All Questions
Tagged with cv.complex-variables ag.algebraic-geometry
395 questions
2
votes
1
answer
197
views
Section 3 of Atiyah's "On analytic surfaces with double points" — some questions
I have some questions about section 3 of Atiyah's "On analytic surfaces with double points," a short 9 page paper. Section 3 is all dedicated to proving lemma 4.
Near the end of section 3, ...
- 129
2
votes
1
answer
213
views
How can one test whether a given analytic curve in the plane is algebraic or not?
Suppose $\Gamma$ is an analytic Jordan curve in the complex plane $\mathbb{C}$. What are ways to test whether $\Gamma$ is (contained in) an algebraic curve, i.e., whether there exists a real ...
- 2,154
2
votes
0
answers
179
views
Analytic continuation of $\int_V (f(x_1,\cdots,x_n))^s dx_i$
Let $V$ be an $n$-dimensional simplex, let $f(\boldsymbol{x}) = f(x_1,\cdots,x_n)\in \mathbb{C}[x_1,\cdots,x_n]$ be a product of linear polynomials that is non-zero in interior of $V$. Also let $E(\...
- 528
3
votes
1
answer
326
views
Holomorphic homotopy conjecture
Let $X$ be a smooth projective variety over the complex numbers, and let $\text{Coh}(X)$ be the category of coherent sheaves on $X$. Consider the dg-category $\text{Perf}(X)$ of perfect complexes on $...
- 31
6
votes
0
answers
340
views
Does the Poincaré lemma (Dolbeault–Grothendieck lemma) still hold on singular complex space?
Let $X$ be a complex manifold, then we have the Poincaré lemma (or say, Dolbeault-Grothendieck lemma) (locally) on $X$, whose formulation is as follows:
( $\bar{\partial}$-Poincaré lemma) If $\...
- 327
7
votes
0
answers
166
views
Example of closed non-exact torsion differential form on variety
I asked this question some time ago on MSE and received close to no interest. I feel it is appropriate for this site:
I am interested in finding a particular example. I would like to find a variety (...
- 513
1
vote
0
answers
133
views
Fundamental set for families of abelian varieties
I'm considering the universal family of principally polarized g-dimensional abelian varieties with level N-structure. Let's briefly recall the usual construction as a quotient of $\mathbb{C}^g \times \...
- 73
16
votes
0
answers
519
views
Gabriel's theorem for complex analytic spaces
Let $X,Y$ be noetherian schemes over $\mathbb{C}$.
Then, it is known that
$$
\text{Coh}(X) \simeq \text{Coh}(Y) \Rightarrow X \simeq Y,
$$
by P. Gabriel(1962).
Are there some results in the case of ...
- 889
3
votes
1
answer
140
views
Is inverse image along finite group quotient $t$-exact for the perverse $t$-structure?
Let $q: \mathbb{C}\to \mathbb{C}$ be the quotient by $\mathbb{Z}/n\mathbb{Z},$ i.e. the map taking $z\mapsto z^n$. In the accepter answer to Operations on perverse sheaves on disk the inverse image of ...
- 1,071
6
votes
0
answers
200
views
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 ...
- 1,286
1
vote
0
answers
198
views
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 $...
- 513
2
votes
0
answers
185
views
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 ...
- 513
3
votes
0
answers
89
views
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 ...
- 658
0
votes
0
answers
121
views
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
1
vote
0
answers
109
views
Monomorphism/Isomorphism of $C_4$-tangent cones for complex varieties
Suppose that $(M,\mathcal{O}_M)$ is a reduced complex analytic space (or complex algebraic variety if you prefer). The tangent linear fiber space $TM$ associated to $M$ is defined as the analytic ...
- 513
1
vote
0
answers
82
views
Finiteness of theta vanishing in the KP direction for locally planar curves
I believe the main question is Question 2 at the end, and for experts it might be completely okay to skip directly to it (assuming I'm not saying any nonsense).
My motivation comes from pure algebraic ...
- 318
3
votes
1
answer
173
views
$n$-th root of meromorphic functions of several complex variables
Let $f:\Omega\rightarrow\mathbb{C}$ be a "nice" meromorphic function of several complex variables on some domain $\Omega$. I wonder if the following claim is true.
Claim. $f$ admits a global ...
- 1,024
2
votes
1
answer
271
views
Irreducibility of an explicit complex projective variety
Let $Y\subset \mathbb P^n_\mathbb C$ be a subvariety defined by a series of homogeneous polynomials $f_1, \ldots, f_t$. Is there an effective way to determine the irreducibility of $Y$ as an algebraic ...
- 279
6
votes
0
answers
160
views
Fourier transform and Hodge-$*$ operator
Suppose I have a full-rank lattice $\Lambda\subset\mathbf{C}$. Then the classical Poisson summation formula says
$$\sum_{\lambda\in\Lambda}f(\lambda)=\sum_{\lambda\in\Lambda'}\widehat{f}(\lambda)$$
...
user516086
2
votes
0
answers
107
views
Two definitions for transverse $(p,p)$ form
Let $V$ be a complex vector space of dimension $n$ and let $V^*$ be its dual. Fix any integer $1\leq p\leq {n-1}.$ A $(p,p)$-form $\alpha\in\bigwedge^{p,p}V^*$ is said to be strictly weakly positive ...
- 295
3
votes
0
answers
87
views
Bounding the degree of the Weierstrass polynomial of a product of a holomorphic function and a polynomial
In brief. For a fixed holomorphic function $v$, I want to bound the degree $q$ of the Weierstrass polynomial $Q$ in the Weierstrass decomposition $v^TP = uQ$, in terms of the degree $p$ of the ...
- 981
1
vote
0
answers
125
views
Canonical basis of cycles of Riemann surfaces
Let $\Gamma$ be the compact Riemann surface defiend by the algebraic curve
$$
f(x, y) = y^n + a_1(x)y^{n-1} + a_2(x) y^{n-2} + \dots + a_{n-1}(x)y + a_n(x) = 0,
$$
where $a_1(x), \dots, a_n(x)$ are ...
- 89
2
votes
1
answer
170
views
A specific question on the Griffiths' paper: the reduction of the pole order
If someone has gone through the Griffiths' paper ``On the periods of certain rational integrals: I,'' could you help me to understand Lemma 8.10?
I don't get why $\eta\in Z^{q,k+1}(l-1)$; although $\...
- 73
5
votes
0
answers
109
views
Does the reduction of the pole order to compute the Poincare residue work?
I am trying to understand the Poincare residue and referring to On Computing Picard-Fuchs Equations, which is cited by Wikipedia's page on the Poincare residue.
On pp. 5--6, he gives a way to compute ...
- 73
1
vote
0
answers
155
views
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
2
votes
1
answer
157
views
Bounding minimal absolute value of a point on a complex algebraic variety
Given a system of complex polynomial equations in $n$ variables, giving rise to an affine variety $V \subseteq \mathbb C^n$, is there a bound $b \in \mathbb R$ such that if $V(\mathbb C) \neq \...
- 373
8
votes
0
answers
333
views
Who introduced the notion of ringed spaces?
My question is very concise, please forgive it.
Who introduced the concept of ringed space?
My first try would be that they were introduced by Cartan in his study of analytic functions with sheaves. ...
- 894
1
vote
0
answers
63
views
Can we perturb a product of linear terms so that we keep the local geometry?
I am trying to understand the following phenomenon. Let me give an example first to elaborate my idea.
Let $Q(x,y)=(y-x)\cdot (y+x)\in \mathbb{C}[x,y]$. If we look at the zero locus of $Q(x,y)$ in the ...
- 111
0
votes
0
answers
111
views
The upper bound of hyperbolic cosine function in complex plane
I want to find the upper bound of the following function :
\begin{equation}
M(\lambda)=\max _{E \in \mathbb{C}}\left|L_{+}-L_{-}\right|
\end{equation}
where
\begin{equation}
\begin{aligned}
& 4 \...
0
votes
1
answer
323
views
An identity for Weierstrass elliptic functions evaluation
Let $\wp(z), \zeta(z)$ and $\sigma(z)$ be the Weierstrass $\wp$, zeta and sigma functions associated to the ODE:
$$\wp'(z)^2 = 4(\wp(z)-e_1)(\wp(z)-e_2)(\wp(z)-e_3)$$
and we assume $e_1=\frac{2-c}3>...
- 43.1k
5
votes
0
answers
136
views
Algebraic dependence of the elliptic functions
Let $\{f_i\}_{i=1}^{n}$ be $n$ elliptic functions in $\mathbb{C}$. We say that $f_1, \dots, f_n$ are algebraic dependent over $\mathbb{C}$ if there exists a polynomial of $n$ variables with constant ...
- 1,706
4
votes
0
answers
214
views
Cartan–Remmert reduction of an algebraic variety
Let $V$ be a normal connected algebraic (say, quasi-projective) variety over complex numbers. Assume that underlying complex analytic space $V^\text{an}$ is holomorphically convex, and thus admits the ...
- 1,170
2
votes
0
answers
109
views
Examples of compact non-Kähler complex manifolds with Kodaira dimension zero
Let $X$ be a minimal compact non-Kähler complex manifold. Suppose that Kodaira dimension $\kappa(X)=0$.
Is there a known example where the canonical bundle is not holomorphically torsion?
For ...
- 1,379
7
votes
1
answer
527
views
Do non-projective K3 surfaces have rational curves?
Define a compact Kähler surface $X$ to be a K3 surface if $X$ is simply connected, $K_X \simeq \mathcal{O}_X$, and $h^{0,1}=0$. If $X$ is projective, then a theorem typically attributed to Bogomolov ...
- 1,379
1
vote
0
answers
170
views
Difference between affine quotient variety and a global quotient orbifold
Given a smooth affine variety $X$ and a finite group $G$ acting by automorphisms on $X$, the quotient space $X/G$ has the structure of an affine variety which is in general not smooth. However, in the ...
- 609
7
votes
0
answers
204
views
Global generation of $S^n \Omega_X$ for a fake projective plane
Let $X$ be a fake projective plane, namely, a compact complex surface with
$$p_g(X)=q(X)=0, \quad K_X^2=9$$ and $K_X$ ample.
Since $K_X^2=9 \chi(\mathcal{O}_X)$, Yau's celebrated proof of the Calabi ...
- 66.3k
5
votes
0
answers
159
views
Higher Cardano formulae in terms of $\Theta$
Consider a polynomial in one variable with complex coefficient
$$f(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$$
we are interested in its roots. Babylonian solved for $n = 2$, and Cardano did it ...
- 5,230
2
votes
0
answers
108
views
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
6
votes
0
answers
252
views
Picard-Lefschetz formula for the quotient of a degenerating family of curves by a cyclic group
$\newcommand{\cD}{\mathcal{D}}\newcommand{\cX}{\mathcal{X}}$(This is a slight rephrasing and modification of the original question)
Let $D\subset\mathbb{C}$ be the complex unit disk. Let $X$ be a ...
- 10.7k
6
votes
1
answer
476
views
Example illustrating necessity of considering birational equivalence and not biholomorphic equivalence in MMP
The minimal model program attempts to classify algebraic varieties up to birational equivalence. For compact Riemann surfaces, Riemann's uniformization theorem tells us that the geometry of the curve ...
- 275
4
votes
0
answers
306
views
Geometric interpretation of Theta functions and the Jacobi inversion problem
A great part of complex geometry and, algebraic geometry has been developed to address the theory of abelian integrals/functions. A very special problem that kept many great mathematicians busy was ...
2
votes
0
answers
79
views
A compact family of holomorphic functions and their corresponding compact ranges?
Let $C = [0,1]^n \subset \mathbb{R}^n$ be the closed unit cube. For some open set $V \subset \mathbb{C}^n$ such that $C \subset V$, denote by $\mathcal{F}$ some compact family of holomorphic functions ...
- 981
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
5
votes
0
answers
284
views
Is there a geography of Hodge numbers for minimal general type algebraic surfaces?
Let $X$ be a minimal smooth projective surface of general type (over $\mathbb{C}$). Let's call such a surface MSPGT. Such a surface has two chern numbers $c_1^2$ and $c_2$. It is known that they are ...
- 10.7k
0
votes
0
answers
321
views
Why are holomorphic $p$-forms parallel?
Let $X$ be a complex compact Kähler manifold, $\Omega_X$ its cotangent bundle and $D$ the Chern connection.
It is, I believe, a standard fact that parallel $k$-forms $\alpha \in \mathcal{A}^k(X)$, i.e....
- 175
5
votes
2
answers
341
views
Can a non-Kähler complex manifold be rationally connected?
Let $X$ be a compact complex manifold. Suppose that $X$ is rationally connected in the sense that any two points lie in the image of a rational curve $\mathbb{CP}^1 \to X$. Are there any non-Kähler ...
- 275
3
votes
1
answer
301
views
Space of algebraic maps, homotopy type of a CW complex
Considering the algebraic maps between two complex varieties denoted by $C_{alg}(X,Y)$, as a subspace of continuous maps with compact-open topology. Does $C_{alg}(X,Y)$ have homotopy type of a CW ...
- 5,901
2
votes
0
answers
134
views
Differential equation of Van Gorder type for zeta of global fields, or: Does the zeta function of a global field satisfy a differential equation?
Let
\begin{equation*}
\zeta(s):=\prod_{p\text{ prime}}\frac{1}{1-p^{-s}}
\end{equation*}
be the Riemann zeta function. Van Gorder has shown that $\zeta$ satisfies a differential equation
\begin{...
- 1,805
4
votes
1
answer
150
views
Constructing proper holomorphic self-mappings of the unit disk with a given set of branch points and corresponding ramification degrees
I was trying to solve the following problem:
Let $f: D \longrightarrow D$ be proper holomorphic (so that means it is a Blaschke product with finitely many factors). Suppose $\{ a_1, ..., a_n \} \...
- 53
2
votes
0
answers
231
views
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