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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4
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2answers
261 views

Is it impossible for the dimension of a topological space to increase under a smooth map?

First let me make a definition. Let $M$ be a smooth manifold and $S \subset M $ a topological subspace of $M$. We say that $S$ has "dimenion" at most $k$ if $S$ is a subset of $$ X_1 \cup X_2 \ldots ...
2
votes
1answer
123 views

Embeddings of of quotient singularities

Let $G$ be a finite subgroup of $SL(n,\mathbb{C})$. Let $G$ act on $\mathbb{C}^{n}$ with the action induced by $SL(n,\mathbb{C})$ and the induced action of $G$ on the unit sphere $S^{2n-1}$ is free. ...
6
votes
1answer
261 views

interpolation with derivative of rational fraction

Studying a problem in conformal geometry, I am facing to the following interpolation problem. Let $P$ and $Q$ two coprime polynomials. Then let $A$ and $B$ two coprime polynomials such that ...
2
votes
0answers
117 views

Is there a way to compute explicitly global sections of tangent sheaf to a projective hypersurface?

Let $f\in \mathbb{C}[z_{0},\ldots,z_{n}]$ be an irreducible homogeneous polynomial and $X=V(f)\subset\mathbb{P}^{n}$ the projective hypesurface associated. I want to find (if any) explicit generators ...
0
votes
1answer
182 views

Opposite complex structure on Kaehler manifold

Hallo, Let $(M,J)$ be a Kaehler manifold. How can one descride the opposite complex structure? What is the precise definition of the opposite complex structure? Can one descride the opposite complex ...
8
votes
1answer
200 views

Dimension of eigenspaces of Laplacian on a compact Riemannian manifold

Let $M$ be a compact smooth manifold, let $g$ a riemannian metric and let $\Delta_{g}$ the Laplacian operator on functions induced by $g$. Is there a (topological?) bound on the dimension of $n$-th ...
2
votes
2answers
182 views

Elementary question about Isotopy (in the definition of a Teichmuller space)

Disclaimer - I don't have much experience in topology/complex geometry, so I apologize if what I'm asking is too elementary for this site. Let $S$ be some orientable surface obtained by removing ...
2
votes
1answer
200 views

Local expression involved in the definition of positivity of vector bundles

This is question follows on from this one. In the linked question, the hermitian form $\theta_E$ on $T^{1,0}X\otimes E$ is defined globally as ...
0
votes
0answers
86 views

Can all varieties with given hilbert polynomial be rigid

Let $X$ be a canonically polarized variety with hilbert polynomial $h$. Does there exist a non-rigid canonically polarized varietz with hilbert polynomial $h$? When is this the case, and when is ...
1
vote
1answer
36 views

Relations between the Veech group and the automorphism group

Let $(X,\omega)$ be a compact (half-)translation surface: $X$ is a compact riemann surface and $\omega$ a (quadratic) differential on it. One can define a distinguished local coordinate $z$ on $X$: ...
4
votes
1answer
102 views

Are there only finitely varieties of general type dominated by a given variety in the following sense

Let $X$ be a smooth projective complex algebraic variety. I believe the answer to the following question should be well-known. It is an instance of the Iitaka conjecture. There are only finitely many ...
3
votes
1answer
174 views

Why is the base change functor faithful

Let $L/k$ be a field extension of algebraically closed fields of characteristic zero. Let $U$ be a smooth quasi-projective variety over $k$. I am trying to understand why the base-change functor from ...
5
votes
1answer
103 views

Fundamental groups of normal complex quasi-projective varieties

I would like to know if there is an explicit example of a finitely presented group that can not be realised as the (topological) fundamental group of a normal complex quasi-projective variety?
0
votes
1answer
114 views

upper bound on product of distances from points on a circle

Let C be a circle of radius 1 in the complex plane with n points on the boundary. Provide an upper bound on the product of the distances of a given point on the circle to the other n points. The goal ...
4
votes
1answer
181 views

Branch locus along a smooth curve

Let $f : S \to X$ be a dominant morphism of smooth complex compact surfaces. Let $C \subset S$ be a smooth curve such that $df$, seen as a map from $T_S \to T_X$, is generically of rank $1$ along $C$ ...
1
vote
1answer
287 views

Decomposition of hermitian form used in the definition of Griffiths/Nakano positivity

Let $E$ be a hermitian holomorphic vector bundle over a complex manifold $X$. Then $\Theta(E)$, the curvature of $E$, is a section of $\bigwedge^{1,1}X\otimes\operatorname{End}(E)$. However, we have ...
4
votes
2answers
354 views

Is there an analogous concept for the degree of a map, when the spaces are singular?

Let $M$ and $N$ be two smooth compact, oriented manifolds and $X\subset M$ an oriented submanifold of $M$ of dimension $k$ (not necessarily closed). Suppose in addition that $\bar{X}-X$ is contained ...
2
votes
0answers
118 views

Kahler cone of a product

Let $(M,J)$ be a complex manifold. We say that $\omega\in H^2(X;\mathbb{R})$ is Kahler if and only if $\omega$ is closed, positive [i.e. $\omega(v,Jv)>0, \forall v\neq0$], and a $(1,1)$-form [i.e. ...
3
votes
1answer
236 views

How to determine “genericness” of an element of a family of algebraic varieties?

Given a (flat) family of complex algebraic varieties $X_t$ (say parametrized by $\mathbb{C}$) and a specific $t_0$, how does one proceed to check if $X_{t_0}$ is a 'generic element'? More precisely, ...
4
votes
0answers
147 views

proper mapping between Stein manifolds

My question is the following: Let $b:X \rightarrow Y$ be a proper holomorphic mapping between two Stein manifolds $X$ and $Y$. For obvious reasons, $b^{-1}(y)$ are finite subsets of $X$. Is the set ...
4
votes
1answer
170 views

What is the discriminant divisor of a surface fibered over a curve?

Let $\pi:X\rightarrow C$ be a flat and proper morphism over $\mathbb{C}$ where $X$ is a smooth projective surface and $C$ is a smooth projective curve. Assume that all the fibers of $\pi$, except ...
1
vote
2answers
191 views

Deforming to decompose vector bundles

After edit: How do we show that for every (holomorphic) vector bundle over a curve, is it possible to deform it to another one which is decomposable (into line bundles)? Before edit: I am not sure ...
18
votes
1answer
355 views

Biholomophic non-Algebraically Isomorphic Varieties

Recently, when writing a review for MathSciNet, the following question arose: Is it true that two smooth complex varieties that are biholomorphic are algebraically isomorphic? The converse is true ...
3
votes
1answer
111 views

What can a quartic surface in $\mathbb{P}^3$ with an ordinary quadruple point look like?

All varieties will be projective and over $\mathbb{C}$. If $S$ is any surface in $\mathbb{P}^3$ of degree 2 that posseses an ordinary double point, it follows easily that $S$ is projectively ...
0
votes
1answer
324 views

First Chern class of canonical bundle ?

This is a somewhat simple question: consider a complex manifold $M$ and its canonical bundle $\omega_X$. It is clear that in $H^2(X,\mathbb{R})$, $$c_1(\omega_X) = - c_1(T_X)$$ (Obvious using ...
0
votes
1answer
332 views

When is an ample line bundle on an abelian variety base point free?

So, any line bundle $L$ on an abelian variety $X$ determines a type $(d_1,\ldots,d_g)$ where $d_i|d_{i+1}$. It's well known that if $d_1\geq 3$ then $L$ defines an embedding, that if $L$ has no fixed ...
0
votes
0answers
93 views

homology class of a rational curve

Let $X\subseteq\mathbb{C}P^n$ be s smooth variety. Let $C\subseteq X$ be an algebraic rational curve [i.e. an algebraic curve which is birational to $\mathbb{C}P^1$]. In what follows, we consider ...
1
vote
1answer
229 views

$P^1$ minus k points

For $k\geq 3$, and $k$ arbitrary points $S=( z_1,\cdots,z_k ) \in \mathbb{P}^1$, we can write $$ P^1 \setminus S \cong \mathbb{H}/G $$ where $\mathbb{H}$ is the upper-half plane and $G\subset ...
7
votes
1answer
229 views

Sheaves on Contractible Analytic Spaces

Let $(X,\mathcal{O}_X)$ be a contractible complex analytic space. Suppose that $\mathcal{F}$ is a coherent sheaf of $\mathcal{O}_X$-modules. Can we invoke the fact that $X$ is contractible to ...
1
vote
1answer
288 views

Hartogs Theorem and Canonical Bundles

Let $X$ be a normal complex affine algebraic variety. Suppose that $Y$ is an open subvariety of $X$, and that the codimension of $X\setminus Y$ in $X$ is at least $2$. One version of the Hartogs ...
2
votes
1answer
276 views

There are many varieties with ample canonical bundle

Let $X$ be a smooth projective connected complex algebraic variety with ample canonical bundle. Let $h$ be the hilbert polynomial of the canonical bundle. Why is the moduli stack of canonically ...
4
votes
0answers
121 views

Vector bundles on Stein Manifolds

This might be standard if true (if so, I shall be grateful if provided with a reference). Given a smooth map from a Stein manifold X to Gr(k,n) (the Grassmannian of k planes in C^n), is there a ...
4
votes
1answer
198 views

Hodge classes and Leray filtration

Let $f :X \to Y$ be a submersion between smooth projective varieties over $\mathbb{C}$ and let $\alpha \in Z^k(X)$ be an algebraic cycle of $X$. Is is true that for all odd numbers $p$ and $q$ such ...
0
votes
1answer
311 views

Dolbeault cohomology

Let $X=S^{2n-1} \times S^1$. I have to compute $H^{(1,0)}_{\bar{\partial}} (X)$ and $H^{(0,1)}_{\bar{\partial}}(X)$ . I don't know how to do this but if we use Kunnet formula we have that ...
2
votes
0answers
262 views

A modification of Maurer-Cartan equation

In deformation theory of complex structure, the Maurer-Cartan equation takes the form $$\bar{\partial}\varphi(t)+\frac{1}{2}[\varphi(t),\varphi(t)]=0.$$ where ...
3
votes
1answer
326 views

A “Riemannian” analogue of Kobayashi metric?

Recall that Kobayashi metric is defined on any complex manifold $M$. This is a pseudo-metric according to which a tangent vector $v$ at $P$ has length at most $1$ if there is holomorphic map from ...
0
votes
1answer
305 views

Question on Hartogs's Extension Theorem

Does Hartogs's extension theorem hold if one replaces the word holomorphic by analytic (of course still in several variables)? For Hartogs's Extension Theorem see here: ...
8
votes
2answers
407 views

Proving that a generic variety with ample canonical bundle has no automorphisms

Let $X$ be a smooth projective connected variety over the complex numbers with ample canonical bundle. If $X$ is generic and $\dim X \leq1$, the automorphism group of $X$ is trivial, see for instance ...
2
votes
1answer
106 views

Reference for the classification of (singular) degree 4 surfaces in $\mathbb{P}^3_{\mathbb{C}}$?

I was told singular quartic algebraic surfaces in $\mathbb{P}^3_{\mathbb{C}}$ have been completely classified and their singularities have been described. Can anyone provide me with a resource where ...
1
vote
0answers
236 views

The relation between the weak Lefschetz theorem and the strong Lefschetz theorem

The Weak Lefschetz Theorem states that for a compact Kahler manifold, $Pic(X) \rightarrow H^{1,1}(X, \mathbb{Z})$ is surjective. The Hard Lefschetz Theorem states that for a compact Kahler manifold ...
6
votes
1answer
236 views

$H^1(X,O_X)$, holomorphic $1$-forms, and $b_1(X)/2$ for normal $X$.

Suppose $X$ is a normal projective variety over $\mathbb C$. In the case $X$ is smooth according to Hodge theory $h^1(X,O(X))$ is the dimension of the space of holomorphic $1$-forms on $X$ and this ...
3
votes
1answer
167 views

Trivial canonical bundle of a Ricci-flat, simplyconnected Kähler manifold

Hallo, I have two questions where I do not really know how to deal with them. Let $(M,J,g)$ be a Kähler manifold, where $g$ is the Riemannian metric and denote by $\omega(\cdot , \cdot) = g(J \cdot ...
1
vote
1answer
100 views

A question on real surfaces on K3 surfaces.

Let $X$ be a K3 surface and $\omega$ be a nowhere vanishing 2-form on $X$. Suppose $Y\subset X$ be a smooth real surface. How can one see that $\omega|_Y=0$ implies $Y$ is a complex submanifold (a ...
1
vote
0answers
147 views

A question on the SYZ mirror symmetry

A toy SYZ mirror symmetry is described as follows. Let $X$ be a $n$-dimensional compact manifold. The contangent bundle $T^*X$ has a natural symplectic structure locally given by the $2$-form ...
4
votes
0answers
118 views

Closed subschemes and the analytification functor

Let $X$ be a scheme of finite type over $Spec(\mathbb C)$ and let $i:Y \hookrightarrow X$ be a closed subscheme, defined by a sheaf of ideals $J \subset \mathcal O_X$. Then there is an induced map ...
0
votes
0answers
79 views

Ricci-flat Kähler metrics on symmetric varieties

Hallo, I have a question on a paper of Azad and Kobayashi "Ricci-flat Kähler metrics on symmetric varieties". Here is the link: ...
0
votes
0answers
97 views

Some question on the paper “Ricci-flat metrics on the complexification of a compact rank one symmetric space”

I am reading the paper "Ricci-flat metrics on the complexification of a compact rank one symmetric space" by Stenzel (here is the ...
1
vote
1answer
73 views

$SU(n)$-invariant subring of $\Lambda^{*}\mathbb{R}^{2n}$

I have the following question: Let $R \subset \Lambda^{*}\mathbb{R}^{2n}$ be the sub-ring of forms which are preserved by $SU(n)$. How can one show that this subring is generated by $\Omega_{0}$ and ...
0
votes
2answers
335 views

A Question on Deformation of Complex Structure

Let's consider a Riemann surface $M$. The $(0,1)$-tangent bundle is locally spanned by $\frac{\partial}{\partial z}$. Suppose we have a deformation of $M$, then the new $(0,1)$-tangent bundle is ...
8
votes
0answers
209 views

Homology classes represented by $J$-holomorphic curves

Let $\Sigma$ be a compact Riemann surface with complex structure $j$. Let $(M,J)$ be an almost complex manifold. A map $u: \Sigma \rightarrow M$ is called $J$-holomorphic if $$ du \circ j = J \circ ...