of Kronheimer “The construction of ALE spaces as hyperkähler quotients”
I want to calculate explicitly the metric on the ALE spaces, obtained by a family of resolution of ...
0
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0
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203
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Definition of motivic/coming from geometry
Suppose $X$ is a projective smooth connected curve, $S\subset X$ is a finite set of points and $U=X\setminus S$.
I encountered the following definiton:
We say a $\mathbb Q$-local system $\mathcal F$ ...
3
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0
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155
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Action of complex Lie group on Dolbeault cohomology
Let $M$ be a compact complex manifold acted holomorphically by a complex Lie group $G$. Let $F$ be a holomorphic $G$-equivariant vector bundle over $M$.
Consider the natural representation of $G$ in (...
0
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0
answers
96
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Torsion free Chern connections and Kähler manifolds
Let $(M,h)$ be an Hermitian manifold and let $\nabla$ be the associated Chern connection. Is it true that $(M,h)$ is Kähler if and only if $\nabla$ is torsion free?
1
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0
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81
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Functional inequality with complex variables
I am interested in considering a function $C(\tau)$ of the complex variable $\tau = t + i\eta$ such that
$C(\tau)$ is analytic for $\Re(\tau)=t>t_0\ge0$
$\exists$ a constant $C_0$ and a function $...