President Lincoln was reluctant to defend Fort Sumter because Virginia and other loyal states may end up seceding.
Fort Sumter was regarded as a property of the United States by those in the North, while those in the South regarded it as a property of the New confederate.
Lincoln's reluctance was borne out of the thought that if the Fort was attacked, certain states like Virginia and North Carolina would secede and join the separatists.
He eventually asked for Union troops after he came up with the plan not to attack except South Carolina attacked them first. The South ended up opening fire first and Lincoln's troop responded.