Tagged Questions

0
votes
1answer
109 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 $H^2(X,\mathbb{R})$, $$c_1(\omega_X) = - c_1(T_X)$$ (O …
0
votes
1answer
113 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 …
6
votes
1answer
162 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
0answers
63 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 fol …
1
vote
1answer
174 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 …
0
votes
1answer
186 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
186 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
193 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
243 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 …
8
votes
2answers
287 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 …
3
votes
1answer
115 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 …
4
votes
0answers
93 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 …
1
vote
0answers
93 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
209 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
149 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?

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