While there are legal precedents to the contrary, he can not cut the tree down since he only owns one half of it. So you would have a good case unless he has a urgent issue to deal with (safety, health, etc..).
Nor can he cut down his one half since it would severly damage your half and might cause it to die and fall (liability issue).
Typically the Cities have no laws regarding the removal of trees in private property.. (but some do so check)
So.. its poop or get off the pot time for you..
1) Write him a letter telling him you do not want the tree removed and cite the shade and the added value to your home as reasons as well as your concern for the safety issue if he only removes his half. Deny him permission in writing.
2) If they start to remove it, you can call the police since it requires tresspassing to cut it down and it also is damaging your property.
3) Or you can go with the flow, and let the tree come down - especially if he is paying for it- and this winter enjoy some seasoned firewood.
Your choice