Preheat the oven to 325°F (160°C). Grease the bottom of the springform pan and line it with parchment paper.

In a mixing bowl, combine the graham cracker crumbs, melted butter, and sugar. Mix until well combined. Press the mixture firmly into the bottom of the prepared springform pan. Bake for 10 minutes, then remove from the oven and let it cool.

In another mixing bowl, beat the softened cream cheese and granulated sugar together using an electric mixer until smooth and creamy. Add the eggs one at a time, mixing well after each addition. Add the vanilla extract, lemon zest, and sour cream; mix until everything is just combined.

Pour the cream cheese mixture over the cooled graham cracker crust, smoothing the top with a rubber spatula.

To make the blackberry sauce, combine the blackberries, sugar, and lemon juice in a medium-sized pot over medium heat. Cook for about 5 minutes until the blackberries begin to break down. Stir in the cornstarch mixture and cook until the sauce thickens. Remove from heat and let it cool slightly.

Drop spoonfuls of the blackberry sauce on top of the cheesecake filling. Use a knife or a toothpick to swirl the sauce into the filling gently.

Bake the cheesecake in the preheated oven for about 50-60 minutes or until the edges are set and the center is slightly jiggly.

Turn off the oven and leave the cheesecake inside for 1 hour with the door ajar.

Remove the cheesecake from the oven and let it cool to room temperature. Cover and refrigerate for at least 4 hours or overnight before serving.

For best results, allow the cheesecake to chill overnight for a firmer texture.

Top with additional fresh blackberries before serving for an elegant presentation.