0
votes
1answer
87 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 …
4
votes
1answer
91 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 cont …
0
votes
1answer
175 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 …
2
votes
1answer
184 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 …
1
vote
1answer
189 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 hol …
0
votes
1answer
240 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 …
4
votes
0answers
92 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 …
3
votes
1answer
111 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 …
8
votes
2answers
284 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 trivia …
1
vote
0answers
91 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 $\varphi(t)\in\Gamma_ …
0
votes
1answer
207 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:
http://en …
0
votes
0answers
146 views
Clifford algebra on almost product structure
Is the algebra defined by $J^2=1$,i.e. (algebra defined on almost product structure ) Clifford algebra?
2
votes
1answer
81 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 wit …
6
votes
1answer
210 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 $1$-form …
0
votes
2answers
266 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 $(0,1)$-ta …

