# 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**

votes

**1**answer

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**

votes

**1**answer

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 $...

**1**

vote

**1**answer

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 ...

**10**

votes

**0**answers

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

**1**answer

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 ...

**2**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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 ...

**9**

votes

**0**answers

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 ...

**5**

votes

**2**answers

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 ...

**1**

vote

**0**answers

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?

**3**

votes

**0**answers

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^...

**8**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**2**answers

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**

votes

**1**answer

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 ...

**10**

votes

**2**answers

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 ...

**5**

votes

**0**answers

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_* \...

**0**

votes

**1**answer

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 $|z|\le 1$ (...

**14**

votes

**2**answers

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 ...

**0**

votes

**0**answers

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 ...

**3**

votes

**1**answer

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....

**1**

vote

**1**answer

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^...

**2**

votes

**0**answers

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 ...

**9**

votes

**2**answers

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**

votes

**2**answers

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**

votes

**0**answers

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

**0**answers

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

**0**answers

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 ...

**0**

votes

**0**answers

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 $H_\mathbb{C}$ (...

**5**

votes

**1**answer

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?

**3**

votes

**1**answer

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**

votes

**0**answers

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

**1**answer

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 ...

**1**

vote

**0**answers

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" ...

**2**

votes

**0**answers

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 ...

**3**

votes

**0**answers

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-...

**2**

votes

**0**answers

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**

votes

**2**answers

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**

votes

**1**answer

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 ...

**3**

votes

**0**answers

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**

votes

**3**answers

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**

votes

**1**answer

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 ...

**1**

vote

**0**answers

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 ...

**7**

votes

**0**answers

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 ...

**1**

vote

**0**answers

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**

votes

**3**answers

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

**3**answers

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 $X$ ...

**8**

votes

**0**answers

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 ...

**2**

votes

**0**answers

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**

votes

**0**answers

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 ...