jump to navigation

Complex analysis poser March 25, 2011

Posted by David Speyer in Uncategorized.
5 comments

The following question occurred to me while writing lectures last night and distracted me from productive work all morning. The least I can do is distract you all in turn.

Let E be an elliptic curve over \mathbb{C}. Let x_1 and x_2 be two points of E such that x_1 - x_2 is not torsion. It is well known that there are no nonconstant meromorphic functions on E which have neither poles nor zeroes in E \setminus \{ x_1, x_2 \}.

Are there any nonconstant holomorphic functions at E \setminus \{ x_1, x_2 \} at all, where we allow essential singularities at z_1 and z_2?

How to think about Hodge decomposition March 23, 2011

Posted by David Speyer in Uncategorized.
5 comments

UPDATE: Greg Kuperberg writes the same things with fewer typos here.

This blog post is meant for me to work through some things I want to present in class tomorrow, so it won’t have as much background as I usually try to include.

Let X be a compact complex manifold with a metric. Then we have operators \partial, \overline{\partial}, \partial^* and \overline{\partial}^*. These go from (p,q)-forms to (p+1,q), (p,q+1), (p-1,q) and (p,q+1)-forms respectively. They obey

\displaystyle{\partial^2=0}, \displaystyle{\overline{\partial}^2=0}, \displaystyle{(\partial^*)^2=0}, \displaystyle{(\overline{\partial}^*)^2=0},

\displaystyle{\partial \overline{\partial} = - \overline{\partial} \partial} and \displaystyle{\partial^* \overline{\partial}^* = - \overline{\partial}^* \partial^*}  (1).

The exterior derivative d is \partial + \overline{\partial}, the Hodge dual formula is d^* = \partial^* + \overline{\partial}^*. We define three Laplacians: \Delta_d = d d^* + d^* d, \Delta_{\partial} = \partial \partial^* + \partial^* \partial and \Delta_{\overline{\partial}} = \overline{\partial} \overline{\partial}^* + \overline{\partial}^* \overline{\partial}.

So, in general,

\displaystyle{ \Delta_d = (\partial+\overline{\partial}) (\partial^* + \overline{\partial}^*) + (\partial^* + \overline{\partial}^*) (\partial+\overline{\partial}) = }

\displaystyle{\Delta_{\partial} + \Delta_{\overline{\partial}} + (\partial \overline{\partial}^* + \overline{\partial}^* \partial) + ( \overline{\partial} \partial^* + \partial^* \overline{\partial})}.

When X is Kähler, we have \Delta_{\partial} = \Delta_{\overline{\partial}} = (1/2) \Delta_{d}. This identity is actually made up out of the following identities, which strike me as more fundamental:

\displaystyle{ \partial \overline{\partial}^* + \overline{\partial}^* \partial =0} and \displaystyle{ \overline{\partial} \partial^* + \partial^* \overline{\partial}=0}   (2).

\displaystyle{ \partial \partial^* + \partial^* \partial = \overline{\partial} \overline{\partial}^* + \overline{\partial}^* \overline{\partial} }   (3).

The two quantities in the third equation are (by definition) \Delta_{\partial} and \Delta_{\overline{\partial}}; we’ll denote them both by \Delta. So \Delta takes (p,q)-forms to (p,q)-forms.

Also, \Delta commutes with all of \partial, \overline{\partial}, \partial^* and \overline{\partial}^*, this is an easy consequence of (1) and (3) (exercise!).

This is all standard. The rest is something that I haven’t seen written down, but strikes me as making the theory much simpler.
(more…)

Follow

Get every new post delivered to your Inbox.

Join 90 other followers