**4**

votes

**2**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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 ...