Preheat oven to 325°F (163°C) and grease a 9-inch springform pan.

Combine graham cracker crumbs, melted butter, and 1/4 cup sugar; press firmly into the bottom of the pan.

Bake crust for 10 minutes, then remove and allow to cool.

In a small saucepan, simmer blackberries, 2 tablespoons sugar, and lemon juice until berries break down. Whisk in cornstarch, cook for 1 minute, then strain through a fine-mesh sieve to remove seeds.

In a large bowl, beat cream cheese and remaining sugar until smooth and aerated.

Incorporate sour cream and vanilla extract.

Add eggs one at a time, beating on low speed until just combined to avoid over-incorporating air.

Pour the cheesecake batter over the cooled crust.

Drop small spoonfuls of the blackberry puree onto the batter and use a thin knife or skewer to create swirl patterns.

Bake for 65 to 75 minutes until the edges are set and the center retains a slight jiggle.

Turn off the oven, crack the door, and let the cheesecake cool inside for 1 hour.

Refrigerate for at least 6 hours or overnight before serving.