We use the Suita conjecture (now a theorem) to prove that for any domain \(\Omega \subset \mathbb {C}\) its Bergman kernel \(K(\cdot , \cdot )\) satisfies \(K(z_0, z_0) = \hbox {Volume}(\Omega )^{-1}\) for some \(z_0 \in \Omega \) if and only if \(\Omega \) is either a disk minus a (possibly empty) closed polar set or \(\mathbb {C}\) minus a (possibly empty) closed polar set. When \(\Omega \) is bounded with \(C^{\infty }\)-boundary, we provide a simple proof of this using the zero set of the Szegö kernel. Finally, we show that this theorem fails to hold in \(\mathbb {C}^n\) for \(n > 1\) by constructing a bounded complete Reinhardt domain (with algebraic boundary) which is strongly convex and not biholomorphic to the unit ball \(\mathbb {B}^n \subset \mathbb {C}^n\).

我们使用Suita猜想（现在是一个定理）证明对于任何域\（\ Omega \ subset \ mathbb {C} \）其Bergman核\（K（\ cdot，\ cdot）\）满足\（K（z_0 ，z_0）= \ hbox {卷}（\ Omega）^ {-1} \）且仅当\（\ Omega \）是磁盘减去a（可能为空）时，某些\（z_0 \ in \ Omega \））的封闭极坐标集或\（\ mathbb {C} \）减去（可能为空）封闭的极坐标集。当\（\ Omega \）受到\（C ^ {\ infty} \）- boundary的限制时，我们使用Szegö内核的零集对此提供了简单的证明。最后，我们证明该定理对于\（n> 1 \）不能成立于\（\ mathbb {C} ^ n \）通过构造一个有界的完整Reinhardt域（具有代数边界），该域对单位球\（\ mathbb {B} ^ n \ subset \ mathbb {C} ^ n \）是强凸的而不是全纯的。