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.
0
votes
0answers
77 views
Kelly's theorem for quadratic polynomials
Let $f_1, \ldots, f_m$ be homogeneous irreducible quadratic polynomials in $\mathbb{C}[x_1, \ldots, x_n]$.
Assume that these polynomials are pairwise coprime.
Denote $P:= f_1 \cdot f_2 \ldots \...
4
votes
0answers
87 views
Continuous function on a complex space that is holomorphic on the complement of a closed subspace
Let $X$ be a complex analytic space and $Y\subseteq X$ a closed complex subspace. Suppose that $f:X\to\mathbb{C}$ is a continuous function that is holomorphic on $X\setminus Y$. Is $f$ holomorphic on $...
-3
votes
1answer
174 views
Holomorphic line bundles with torsion Chern class [closed]
Suppose you have a holomorphic line bundle $L$ such that $L^{n}$ is a trivial holomorphic line bundle and the base complex manifold $M$ has no torsion cohomology classes in second degree (i.e. $H^{2}...
6
votes
1answer
247 views
Rationality of the moduli space of genus g curves
I'm not an expert in this topic, so please excuse my negligence. I'd also appreciate references to the literature. Throughout, I will work over the complex numbers, although the analogous questions ...
3
votes
0answers
61 views
Status of global spherical shell conjecture for minimal complex surfaces?
A class VII surface is a compact complex surface $M$ such that $b_1(M)=1$ and $kd(M)=-\infty$. Class VII surfaces with vanishing second Betti number have been classified by Bogomolov (and are either ...
0
votes
1answer
211 views
Kähler form on complex projective algebraic variety [closed]
I am not very familiar with the notion of projective algebraic varieties, I work mostly from an algebraic topology/differential geometry point of view, but I am trying to find a prove for the ...
2
votes
0answers
61 views
Does there exist a leaf of this holomorphic foliation with non trivial holonomy?
Let's $\mathcal{F}$ be the holomorphic foliation of $\mathbb{C}^2$ tangent to the kernel of $\alpha=(sin x) dx -(cos x)dy$.
Are all leaves of $\mathcal{F}$ simply connected? If the answer is no, ...
5
votes
2answers
398 views
Embedding of a complex submanifold in projective space
Suppose you have a projective manifold $M$, a very ample bundle $\scr L$ and a transverse holomorphic section $s \in H^0(\scr L)$. Then the zero set of $s$ is a complex submanifold $S_M$.
Can we ...
2
votes
1answer
52 views
Set of sections whose zeroes avoid a given divisor is (Zariski) dense?
Let $X$ be a smooth complex projective variety of dimension $n$, and let $\mathcal{F}$ be a globally generated rank $n$ vector bundle on $X$. Let $D$ be a smooth divisor on $X$.
Is it true that ...
1
vote
0answers
26 views
What separates a cyclic polytope from a projective polytope?
I am having trouble understanding the difference between a cyclic polytope and a convex projective polytope as positive geometries.
The link https://arxiv.org/pdf/1703.04541.pdf is the source of ...
6
votes
1answer
315 views
Are there non-projective, but algebraic, hyperkahler varieties?
Let $k$ be an algebraically closed field of characteristic zero. I am not sure what the right definition of a hyperkahler variety over $k$ is, but I think the following might be close enough.
...
7
votes
2answers
421 views
The Ricci Form and the First Chern Class
Let $(M, \omega)$ denote a compact Kähler manifold. Since $d\omega =0$, $\omega$ represents a cohomology class in $H^2(M, \mathbb{R})$. Let $\rho$ denote the Ricci form of $M$, in local coordinates, ...
5
votes
0answers
122 views
Structure of the Kähler cone
In Calabi's Extremal kahler metrics paper, MR0645743, on page 262, the author mentioned that "It is conjectured that the structure of Kähler cone is determined by a finite number of real analytic ...
2
votes
0answers
107 views
Degeneration of a metric
I want to understand how the metric degenerate on a family of projective varieties (mainly for abelian varieties.).
Let $X$ be a smooth projective variety over $\mathbf{C}$.
Let $B$ be a smooth ...
2
votes
2answers
191 views
When is the space of holomorphic sections of the tensor product of two line bundles given by the span of the tensor product of the basis?
Let $S$ be a compact complex manifold and $L_1, L_2 \longrightarrow S$
be two holomorphic line bundles. Under what conditions (hopefully something that is easy to check) on $L_1$ and $L_2$ is the ...
1
vote
0answers
144 views
(Real) holomorphic vector fields on compact Kähler manifolds
I am trying to prove Proposition 2.1.1 of Gauduchon's note on Kähler extremal metrics (page 67). In order to show that, for compact Kähler manifolds, the complex Lie algebra of real holomorphic vector ...
4
votes
2answers
429 views
Criteria for a coherent sheaf pushing forward from the universal cover
I would like to prove (or find a counterexample to) the following statement:
Let $X$ be a complex analytic scheme and let $\pi: Y \to X$ be its universal cover. Let $F$ be a coherent sheaf on $X$ and ...
4
votes
1answer
131 views
Can an algebraic variety over a field $k$ be the union of proper closed subsets $(S_i)_{i\in I}$ with $I < k$
Let $k$ be an algebraically closed field (of characteristic zero, if it helps).
Let $X$ be an algebraic variety over $k$. Let $I$ be an index set such that the cardinality of $I$ is smaller than the ...
4
votes
1answer
106 views
Examples of surfaces with negative Kahler curvature operator
Compact ball quotients are examples of compact Kahler surfaces with negative curvature operator.
Are there any other examples ? What about nonpositive (other than the product of two Riemann surfaces ...
3
votes
0answers
94 views
Extending polarization (or Kähler metric) from the central fiber to the nearby fibers (or the total space)
Let $\pi: \mathcal{X}\to B$ be a complex analytic family of compact complex manifolds, i.e. $\pi$ is a surjective, proper submersion between complex manifolds. For simplicity, we assume $B$ is the ...
2
votes
0answers
218 views
Siu's arguments on Calabi-Yau theorem?
In Siu's lecture note Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics, he shows the $C^0$ and $C^2$ estimates of the complex Monge-Ampère equation on a Riemannian ...
3
votes
1answer
236 views
About the isotriviality of pencils of plane curves
Let $F$ and $G$ be coprime complex homogeneous polynomials in three variables of the same degree $d\geq 4$. Suppose that a general member of the pencil $\{F+tG=0\}\subset \mathbb{P}^2$ is smooth.
...
6
votes
1answer
199 views
Obstructions to simultaneous resolution
The following question on simultaneous resolutions is a follow-up to earlier questions posed here (e.g. Resolution of singularities for flat families.).
What I'm interested in is an "obstruction ...
1
vote
0answers
150 views
Can one construct a ramified cover over prescribed divisors?
Suppose $X/\mathbb C$ is a smooth projective variety of dimension $d>1$, and $D \subset X$ is a simple normal crossing divisor. For any $k>0$, can I construct a ramified $K$-th cover $f: \tilde ...
3
votes
1answer
257 views
Can an abelian variety dominate a variety of general type?
Let $X$ be a projective (not necessarily smooth) normal variety of general type over $\mathbb{C}$. Let $A$ be an abelian variety and let $A\to X$ be a surjective morphism.
Is $X$ zero-...
4
votes
2answers
172 views
Cycle class of zeroes of a global section
Let $\mathcal{F}$ be a locally free sheaf of rank $n$ on an $n$ dimensional complex manifold $X$. If the zero locus of a generic global section of $\mathcal{F}$ is $0$ dimensional, then its cycle ...
5
votes
1answer
219 views
The logarithm of Kähler metric is not globally defined
In reducing the existence of Kähler-Einstein metrics to the complex Monge Ampere equation, the logarithm $$-\log \det (\omega + \partial \overline{\partial} \phi)$$ appears, where $\omega$ is a Kähler ...
3
votes
0answers
180 views
Dolbeault cohomology of complex manifolds
Let $X$ be a compact complex manifold of dimension $n$ and let $D$ be a smooth hypersurface. Then the restriction map gives rise to a homomorphism in Dolbeault cohomology:
$$
\rho:H^{p,q}_{\bar{\...
6
votes
0answers
171 views
Differential topology on arbitrary fields
What do the differential topology theories on arbitrary fields have in common?
Different differential topology theories
There is "ordinary" differential topology on real manifolds, with its rich ...
1
vote
0answers
147 views
How to see the Hodge filtrations vary holomorphically directly
I have asked this question on Stack Exchange, "Why the Hodge filtrations vary holomorphically", but I have not got any reply, therefore I have revised this question a little bit, and post it here on ...
1
vote
0answers
47 views
Do we have a $4$-term decomposition of $\bar{\partial}_M$ for a holomorphic fiber bundle $M\to B$?
First let us consider a Riemannian fiber bundle, i.e a fiber bundle $\pi: M\to B$ of oriented Riemannian manifolds. We denote by $T(M/B)$ the bundle of vertical tangent vectors and assume that the ...
4
votes
1answer
251 views
A question on Steenbrink's paper, limit of Hodge structures
Steenbrink in his paper "Limit of Hodge Structures", (supplemented by the book "Mixed Hodge Structures" by Peters and Steenbrink) discuss the limit mixed Hodge structures for a fibration over the unit ...
5
votes
1answer
137 views
$S^{2}$-bundles over complex projective varieties
Is there an example of a smooth complex projective variety and an $S^{2}$-bundle over it which is not diffeomorphic to a complex projective variety?
5
votes
0answers
195 views
On the Choice Content of Carathéodory's Conformal Mapping Theorem
The Schoenflies theorem, as a variant of the well-known Jordan curve theorem, states that the interior and the exterior planar regions determined by a simple closed curve (aka Jordan curve) in $\...
5
votes
0answers
123 views
Do non-constant maps specialize to non-constant maps?
Let $R$ be a dvr with fraction field $K$ and residue field $k$. Let $\mathcal{X}\to \mathcal{Y}$ be a morphism of $R$-schemes such that $\mathcal{X}_K\to \mathcal{Y}_K$ is non-constant.
Is the ...
5
votes
0answers
113 views
Criterium for algebraicity of an analytic map
Let $X$ and $Y$ be algebraic varieties over $\mathbb{C}$. Let $f:X^{an}\to Y^{an}$ be a holomorphic map.
Is the following statement correct?
If there is an algebraic variety $V$ over $\mathbb{...
2
votes
0answers
112 views
Do the adjoints of the Lefschetz operators always commute?
Let $M$ be a projective nonsingular complex variety. Let $I$ and $J$ be two complex structures on M. We then have the corresponding Kälher classes $\omega_I$ and $\omega_J$ in $H^2(M, \mathbb{R})$, ...
1
vote
0answers
123 views
Biholomorphic maps between cotangent bundles with non-standard complex structures
Let $X$ be a compact Kähler manifold. Let $\omega_i$ (i=1,2) be Kähler forms on $X$. Assume that $\psi:X\rightarrow X$ is a diffeomorphism such that $\psi^*\omega_2=\omega_1$.
Recall that each $\...
4
votes
1answer
132 views
Equivalence of complex structures on flag manifold
Let $G$ be a compact Lie group and $T$ a maximal torus of $G$. One way to construct a complex structure on $G/T$ is to choose a nilpotent subalgebra $\mathfrak{n}^+$ corresponding to some choice of ...
6
votes
0answers
166 views
A general definition of an equisingular family of singular varieties?
This question is about the existence of a definition. I'm far from being an expert in the field in question I apologize in advance for any inaccuracies or stupid and wrong assumptions.
Let $X$ be a ...
4
votes
0answers
110 views
Lagrangian foliation for a holomorphic symplectic manifold
I am interested in gathering as many examples as possible for Lagrangian foliations of holomorphically symplectic manifolds $(X, \omega)$, where $X$ is a $2n$-dimensional complex manifold equipped ...
4
votes
0answers
240 views
Proof of Chow's theorem in special case
I'm trying to prove Chow's theorem (closed analytic subvarieties of projective space are algebraic) in a special case by the following differential geometric method (imitating the argument for the ...
3
votes
2answers
266 views
What is the “dual” of the space of currents?
On a smooth maniflod $M$ of dimension $n$, a current of degree $n-p$ is a functional on the space of compactly supported differential $p$-forms which is continuos. We denote the space of currents of ...
6
votes
0answers
125 views
The algebraic structures on $H^{1,1}(X,\mathbb C)$ induced by Kahler classes
Let $X$ be a compact Kähler manifold of dimension $n$. Each Kähler class $\omega$ on $X$ defines an adjoint Lefschetz operator $\Lambda$, and using this we can make $H^{1,1}(X,\mathbb C)$ into an ...
0
votes
0answers
135 views
Classifying $PGL(n,\mathbb{C})$-bundles over a compact Riemann surface
Let $X$ be a connected compact Riemann surface. How does one go on proving that the set of PGL($n,\mathbb{C}$)-bundles over $X$ is topologically classified by $\pi_1(PGL(n,\mathbb{C}))$? Is it true ...
4
votes
1answer
169 views
Reference to the conjecture about injectivity of Abel-Jacobi map
Suppose $k$ is a number field, and $\sigma:k \rightarrow \mathbb{C}$ is an embedding. Then there is the (generalised) Abel-Jacobi map
\begin{equation}
\text{CH}^j(X)_0 \rightarrow \frac{H^{2j-1}((X \...
2
votes
0answers
117 views
Distribution of random hyperplanes in projective spaces
Let $X\subset \mathbb{CP}^{N-1}$ be a smooth subvariety of dimension $n$. Assume that $X$ is not contained in a hyperplane of $\mathbb{CP}^{N-1}$.
Let $\mu$ be a smooth probability measure on $X$. ...
1
vote
0answers
101 views
Explanation of proposition 6.7 (a) of Fulton's Intersetion Theory
Suppose $X$ is a smooth variety over a field $k$ of characteristic zero, and $Z$ is a smooth subvariety of codimension d. Now let $\tilde{X}$ be the blow-up of $X$ at $Z$, and let the exceptional ...
1
vote
0answers
71 views
Translates of a line bundle on a complex $n$-torus
Suppose $\mathbb T:=V/\Gamma$ is a complex $n$-torus (i.e., $V$ is an $n$-dimensional $\mathbb C$-vector space and $\Gamma$ is a rank $2n$ lattice in $V$). Fix a holomorphic line bundle $L\in\text{Pic}...
5
votes
0answers
109 views
Complex algebraic submersions
Let $X$, $Y$ and $Z$ be smooth complex algebraic varieties and let $f:X\to Y$ and $g:X\to Z$ be two morphisms. Suppose that $f$ is surjective, that $df_x:T_xX\to T_{f(x)}Y$ is surjective for all $x\in ...
