“High crimes and misdemeanors” is not strictly a legal concept. The general thrust of the impeachment inquiry is that Trump was using his office in an inappropriate way, rather than he violated specific provisions of the US code.
That said, there are several possible areas of criminal violation.
Federal campaign finance law makes it illegal to “solicit, accept, or receive a contribution or donation” from a foreign national. The question of whether or not the president ran afoul of that law certainly seems relevant in light of what we saw in the transcript. Legality would likely end up hinging on whether or not an investigation into Hunter Biden constitutes a “thing of value” within the meaning of the statute.
There is also federal bribery law, which may cause problems for any public official who “corruptly demands, seeks, receives, accepts, or agrees to receive or accept anything of value personally … in return for … being influenced in the performance of any official act.” This raises the same “thing of value” question as the campaign finance charge. What’s more, the Supreme Court’s 2016 decision in McDonnell v. United States adopted an extremely narrow view of what constitutes an “official act.” A prosecutor would likely need to show a very direct and explicit linkage of military aid for Ukraine to Trump’s demands, while his defense could always note that the aid did eventually flow even without a Biden investigation.
What Trump was doing on the call seems to fit the ordinary language description of extortion pretty well, but the relevant federal statute here is the Hobbs Act. This specifically references “obtaining of property from another, with his consent, induced by wrongful use of actual or threatened force, violence, or fear, or under color of official right.” Making the case that a Biden investigation would be a form of “property” seems harder than the “thing of value” requirement for the first two options.
Last, but by no means least, there is always the old standby of obstruction of justice. Trump has engaged in an array of counter-investigative techniques — from storing the transcript of the call on a codeword-classified server to initially blocking the release of the whistleblower complaint to threatening the whistleblower to refusing to cooperate with the impeachment inquiry — that could potentially be construed as efforts to obstruct justice.
All of these seem somewhat tenuous as legal cases. An impeachment proceeding, however, is not a legal case. Congress is under no particular obligation to stick to the Supreme Court’s narrow view of official acts, be fussy about the meaning of “thing of value,” or take the view that property is the only thing that can be extorted.
House Democrats ultimately settled on an obstruction charge plus a catch-all claim of “abuse of power,” arguing not that Trump’s conduct violated a specific provision of the criminal code but rather his overall constitutional responsibilities.