This is something I always forget exists and has a name, so I end up reproving it. Since this sequence of posts is a hodge-podge of things to help me take a differential geometry test, hopefully this will lodge the result in my brain and save me time if it comes up. I'm not sure… Continue reading Naturality of Flows

# Category: manifolds

## Lie groups have abelian fundamental group

Last year I wrote up how to prove that the fundamental group of a (connected) topological group was abelian. Since Lie groups are topological groups, they also have abelian fundamental groups, but I think there is a much neater way to prove this fact using smooth things. Here it is: Lemma 1: A connected Lie… Continue reading Lie groups have abelian fundamental group

## PDE’s and Frobenius Theorem

I've started many blog posts on algebra/algebraic geometry, but they won't get finished and posted for a little while. I've been studying for a test I have to take in a few weeks in differential geometry-esque things. So I'll do a few posts on things that I think are usually considered pretty easy and obvious… Continue reading PDE’s and Frobenius Theorem

## The Cohomology Computation

Alright, I'm in a sort of tough spot. Yesterday I started typing this up, but I just don't have the motivation. There are lots of tedious details that no one is going to read and will not come up in our study after this. It is all incredibly standard chasing Fourier coefficients around, so I'm… Continue reading The Cohomology Computation

## Cohomology of Abelian Varieties II

Two posts in the same week! Before we get started today, we need to introduce one more piece of new notation. Let $latex {\overline{T}}&fg=000000$ be the $latex {\mathbb{C}}&fg=000000$-antilinear maps $latex {V\rightarrow \mathbb{C}}&fg=000000$. Our goal is to prove that $latex {H^q(X, \mathcal{O}_X)\simeq \bigwedge^q\overline{T}}&fg=000000$ and $latex {H^q(X, \Omega^p)\simeq \bigwedge^pT\otimes \bigwedge^q\overline{T}}&fg=000000$. To do the calculation we will use… Continue reading Cohomology of Abelian Varieties II

## Cohomology of Abelian Varieties

Hopefully I'll start updating more than once a month. Since it's been awhile and the previous post was tangent to what we're actually doing, I'll recap some notation. $latex {X}&fg=000000$ will be a compact complex (connected) Lie group of dimension $latex {g}&fg=000000$. We showed that we have an analytic isomorphism $latex {X\simeq (S^1)^{2g}\simeq (\mathbb{R}/\mathbb{Z})^{2g}}&fg=000000$. Let… Continue reading Cohomology of Abelian Varieties

## Complex Lie Group Properties

Today we'll do two more properties of compact complex Lie groups. The property we've already done is that they are always abelian groups. We go back to the notation from before and let $latex X$ be a compact complex Lie group and $latex V=T_eX$. Property 1: $latex X$ is abelian. Property 2: $latex X$ is… Continue reading Complex Lie Group Properties