In a medium bowl, whisk together 1 cup pumpkin puree, ½ cup heavy cream, 2 teaspoons vanilla extract, 1 teaspoon cinnamon, ½ teaspoon ginger, ¼ teaspoon nutmeg and ¼ teaspoon salt. Set aside.
In another medium bowl, beat 1 ¼ pounds cream cheese at medium low speed until creamy, about 2 minutes, scraping down the sides of the bowl as necessary. Gradually add ½ cup granulated sugar, ½ cup brown sugar and beat until blended. Add the pumpkin mixture and mix again until blended. Add 1 Tablespoon cornstarch and mix just until combined. Add 4 eggs one at a time, mixing just to incorporate after each addition and scraping down the sides of the bowl as necessary. After all the eggs are incorporated, continue to beat the filling for an additional minute to ensure the mixture is smooth.
Scrape the batter over the cooled crust. Place the wrapped pan in a roasting pan or large baking pan. Pour enough hot water into the roasting pan to come up just 1 inch of the sides on the springform pan. Bake the cheesecake in the water bath for 65-75 minutes, until the center of the cake is set but still slightly wobbly (the cake will set completely as it cools).
Remove the cake pan from the water bath, place the pan on a wire rack, and carefully loosen the foil. Let the cake cool completely.
Refrigerate the cheesecake for at least 4 hours before serving, but 24-48 hours is ideal.