Preheat oven to 325°F (165°C) and prepare a 9-inch springform pan by wrapping the outside with heavy-duty aluminum foil.

In a bowl, combine graham cracker crumbs and sugar, then stir in melted butter until evenly moistened.

Press mixture firmly into the bottom and 1-inch up the sides of the springform pan.

Bake for 10 minutes, then remove and cool while preparing the filling.

Combine blackberries, sugar, and lemon juice in a small saucepan over medium heat.

Simmer for 5 minutes, then mash berries with a fork.

Whisk cornstarch with 1 tablespoon water and stir into the blackberry mixture.

Cook for another 2 minutes until thickened, then strain through a fine mesh sieve.

Cool completely before using.

In a large bowl, beat cream cheese until smooth and creamy.

Add sugar and beat until well combined.

Add eggs one at a time, mixing well between each addition.

Mix in sour cream, vanilla extract, lemon zest, and lemon juice.

Pour filling over cooled crust.

Drop spoonfuls of blackberry puree on top and swirl with a knife or toothpick.

Place the foil-wrapped springform pan in a larger roasting pan and add hot water halfway up the sides.

Bake for 60-70 minutes until the center is almost set but still slightly jiggly.

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

Remove from oven, run a knife around the edges, and cool completely.

Refrigerate for at least 4 hours or overnight before serving.