# Tagged Questions

Complex geometry is the study of complex manifolds and complex algebraic varieties. It is a part of both differential geometry and algebraic geometry.

417 views

97 views

### What's Griffiths' ampleness in modern language?

When reading Griffiths' paper "Hermitian differential geometry, Chern classes and positive vector bundles", I found his definition of ampleness: Let $X$ be a compact complex manifold, $E\to X$ a ...
262 views

100 views

### Condition for a complex vector bundle to be holomorphic?

Suppose that $(E,\pi,M)$ is a complex vector bundle. I've seen it suggested in a few places that if $E$ and $M$ are complex manifolds, and $\pi$ is a holomorphic map, then $(E,\pi,M)$ is in fact a ...
293 views

### Exercise $3.1.7$ from Huybrechts' Complex Geometry: An Introduction

I am trying to do the following question: Show that $L$, $d$, and $d^*$ acting on $\mathcal{A}^*(X)$ of a Kähler manifold $X$ determine the complex structure of $X$. Here $L$ is the Lefschetz ...
72 views

### Automorphism on a kahlerian variety which acts as -id on $H^2(X,\mathbb{Z})$

I've read this proposition: "there is no automorphism (biholomorphic map) of a Kahlerian complex manifold $X$ which acts on $H^2(X,\mathbb{Z})$ as $-$identity" so I tried to prove this statement and ...
216 views

### Why do algebraic varieties contain curves passing through two given points

Let $X$ be an algebraic variety over the complex numbers. My definition of an algebraic variety is a finite type separated $\mathbf C$-scheme. Someone told me that such varieties have the following ...