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.

Filter by
Sorted by
Tagged with
2
votes
1answer
72 views

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 ...
4
votes
0answers
82 views

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 ...
2
votes
0answers
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, \...
3
votes
0answers
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 ...
1
vote
1answer
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 $F=\sqcup_{\alpha \in A} F_{\alpha}$ ...
5
votes
0answers
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 ...
3
votes
0answers
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?
5
votes
0answers
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 ...
5
votes
1answer
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 ...
5
votes
1answer
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 ...
-1
votes
0answers
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 ...
1
vote
0answers
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 $\theta>0$...
3
votes
0answers
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 ...
2
votes
0answers
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 ...
8
votes
1answer
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 $$ \...
1
vote
1answer
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 ...
2
votes
1answer
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}...
2
votes
0answers
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 ...
1
vote
0answers
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 ...
1
vote
0answers
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)= \{...
7
votes
0answers
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 ...
3
votes
1answer
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 ...
2
votes
1answer
109 views

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 ...
3
votes
1answer
89 views

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{...
0
votes
0answers
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 $...
8
votes
1answer
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 ...
5
votes
1answer
130 views

“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^...
3
votes
1answer
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?
1
vote
0answers
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.
0
votes
0answers
34 views

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 ...
8
votes
2answers
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. ...
2
votes
0answers
72 views

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 \...
4
votes
1answer
149 views

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?
-1
votes
1answer
62 views

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 ...
1
vote
0answers
37 views

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 $\...
1
vote
1answer
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,...
3
votes
0answers
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)^...
3
votes
0answers
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 ...
11
votes
2answers
269 views

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 ...
4
votes
1answer
88 views

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 ...
2
votes
0answers
80 views

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 (...
3
votes
1answer
162 views

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}\...
2
votes
1answer
145 views

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 ...
14
votes
0answers
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 $\...
2
votes
1answer
95 views

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 $$\...
3
votes
1answer
99 views

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 ...
4
votes
1answer
197 views

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 ...
7
votes
2answers
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 ...
4
votes
1answer
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 ...
1
vote
0answers
58 views

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 ...

1
2 3 4 5
46