Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry.

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13
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1answer
295 views

List of Applications of the $\partial\overline{\partial}$-lemma

Quoting from Huybrecht's book Complex Geometry on the $\partial\overline{\partial}$-lemma for Kaehler manifolds: Although it looks like a rather innocent technical statement, it is crucial for ...
6
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2answers
156 views

$G$-invariant holomorphic vs. polynomial functions

Let $X\subseteq\Bbb C^n$ be a smooth affine variety and $G$ a complex reductive group acting algebraically on $X$. Let $x_0\in X$. If there is a non-constant $G$-invariant holomorphic function ...
6
votes
1answer
273 views

Is polynomial convexity a topological invariant?

Is the property of being polynomially convex a topological invariant? In other words, let $M$ and $N$ be two homeomorphic, compact subsets of $n$-dimensional complex Euclidean space, and assume in ...
5
votes
1answer
95 views

Density of non-algebraic leaves in the characteristic foliation

Let $X$ be a compact complex manifold equipped with a holomorphic symplectic form $\omega$. Let $D$ be a smooth divisor on $X$. At each point of $D$, the restriction of $\omega$ to $D$ has ...
1
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0answers
63 views

Decompositon of the Euler class in the ideal generated by Weyl-invariant polynomials

Let $G$ be a complex reductive Lie group, $B$ be a Borel subgroup, $T\subset B$ be a maximal torus, $W$ be the Weyl group. Then the space $X:=G/B$ is a complex manifold of dimension $n$, denote by ...
0
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1answer
470 views

Is there any open Ricci-flat ALE 4-manifold other than Hyper-Kahler ALE 4-manifolds?

Concerning my previous question Non simply connected HyperKähler 4-manifolds without ALE metrics the following question occurred to me: Is there any open Ricci-flat ALE 4-manifold other than ...
16
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1answer
376 views

Is there an integrable complex structure on $\mathrm{SU}(3)$?

Is there a complex manifold diffeomorphic to $\mathrm{SU}(3)$? This question arises in a StackExchange discussion by HK Lee, Ted Shifrin and Jason DeVito: ...
6
votes
1answer
321 views

The Hypercomplex Structure of $SU(3)$

(A) In this really stylish answer it is shown that one can define a family of complex structures $J_{\lambda}$ on the Lie group SU(3), dependent on the parameter $\lambda \in {\mathbb C}\backslash ...
4
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2answers
525 views

Alternative Almost Complex Structures

Originally posted on Maths Stack Exchange. Let $V$ be a real vector space. An almost complex structure on $V$ is a map $J : V \to V$ such that $J^2 = -\mathrm{id}_V$. An almost complex structure ...
5
votes
1answer
364 views

Holomorphic vector bundles over $\mathbb{C}^{n}\setminus 0$

Is it true that every holomorphic vector bundle over $\mathbb{C}^{n}\setminus 0$ is trivial? If not true, how can one construct a counterexample? And just a small note here (wrong): For $n\leq 2$, ...
15
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1answer
1k views
3
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1answer
230 views

How to tell if it's a Moishezon morphism

Suppose that $f \colon X\rightarrow S$ is a proper morphism of reduced and irreducible complex spaces and $f$ is a smooth deformation in the sense of Kodaira and Spencer. If we know each fiber $X_s$, ...
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0answers
38 views

How does x+jy represent a 2D coordinate? [closed]

Let $z = x + jy = |\overrightarrow{z}|e^{j\theta}$ where $j = \sqrt{-1}$ and $\angle{\overrightarrow{z}} = \theta$ How does $z$ represent a 2D coordinate? Where does the exponential $e$ come from? ...
0
votes
1answer
396 views

Positivity of forms

I'm getting stuck with a proposition. Please can someone help me? Let $E$ be a holomorphic vector bundle with hermitian metric $h$ over a connected compact Kähler manifold $X$ with Kähler form ...
3
votes
0answers
67 views

Open Period Integrals of Elliptically Fibered K3 surfaces

Let M be the period domain for elliptic K3 surfaces $(X,\Omega)$ with a holomorphic two-form. Denote the fiber class $f$. Then $$M=\{\Omega\in f^\perp\otimes \mathbb{C}\,:\, \Omega\cdot \Omega=0, ...
4
votes
1answer
160 views

Hyper-Kaehler Strucutre for Compact Lie Groups?

We know from the classy work of Joyce that "any compact Lie group becomes hypercomplex after it is multiplied by a sufficiently big torus". The quote comes from the Wikipedia page. I am asking if it ...
2
votes
1answer
170 views

Deformation of $\mathbb{P}^1 \times \mathbb{P}^1$

I learnt that $\mathbb{P}^1 \times \mathbb{P}^1$ is rigid, but can be deformed to a non-rigid Hirzebruch surface $S$. Suppose $\pi: M \to B$ is such deformation such that $\mathbb{P}^1 \times ...
2
votes
0answers
153 views

Normal bundle to fibers of a rational morphism

Let $f:X\dashrightarrow C$ be a rational fibration from a 3-dimensional variety $X$ to a curve $C$ such that generic fiber is smooth and different fibers intrsect in smooth curves. Take $S$ to be a ...
1
vote
1answer
69 views

Find the Picard Fuchs operator of a four parameter fundamental period

In my research, we have constructed a Calabi-Yau as a hypersurface of a toric variety and we could compute the fundamental period of the complex moduli, which is a series in four parameters. Say ...
2
votes
1answer
215 views

Relative tangent bundle and trivilization, tautological foliation

Let $T_{X}\rightarrow X$ be the tangent bundle over a complex manifold $X.$ Let $\pi:PT_{X}\rightarrow X$ be a projectivization of that bundle. Let $L$ be the tautological line bundle of $PT_{X}.$ ...
4
votes
1answer
235 views

Algebraicity and non-algebraicity of leaves of the characteristic foliation

Let $X$ be a compact complex manifold equipped with a holomorphic symplectic form $\omega$. Let $D$ be a smooth divisor on $X$. At each point of $D$, the restriction of $\omega$ to $D$ has ...
30
votes
2answers
919 views

Complex manifold with subvarieties but no submanifolds

I previously asked this question on MSE and offered a bounty but received no responses. There are examples of compact complex manifolds with no positive-dimensional compact complex submanifolds. ...
5
votes
1answer
72 views

Complex manifolds with spanning sets of holomorphic tensor fields

This question is an extension of this one. Consider a complex manifold $(M^{2n}, J)$. Fix $1 \leq p \leq n-1$, and suppose that the space of holomorphic sections of $\Lambda^{p,0}$ spans ...
1
vote
0answers
43 views

Hypercomplex, hyperKahler, or quaternion-Kähler from Joins/Connected Sums

I am looking for examples of (compact) hypercomplex, hyperKahler, or quaternion-Kähler manifolds which can be constructed as joins/connected sums of manifolds which do are not hypercomplex, ...
5
votes
3answers
640 views

Is there a “by hand” proof on the symmetry of the Atiyah class of $TX$?

Let X be a complex manifold and $TX$ its tangent bundle. The Atiyah class $\alpha(E)\in \text{Ext}^1(E\otimes TX, E)$ for a vector bundle $E$ is defined to be the obstruction of the global existence ...
3
votes
2answers
212 views

Chern classes and singular hermitian metrics on vector bundles

Let $L$ be a holomorphic line bundle on a complex manifold $X$, and assume it is equipped with a singular hermitian metric $h$ with local weight $\varphi$. Then, one can show that the de Rham class of ...
2
votes
1answer
155 views

Kahler Ricci flow in Fano fibration

Let $f:X\to Y$ be a Fano fibration of Kahler manifolds $X, Y$. Then why the Kahler Ricci flow $$\frac{\partial \omega}{\partial t}=-Ric(\omega(t))$$ starting of $[\omega_0]=f^*(\omega_Y)+c_1(X)$ ...
2
votes
2answers
156 views

Zeroes of trigonometric-like function

Consider a function $f(z)=\cos(z)\cosh(az)+\sin(z)\sinh(bz)$ for $z\in \mathbb{C}, a,b \in \mathbb{R}$. Denote $D\subseteq \mathbb{R}^2$ being the set of such pairs $(a,b)$ of parameters so that NOT ...
7
votes
2answers
1k views

Riemannian metrics as sections of a vector bundle

Let $\pi \colon E \to M$ be a smooth vector bundle. A Riemannian metric on $E$ can be regarded as a global section of the vector bundle $(E\otimes E)^{\ast}$, or more specifically, the subbundle ...
1
vote
1answer
129 views

A necessary condition for existence of Ricci flat metric on pair (X,D)

Let $X$ be a complex compact manifold with simple normal crossing divisor $D$. Is the condition $K_X +D = 0$ necessary for the existence of Ricci-flat metric?
13
votes
2answers
455 views

regular polygon question

Let $a_1,a_2,\ldots,a_n$ be distinct points on the complex plane $\mathbb{C}$ and $L$ be a circle in $\mathbb{C}$ such that $$f(z):=\sum_{i=1}^n|z-a_i|^{2n-2}$$ is constant on $L.$ Could somebody ...
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0answers
96 views

Applications of Iss'sa's theorem on homomorphisms between algebras of meromorphc functions

In Remmert's book Funktionentheorie II, the following theorem, apparently due to Hironaka under the pseudonym Iss'sa, is proved: Let $U,V \subseteq \mathbb{C}$ be open subsets. Every ...
2
votes
0answers
116 views

Kodaira fibration and moduli space of Riemann surfaces

Here we mean Kodaira fibration $f: X \rightarrow C$ where $f$ is a holomorphic submersion with maximal rank everywhere, but not a complex fiber bundle map. Such a surface has been constructed by ...
1
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0answers
47 views

Hyper-Complex Connected Sums of Grassmannians?

As we all know, the Grassmannians are Kaehler manifolds. Is there anyway to take connected sums of Grassmannians and produce a examples of hypercomplex manifolds, hyper-Kaehler or Calabi--Yau exmples ...
5
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0answers
63 views

On K-theory of blow-ups of compact complex manifolds

Is there a long exact sequence for the K-theory of (coherent sheaves on) blow-ups of compact complex manifolds? Does it split? What can one say on (possibly, singular) complex analytic spaces here? ...
5
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0answers
147 views

When a proper smooth fibration is isotrivial?

Let $\pi : X \to Y$ be a proper smooth morphism between complex quasi-projective varieties. Assume there exists a connected and Zariski dense analytic subvariety $Z$ of $Y$ such that for any two ...
0
votes
1answer
61 views

Sequence of translation surfaces and length of saddle connections

A sequence of translation surfaces $(X_n,\omega_n)$ is said to "go to infinity" if it leaves every compact set in the space of translation surfaces as $n$ goes to infinity. I know that this is ...
8
votes
1answer
1k views

How much of modern algebraic geometry is there in modern complex(algebraic, analytic, differential) geometry?

Good day to you, people of mathoverflow. I'll get to the point. I wonder how much of modern abstract algebraic geometry is there in modern complex geometry? What do I mean by complex geometry? ...
6
votes
1answer
138 views

First Chern class vanishes on a Lagrangian submanifold

Suppose $(M,\omega)$ is a symplectic manifold and $L \subset M$ is a compact Lagrangian submanifold. Is it the case that the first Chern class $c_1(TM) \in H^2(M)$ vanishes when restricted to the ...
8
votes
3answers
2k views

Why are local systems on a complex analytic space equivalent to vector bundles with flat connection?

Let $X$ be a complex analytic space. It is a 'well known fact' that the categories of local systems on $X$ (i.e. locally constant sheaves with stalk $C^n$), and of (holomorphic) vector bundles on $X$ ...
25
votes
2answers
5k views

The error in Petrovski and Landis' proof of the 16th Hilbert problem

What was the main error in the proof of the second part of the 16th Hilbert problem by Petrovski and Landis? Please see this related post and also the following post. Added : According to their ...
5
votes
3answers
340 views

Families of abelian varieties on the line (or more generally simply connected varieties)

I'm curious whether the following is true: Question 1: Let $V/\mathbb{C}$ be a smooth connected variety such that $V^\text{an}$ is simply connected. Then, is every abelian scheme $f:\mathscr{A}\to ...
8
votes
0answers
157 views

Holomorphic contractibility of GL(H)?

Kuiper's theorem is well-known to give the triviality of the homotopy groups of ${\rm GL}(\mathcal{H})$ for $\mathcal{H}$ a (separable) infinite-dimensional complex Hilbert space. Work of Palais later ...
3
votes
1answer
912 views

Hamiltonian potentials of holomorphic vector fields on modifications of Kahler manifolds

let $(M,\omega)$ be a compact Kähler manifold. Let $\mathfrak{g}=H^{0}(M,T_{M})$ be the Lie algebra of holomorphic vector fields on $M$.We can decompose $\mathfrak{g}$ as ...
2
votes
1answer
157 views

The spectrum and the tangent space of the algebra of holomorphic functions on a Stein manifold

Let $A$ be a Fréchet algebra over ${\mathbb C}$, and let us call the spectrum ${\tt Spec}[A]$ of $A$ the set of all characters, i.e. continuous multiplicative linear functionals $s:A\to{\mathbb C}$, ...
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0answers
119 views

Atiyah-Guillemin-Sternberg Theorem for current

The Atiyah-Guillemin-Sternberg theorem says that the image of a moment map of a toric action is independent of the choice of symplectic form in the cohomology class. So all is well for a smooth ...
0
votes
0answers
77 views

How to investigate the harmonocity of holomorphic vector fields?

Let $(M,g,J)$ be a Kahler manifold and $\nabla$ be its Levi-Civita connection. We know that $\Delta _gX=||\nabla X||^2X$ is the characterizing equation for harmonic unit vector fields. I dont know ...
22
votes
3answers
3k views

Which Kahler Manifolds are also Einstein Manifolds?

Is it known which Kahler manifolds are also Einstein manifolds? For example complex projective spaces are Einstein. Are the Grassmannians Einsein? Are all flag manifolds Einstein?
2
votes
1answer
102 views

Kernel of projection formula

For a closed embedding of compact complex manifolds $$ \iota : Y \hookrightarrow X $$ and any $\alpha \in H^*(X,\mathbb Q)$, we have trivially: $$ \iota^*(\alpha)=0\quad \Rightarrow ...
1
vote
0answers
72 views

How many real-analytic forms exist on a real-analytic manifold?

I hope my question isn't too vague: Let $M$ be a real-analytic compact manifold (say surface for simplicity). How big is $\Omega^k(M)$ (space of global real-analytic $k$-forms)? Let me explain ...