# 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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### Is there an algorithm on an infinite time Turing machine to compute a dense subset of $M_g$ for large $g$?

Denote by $M_g$ the coarse moduli space of connected smooth projective complex curves of genus $g$. If $g$ is a random large integer (e.g. $g=100$) does there exist an algorithm on an infinite time ...

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### Holomorphic tubular neighborhood of divisors at infinity

For the discussion of holomorphic tubular neighborhoods and some criteria for their existence see this question.
Let $X$ be a smooth quasi-projective variety over $\mathbb{C}$. Hironaka tells us that ...

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84 views

+50

### Anti-self-dual Ricci-flat and Kähler Ricci-flat manifolds

Let $(M^4,g)$ be a complete Riemannian $4$-manifold with anti-self-dual (i.e. $W_+=0$) and Ricci-flat metric $g$.
Can we find a finite cover $(\tilde M, \tilde g)$ of $(M,g)$ such that $(\tilde M, \...

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163 views

+100

### A density criterion and a submersion map of a Hodge bundle

In Voisin's excellent book 《Hodge theory and complex algebraic geometry II》5.3.4 - a density criterion, there is a important theorem:
Let $X$ be a compact Kähler manifold, $\pi:\mathcal X \rightarrow ...

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98 views

### Fixed locus of a Kahler $S^1$-action

Given a compact Kahler manifold $M$ with an $S^1$-action by Kahler isometries, we know that
Its fixed loci $F=M^{S^1}$is a smooth Kahler submanifold.
It splits $F=\sqcup_{\alpha \in A} F_{\alpha}$ ...

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194 views

### Galois action on torsion in homotopy groups not induced by homotopy equivalences

Let $V$ be a simply connected smooth projective complex variety defined over the rationals. Then for any integer $n\geq 2$ the group $\pi_n(V)$ is finitely generated abelian so profinite completion ...

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173 views

### A homotopy equivalence from a variety to itself that is not homotopic to a homeomorphism

Let $V$ be a simply connected smooth projective complex variety. Can there be a homotopy equivalence $V\to V$ that is not homotopic to a homeomorphism?

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323 views

### Complex conjugation inducing a trivial map on the fundamental group

Let $V$ be a smooth projective complex variety defined over the reals such that $G=\pi_1(V)$ is a non-abelian finite simple group. Assume that $V$ has a real point. Can the map $G\to G$ induced by ...

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255 views

### Stably trivial non-trivial vector bundles

I have two related questions. Can there be a stably trivial non-trivial holomorphic vector bundle over a closed complex manifold? Can there be a stably trivial non-trivial algebraic vector bundle over ...

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395 views

+500

### Building all holomorphic vector bundles from the tangent bundle

Let $V$ be a smooth projective complex variety such that the canonical bundle is not trivial. We can construct some vector bundles over $V$ by starting with the tangent bundle and applying tensor ...

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99 views

### From resolution to normal crossing singularity

Let $X$ be a variety in $\mathbb{P}^n_{\mathbb{C}}$, and let $x$ be an isolated singular point of $X$ such that:
locally in $x$, $X$ has $1\leq m\leq n$ smooth branches; i.e. there exists an open ...

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46 views

### Are positivity for forms and that for currents consistent when talking about smooth forms?

Let $X$ be a complex manifold and $\theta$ a smooth $(1,1)$-form on $X$.
(1) If $\theta>0$ in the sense of currents, then can we deduce that $\theta>0$ in the sense of forms?
(2) If $\theta>0$...

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46 views

### Complex Monge Ampere equal with zero right hand side

Let $(X, \omega)$ be a compact Kaehler manifold, with $K_{X}$ numerically effective. Suppose that $[\zeta] = c_1(K_X)$ and $\int_{X}\zeta^n = 0$. I am interested in solving
\begin{equation}
(\zeta + i ...

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66 views

### A complex analytic interpretation of multiplicity on the special fiber of a flat family

Let $X$ be a variety over $\mathbb C$ and $\pi: X\to \Delta$ be a flat morphism over the unit disk $\Delta=\{z:|z|<1\}$. Let $Z$ be a component of $X_0=\pi^{-1}(0)$. The multiplicity of $Z$ is ...

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160 views

### Complex structures on Hermitian symmetric space

Let $(M_1,g_1,J_1)$ and $(M_2,g_2,J_2)$ be two simply-connected Hermitian symmetric spaces, which are isometric as two Riemannian manifolds.
Can we find an isometry $\varphi:M_1 \to M_2$ such that
$$
\...

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173 views

### Non-isotrival fiber bundle over compact Riemann surface

In this paper, Kodaira constructed a fiber bundle $\Phi:M_{m,n}\to S$ from a compact complex surface $M_{m,n}$ to a compact Rieman surface $S$ of genus $>0$. In particular, (on p.212) for any point ...

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260 views

### Relationship between volume and area

Let $\mu(z) dV_n$ be a measure in $\mathbb{C} ^n$.
Let $B_n(r) := \{z \mid \|z\| < r\}$ be the ball of radius $r$ in $\mathbb C^n$, and $\partial B_n(r) $ be the corresponding sphere.
In $\mathbb{C}...

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90 views

### Can logarithmic connection operate on currents?

Let $(X,\omega)$ be a compact Kähler manifold of dimension $n$, and let $D=\sum_{i=1}^r D_i$ be a simple normal crossing divisor on it, i.e., a divisor with smooth components $D_i$ intersecting each ...

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74 views

### Can logarithmic connection on holomorphic vector bundle induce logarithmic connection on dual bundle?

Let $(X,\omega)$ be a compact K"ahler manifold of dimension $n$, and let $D=\sum_{i=1}^r D_i$ be a simple normal crossing divisor on it, i.e., a divisor with smooth components $D_i$ intersecting ...

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77 views

### Saturated ideals in computational algebra

Let $R$ a commutative ring with one and $I, J \triangleleft R$ two ideals.
The saturated ideal $I^{sat}_J$ with respect $J$ is the ideal
$$(I : J^\infty )= \cup_{n \geq 1} (I:J^n)$$
where $(I:J^n)= \{...

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139 views

### Physical and mathematical significance of the NS-2 brane

This question is about topological string theory and it was also posted in Physics Stack Exchange.
The existence of a new brane called "an NS-2 brane" is predicted in (the second paragraph ...

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65 views

### Bounded form in complex complete manifold

If $\alpha$ is a bounded form in a complex complete manifold $X$ (i.e $\sup_X|\alpha (x) |<\infty$, then $d\alpha$ is it also bounded?
Rq: if $d\alpha$ is bounded then \alpha is not necessary ...

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### Indecomposability of local systems and vector bundles

Let $M$ be a (connected) complex manifold, $L$ be a local system on $M$ and $\mathcal{L}$ the vector bundle associated to $L$. If $L$ is indecomposable, does it imply that $\mathcal{L}$ is also ...

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### Conformal mapping between two right-angled triangles

I want to derive a conformal mapping $f\!:\!A\!\to\! B$ where $A=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq x \}$ and $B=\{ (x,y)\ |\ x\!\in\![0,1]\ \text{ and }\ 0 \leq y \leq \frac{...

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147 views

### Recipe for resolving a coherent sheaf

Let $X$ be a complex manifold and let $V\subset X$ be a subvariety. Let $F\rightarrow V$ be a holomorphic vector bundle over $V$ and let $\mathcal{S}=\Gamma(F)$ be the sheaf of holomorphic section of $...

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374 views

### A diffeomorphism of complex surfaces mapping subvarieties to subvarieties

Let $X$ and $Y$ be smooth projective complex surfaces. If a diffeomorphism from $X$ to $Y$ maps subvarieties to subvarieties does it have to be holomorphic or antiholomorphic? Can we at least verify ...

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### “Holomorphic line bundle” + “algebraic after finite cover” implies “algebraic”?

Let $X$, $Y$ be complex affine algebraic manifolds (closed submanifolds of $\mathbb{C}^n$), let $f\colon Y\to X$ be a finite covering. Let $\mathcal{L}$ be a holomorphic line bundle on $X$. Suppose $f^...

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244 views

### A complex variety with a finite non-abelian simple fundamental group

Does there exist a complex smooth proper variety whose fundamental group is finite non-abelian simple?

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49 views

### Kähler-hyperbolic and Kobayashi-hyperbolic complex manifold

I look for a clear proof of the implication Kähler-hyperbolic in the sense of Gromov implicate Kobayashi-hyperbolic for compact complex manifolds.

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### Exponential of mixed-type End-valued differential form

Let $E\rightarrow \mathbb{P}^1$ be a complex vector bundle and let $a_{(0,0)},a_{(1,0)},a_{(0,1)},a_{(1,1)}$ be differential forms
such that $a_{(i,j)}\in\Omega^{i,j}(\mathbb{P}^1,End(E))$. I would ...

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289 views

### Morphism with connected fibers induce surjection on fundamental groups?

Let $X,Y$ be path-connected finite CW complexes with base points $x_0,y_0$, let $f\colon X\to Y$ be a surjective continuous map, such that for every $y\in Y$, the fiber $f^{-1}(y)$ is path connected. ...

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### Estimates for tensors using local coordinates

Suppose that we have a Kähler manifold $(M, \omega_0)$ and another $(1,1)$ form $\eta$ on $M$. Let $\varphi$ be a smooth function such that $\omega_{\varphi} = \omega_0 + i \partial \bar \partial \...

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### Existence of plurisubharmonic functions on complex manifolds

Let $X$ be a noncompact complex manifold which contains no positive dimensional compact analytic sets.
Conjecture: There must be strictly plurisubharmonic functions on $X$ .
Is it true?

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### Image of transcendental meromorphic functions

Let $f$ be a trancendental meromorphic function such that $f'(z) \ne 0$ for all $z \in \mathbb{C}$. Let $\Pi$ be the stereoprojection map from the north pole on the unit sphere. My question is the ...

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### Geometry of the complex Gauge group

Let $E\rightarrow X$ be holomorphic vector bundle on a complex manifold $X$. Denote by $\mathcal{G}=\Gamma(Aut(E))$ the group of complex smooth automorphisms of $E$.
Is there a way to endow $\...

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81 views

### Relations between Dolbeault cohomology and the corresponding $L^2$-cohomology

Let Dolbeault cohomology and the corresponding $L^2$-cohomology be denoted by $H^{p,q}(X) $
and $H^{p,q}_{(2)}(X)$ respectively.
As is well known, on a compact complex manifold $X$, $H^{p,...

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66 views

### Complex Monge-Ampere equation with degenerate right hand side

Given a Kahler manifold $(X, \omega_0)$, and a smooth function $f$, suppose that I have a smooth solution to the following complex Monge-Ampere equation:
$(\omega_0 +i \partial \bar \partial \varphi)^...

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96 views

### Analytic and algebraic definitions of intersection multiplicity of two complex algebraic curves coincide

There are two definitions of intersection multiplicity of two complex algebraic curves. One is given in An introduction to algebraic curves by Griffiths. Let $t \mapsto (t^k, y(t^k) )$ be the local ...

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### Examples of Stokes data

I'm trying to learn Stokes data but can't find an example to get my teeth into it.
Background. It's well-known that on a complex manifold $X$, there is the Riemann Hilbert equivalence
$$\text{regular ...

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### minimizing weighted length of closed curves

Let $\mathcal{A}$ be the family of closed smooth curves in the right half of the complex plane $\mathbb{C}$ such that any curve in the family must enlose the point $z=1$ and tangent to the $y$-axis at ...

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### Equivariant resolution of singularities with equivariant centres

From what I understand, given a complex projective variety X inside a compact complex manifold Y, according to Hironaka, there is a sequence of $r$ blowups $Y_i$ of Y along complex submanifolds (...

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### Stokes's Theorem with singularities on projective line

Let $X$ be a complex manifold and $\omega\in \Omega(X\times \mathbb{P}^1)$ a form. I met the following identity:
$$\int_{\mathbb{P}^1}(\partial_z\bar{\partial}_z\omega) \log|z|^2=\int_{\mathbb{P}^1}\...

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### Curves on a Kahler manifold

Let $(X,\omega )$ be a compact Kahler manifold. For any $d>0$ are there only finitely many families of curves $C_i$ such that $C_i\cdot \omega <d$? (More precisely, if $C$ is any curve such that ...

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203 views

### Moments of Plücker coordinates on complex Grassmannian

Consider the Grassmannian $Gr(k,N)\simeq U(N)/(U(k)\times U(N-k))$ which parametrizes $k$-dimensional subspaces of $\mathbb{C}^N$. Let us put on it the $U(N)$-invariant probability measure. Let $\...

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### Simple example of non-integrable holomorphic connection

Let $X$ be a complex manifold with complex dimension $d$ and structure sheaf $\mathcal{O}_X$. Let $E$ be a locally free sheaf on $X$. A $holomorphic$ connection on $E$ is a morphism of sheaves
$$\...

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### Tangent bundle with Sasaki metric is Kähler iff $M$ is locally flat

I'm having a hard time proving the following:
If $M$ is an $n$-dimensional indefinite Riemannian manifold whose metric $g$ has index $s$, then the metric of Sasaki $g^{D}$ is an indefinite metric ...

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### Critical points of polarized endomorphisms of algebraic varieties

My main question is the following:
Let $f: \mathbb{CP}^n \to \mathbb{CP}^n$ be a holomorphic endomorphism of degree $d \ge 2$ of $\mathbb{CP}^n$ .
1. Let $X \subset \mathbb{CP}^n$ be an irreducible ...

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258 views

### Vector field with constant divergence around embedded submanifold

Let $M$ be a smooth $n$-dimensional manifold and $N\subset M$ be a closed embedded submanifold of codimension at least $2$. Furthermore, let $\mu$ be a volume form on $M$.
Question: Does there ...

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209 views

### Fun examples relating to Hopf surfaces

A Hopf surface is a compact complex surface whose universal cover is complex analytically isomorphic to $\mathbb{C}^2 \setminus \{ 0 \}$. I would like to know whether anyone has any of the following ...

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### zero extension of positive currents are always positive

In Demailly's Complex Analytic and Differential Geometry page 139:
He said the trivial (zero) extension of the positive current $T$ (on $X\setminus E$), which denoted by $\tilde T$ is always positive ...