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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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 …

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_ …

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 …

Is the algebra defined by $J^2=1$,i.e. (algebra defined on almost product structure ) Clifford algebra?