In .NET, talking to a database is handled with ADO.NET, which uses something called "connection pooling". The connection pool is basically a collection of open connections to your database that ADO.NET manages for you. In your code, when you create and open a Connection object, ADO.NET first looks in the connection pool to see if it already has an open connection to your data source, and if it finds one it uses that (instead of actually creating and opening a new connection). When you close your connection, ADO.NET does not really close it, but instead returns it to the connection pool.
Therefore, you do not need (and in face do not want) to maintain open connection objects inside your application (in a global variable or anywhere). The correct approach with data access in ADO.NET is to create and Open a Connection object, do whatever you need to do with the database, and then Close and Dispose your Connection.