Preheat the oven to 325°F (163°C).

Wrap the bottom and sides of a 9-inch springform pan in a double layer of heavy-duty foil (this keeps water out if you do a water bath).

In a bowl, combine graham cracker crumbs, sugar, melted butter, and a pinch of salt until the crumbs are evenly moistened.

Press the mixture into the bottom (and slightly up the sides) of the springform pan. Use the bottom of a measuring cup to pack it down firmly.

Bake the crust for 8–10 minutes, then let it cool while you make the filling.

In a small saucepan, combine blackberries, raspberries, sugar, and lemon juice. Cook over medium heat, stirring occasionally, until berries break down and the mixture is saucy (about 6–8 minutes).

If you prefer a thicker swirl, stir in the cornstarch slurry and cook 1–2 more minutes until glossy and slightly thickened.

Press the mixture through a fine-mesh sieve to remove seeds for a smooth puree, or skip this step if you like texture.

Chill the puree while you prepare the filling (cold swirl will hold better when layered).

Beat the room-temperature cream cheese and sugar together on medium speed until smooth and creamy. Scrape down the sides so nothing is lumpy.

Add the eggs one at a time, mixing on low speed just until combined after each addition. Overbeating will add air and can cause cracks.

Mix in sour cream, vanilla, and lemon zest until silky. Finish with a gentle scrape of the bowl.

Pour about two-thirds of the filling into the cooled crust.

Spoon about 1/3 cup of berry puree into the filling in several spots. Add the remaining filling, then drizzle more puree on top.

Use a thin knife or skewer to gently swirl the berry puree through the top of the cheesecake. Don’t overdo it — you want pretty ribbons, not marble chaos.

Place the foil-wrapped springform pan into a larger baking dish or roasting pan. Place in the oven, and carefully pour hot water into the larger pan to reach about halfway up the side of the springform (this creates a gentle, even bake).

Bake for 50–65 minutes, until the edges are set and the center has a slight jiggle (it will set more as it cools).

Turn off the oven and crack the door open. Let the cheesecake rest in the oven for 30 minutes to cool gradually — this reduces cracks.

Remove from the oven, lift the cheesecake from the water bath, and let it cool to room temperature on a wire rack. Then chill in the fridge for at least 6 hours or overnight.

Run a thin knife around the edge of the pan before releasing the springform.

Top with fresh berries and a dusting of powdered sugar or a light berry glaze if desired.

Slice with a knife dipped in hot water and wiped dry between cuts for clean edges.