Prepare the Crust : Preheat the oven to 325°F (165°C). Grease a 9-inch springform pan. Combine graham cracker crumbs and melted butter in a bowl until well mixed. Press the mixture firmly into the bottom of the pan to form the crust. Set aside.

Make the Blueberry Swirl : In a small saucepan over medium heat, combine blueberries, water, and sugar. Stir occasionally until the mixture simmers. Dissolve cornstarch in 1 tablespoon of water and add it to the blueberry mixture. Stir until thickened. Remove from heat and allow to cool.

Prepare the Filling : In a mixing bowl, beat the cream cheese and sugar until smooth and fluffy. Add sour cream, eggs, and vanilla extract, mixing until fully incorporated. Stir in black food coloring until the mixture reaches a rich, deep hue.

Assemble the Cheesecake : Pour the filling over the crust in the springform pan. Spoon dollops of the cooled blueberry mixture onto the filling and swirl gently with a toothpick or knife for a marbled effect.

Bake and Cool : Bake for 50 minutes, or until the edges are set and the center jiggles slightly when shaken. Turn off the oven and leave the cheesecake inside with the door slightly ajar for 1 hour to cool gradually. Transfer to the refrigerator to chill for at least 4 hours or overnight.

Serve and Enjoy : Carefully remove the cheesecake from the pan and slice it into portions. Garnish with fresh blueberries or a drizzle of the remaining blueberry compote for added flair.