My understanding is that, since the Senate is in recess, it would take unanimous consent for the Senate to start a trial before January 20. Assuming the House moves ahead with impeachment, Chuck Schumer might try to get unanimous consent at a scheduled Senate pro forma session next Thursday, but if so, a Republican will almost inevitably object.
This would push an impeachment trial past Trump’s presidency, and mean that Senate Democrats would have to largely spend the initial days and weeks of Biden’s presidency litigating Donald Trump’s misconduct, rather than confirming Biden appointees or passing Biden legislation.
They may decide that it’s worth it anyway, but it will mean that Biden’s promise to turn the page from Trump and return normality to Washington would be delayed, when a president usually wants to rush to get things done in the early days when his influence is at its highest.