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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.

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1answer
79 views

Analytic sections of a GIT quotient lying in the Kempf-Ness set

I would like to understand whether the following is true. Given a complex reductive group $G$ (for me $G = \operatorname{PGL}_n(\mathbb{C})$) acting on a vector space $V$ (for me $V = M_n(\mathbb{C})^{...
10
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1answer
751 views

Is the quotient of a toric variety by a finite group still toric

I have asked this question on Math.StackExchange, but haven't got any reply. Suppose $X$ is a toric variety with fan $\Sigma$, and the lattice of one-parameter subgroups of its torus is $N$. Suppose $...
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vote
1answer
100 views

Bound of the measure of the support of a set of divisors in a fixed linear system

Let $(X,L)$ be a compact polarized complex manifold of dimension $n$. Let $\varphi$ be a smooth positive metric on $L$. Define $\omega=dd^c\varphi$. We shall use $MA(\varphi)=\omega^n$ as the measure ...
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267 views

Does Lefschetz-type theorems imply ampleness?

Let $X$ be a smooth $n$-dimensional complex projective variety and $D \subset X$ a smooth (effective) divisor. Consider the following properties: $D$ is ample. (Positivity) For any $k$-dimensional ...
9
votes
1answer
266 views

Do all Fano threefolds have effective $c_2$?

Let $X$ be a smooth complex projective Fano threefold. Then the class $c_1(X)$ can be realised as an effective divisor in $X$. It is it true that the class $c_2(X)$ can be realised as an effective ...
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0answers
121 views

When is a minimal immersion holomorphic?

Let $(X,g_X)$ be a Riemann surface and $(Y,g_Y)$ a Kahler manifold. Let: $\phi\colon X\to Y$ be a minimal immersion, that is, a conformal harmonic smooth map with respect to $g_X$ and $g_Y$. If I am ...
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0answers
107 views

Differentials on tori realised as double of annuli

In this question it was described how to realise a torus as the double of an annulus Explicit construction of mirror surface and complex double for an annulus. In short, the torus is realised ...
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0answers
203 views

Local meaning of the Pfaffian of the curvature

The Ricci and scalar curvatures have very nice pointwise interpretations (using the local expression for the volume form for example). So, (at least Ricci) having special metrics (like Einstein) can ...
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2answers
260 views

Interesting examples of vector bundles on hyperkahler varieties

I'm looking for a few concrete examples of vector bundles on hyperkahler varieties of dimension $\ge 4$. Here are a few examples I know already: For $X$= the Hilbert scheme of points $S^{[n]}$ on a ...
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0answers
117 views

Path components of complex analytic spaces [closed]

Let $(X,\mathcal{O}_X)$ be a complex analytic space. Are the path components of $X$ the same as the connected components?
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0answers
220 views

Algebraic vs analytic de Rham cohomology

Let $X$ be a smooth projective variety over $\mathbf{C}$, $\Omega^{\bullet}_X$ its algebraic de Rham cohomology. Let $p : X_{\rm an}\to X_{\rm Zar}$ the obvious morphism of sites. We have $p^*\Omega^...
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288 views

Bloch Ogus spectral sequence

Let $X$ be a smooth projective variety over $\mathbf{C}$, and $p : X_{\rm an}\to X_{\rm Zar}$ the obvious map of sites. The Leray spectral sequence $$H^r(X_{\rm Zar}, R^sp_*\mathbf{C})\Rightarrow H^{...
4
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1answer
137 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 ...
9
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2answers
393 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 ...
5
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1answer
99 views

A condition for a dga to be minimal

I'm reading a book "Complex Geometry" by Daniel Huybrechts. In this book he says that a simply connected dga satisfying some conditions must be minimal. (p.147, Remark 3.A.13) I tried to prove this ...
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2answers
485 views

Two smooth tangent almost complex curves in a $4$-manifold

I would like to know if following is correct. Statement. Suppose we have a smooth (i.e., $C^\infty$) almost complex structure on $\mathbb R^4$ and $C_1, C_2$ are two $J$-holomorphic curves passing ...
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0answers
122 views

The splitting in the decomposition theorem

A special case, which I think is much older, of the Decomposition Theorem states that, for a projective smooth morphism $f: X \to Y$ of complex algebraic varieties, the higher direct image $Rf_* \...
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1answer
196 views

Bounding the derivative of a holomorphic function on a disk by its absolute value

Let $f(z)$ be a holomorphic function defined on the disk $|z|\le 2$. Suppose $|f(z)|<1$ for $|z|\le 2$. It looks like there is a constant $c>0$ such that $|f(z)'|<c$ on the disk $|z|\le 1$ (...
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2answers
1k views

Deformations of Calabi-Yau manifolds

Let $X$ be a compact complex smooth manifold with holomorphically trivial canonical class. It is true that any (sufficiently small?) deformation of the complex structure of $X$ also has ...
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0answers
122 views

Re-expressing $\operatorname{Hess}_{\rho}(L,N)\cdot(\nabla_N\overline L)\rho$

Let $\Omega\subseteq\Bbb C^n$ be a pseudoconvex domain. Let $r,\rho$ two defining functions for $\Omega$. Then it is known that they are related by $\rho=re^{\psi}$ for a suitable real smooth ...
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1answer
129 views

Clarification on Beltrami Differentials

I have troubles with the theory of existence of quasi-conformal homeomorphisms realizing Beltrami coefficients. Let $X$ be a (compact) Riemann surface and $f \colon X \rightarrow \mathbb{C}$ be smooth....
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1answer
73 views

Upper bound of the dimension of automorphism group of compact Kähler manifolds

It is well-known that the dimension of the isometry group of an $n$-dimensional compact Riemannian manifold is no larger than $\frac{1}{2}n(n+1)$, which is attained precisely by $S^n$ and $\mathbb{R}P^...
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0answers
59 views

Notation and geometry facts in a paper on the Diederich-Fornæss index

I am reading this article by Bingyuan Liu on the Diederich-Fornæss index. I am having some problems with both the notation and the geometrical side. 1)I don't know what kind of objects $N,L$ are ...
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2answers
418 views

When is a formal deformation convergent?

Let $X$ be a finite type scheme over $\mathbb{C}$ and let $ \mathcal{X} \to Spf(\mathbb{C}[[x]])$ be a formal deformation of $X$. Which of the following assumptions (or combinations thereof) are ...
3
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2answers
373 views

Topology of the blowup of a surface at a point (connected sum)

Let $S$ be a complex algebraic (smooth) surface and $\widetilde{S}$ be the blowup of $S$ at a point $p\in S$. I would like to understand the statement: As a topological manifold, $\widetilde{S}$ ...
5
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0answers
713 views

Fractal covering of a plane with complex-base numeral systems - is periodicity necessary?

Taking a base $z$ positional numeral system with digits $a_k\in \{0,\ldots,n-1\}$: $$s:\left\{(a_k)\in\{0,\ldots,n-1\}^{\mathbb{Z}}: \exists_K \forall_{k>K} \ a_k=0\right \}\to \sum_{k\in\mathbb{...
3
votes
0answers
62 views

Transfer modules and Weyl algebra

Let $V$ be a $\mathbb{C}$-vectorial space of dimension $n$ and $V^*$ the complex dual space. I would like to understand the following isomorphism $$D_{V^* \leftarrow V \times V^*} \overset{L}\otimes_{...
5
votes
0answers
79 views

Holomorphic vector fields and derivations

Let $M$ be a complex manifold and $U\subset M$ a domain. Question: Is every derivation of the complex algebra of holomorphic functions $\mathcal{O}(U)$ induced by a holomorphic vector field defined ...
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0answers
193 views

Group Action from $\mathbb{C}^*$ and Hodge decomposition

In Claire Voisin's book (Hodge Theory and complex algebraic geometry), at the first exercise of chapter 6, the author claims that if we have a continuous action from $\mathbb{C}^*$ on $H_\mathbb{C}$ (...
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1answer
327 views

Are holomorphic vector bundles over Kähler manifolds Kähler

Let $X$ be a Kähler manifold and $E\to X$ a holomorphic vector bundle. Is there a Kähler structure on $E$ compatible with its complex structure?
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1answer
108 views

Closure of orbit in complex manifold

Let $G$ be a complex algebraic group (reductive if necessary) acting holomorphically on a complex manifold $X$. Does the closure of every $G$-orbit contain a closed orbit? If $X$ is a complex ...
4
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0answers
111 views

homologically trivial $1$-cycles and surfaces

Let $X$ be a smooth (complex) threefold and $\gamma\in {\rm CH}_1(X)$ a homologically trivial $1$-cycle. Is there a way to construct a (singular) surface $S\subset X$ supporting $\gamma$ such that, ...
3
votes
1answer
192 views

Do Degree Zero Pseudo-Differential Operators on a Manifold Send Smooth Functions to Smooth Functions?

I'm not an analyst, so forgive me if what I'm asking is not suitable for Mathoverflow. For convenience, let $X$ be a compact complex manifold, and $E$ a holomorphic vector bundle on $X$. Let $H$ be ...
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0answers
104 views

Integral coniveau spectral sequences in Hodge Theory

Let $X$ be a smooth projective complex analytic space. We name $\mathcal{H}^*(\mathbf{Z}(n))$ the Zariski sheafification of Betti cohomology with $\mathbf{Z}(n)$ coefficients. We have a "coniveau" ...
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0answers
74 views

Norm function is proper on subsets of positive line bundle

Let $X$ be a compact complex manifold and $L$ a positive line bundle on $X$. By Kodaira's embedding theorem, $X$ is projective algebraic. If $S\subseteq L^*$ is closed in the Zariski topology and ...
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0answers
152 views

What properties does the inertia stack $I_X$ of an algebraic stack $X$ inherit from $X$?

Let $S$ be a noetherian scheme and let $X$ be a "nice" algebraic stack over $S$. For instance, let's say $X$ is a finitely presented algebraic stack over $S$, or that $X$ is a finite type separated DM-...
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0answers
104 views

Duality of Mixed Hodge Structures without compactness

Let $X$ be a smooth separated algebraic variety over $\mathbb{C}$ and $Z \subset X$ a subvariety of codimension $p$. There are no compactness assumptions. I am looking for an isomorphism of mixed ...
2
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2answers
233 views

Image of curve along a finite etale Galois map

Let $f:X\to Y$ be a finite etale Galois morphism of varieties over $\mathbb{C}$. Let $C$ be a smooth quasi-projective connected curve in $X$. Is $f(C)$ a smooth curve?
2
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1answer
268 views

Is $\mathbb{C}[x^{\pm 1},y]/\langle y^3-(x^2+ax+b) \rangle$ a $n$-point ring?

Definition: Let $\{a_1,\dots,a_n\}$ be any $n$ distinct points on the Riemann sphere $\mathbb{C}\cup\{\infty\}$ with coordinate $s$, and let $R$ be the ring of rational functions with poles ...
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0answers
86 views

Geometric or topological flavored proof of Nevanlinna five valued theorem?

In a very early state of the development of the Nevanlinna theory, Nevanlinna proved what is now called Nevanlinna five valued theorem, Let $f$ and $g$ be two transcendental meromorphic function. ...
7
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3answers
604 views

Examples of manifolds that do not admit scalar flat metrics

The Kazdan-Warner trichotomy states that for $n\ge 3$, a compact $n$-manifold falls into one of three categories: (A) Every (smooth) function is a scalar curvature. (B) The manifold is strongly ...
15
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1answer
608 views

Holomorphic line bundles on $\mathbb{P}^1$ from gluing data

Let $g$ be an $n \times n$ matrix of functions $g_{ij}(z)$ in $\mathbb{C}(z)$. Suppose that the $g_{ij}(z)$ have no poles on the annulus $1-\epsilon < |z| < 1+\epsilon$ and that $\det g(z)$ is ...
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0answers
187 views

Absolute Hodge Cycles over $\mathbf{Q}$

In the 1986 notes by Milne "Hodge cycles on abelian varieties", Deligne defines the notion of absolute Hodge cycles. For a smooth projective variety defined over $k\subset\mathbf{C}$ non ...
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0answers
152 views

Curvilinear locus in the Hilbert scheme of points

Let $X$ be a smooth complex projective variety of dimension $d$. Then the Hilbert scheme of $n$ points $X^{[n]}$ is not irreducible in general, but it has always the main component $X^{[n]}_{sm}$ of ...
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0answers
76 views

A singular foliation analogy of the Riemann Hilbert problem

Note: In this question by $\mathbb{C}P^1 \subset \mathbb{C}P^2$ we mean that we choose the line at infinity in the form $\{[0,y,z]\in \mathbb{C}P^2\} $ which is identified by $\mathbb{C}P^1$. ...
7
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3answers
353 views

When do automorphisms on open subsets extend

Let $X$ be a normal affine variety of dimension at least two over $\mathbb{C}$ and let $U\subset X$ be a dense open. Assume that $\mathrm{codim}(X\setminus U) \geq 2$. I think Hartog's lemma implies ...
18
votes
3answers
530 views

If a variety $X$ has finite automorphism group, is the same true for its $n$-fold self-products?

Let $X$ be an algebraic variety over $\mathbb{C}$. Let $n\geq 1$ be an integer and let $X^n$ be the $n$-fold self product of $X$. Q. Is there an integer $n\geq 1$ and an algebraic variety $X$ ...
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0answers
197 views

Analytic space not embeddable in any complex manifold

I am looking for an example of a compact complex analytic space, reduced and irreducible, which does not admit any holomorphic embedding into any (smooth) complex manifold (possibly non-compact). I ...
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0answers
95 views

Why do “large” opens of abelian surfaces have “small” canonical bundle?

Let $A$ be an abelian surface, and let $S$ be a finite set of points. Let $U=A\setminus S$. Note that $U$ is a "large" open of $A$. Let $B\to A$ be a proper birational surjective morphism with $B$ ...
2
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0answers
102 views

Do integral curves on simple abelian surfaces define big line bundles?

Let $A$ be a simple abelian surface over $\mathbb{C}$. Let $C\subset A$ be an irreducible and reduced one-dimensional closed subscheme. Since $A$ is simple, the normalization of $C$ is of genus at ...