AI 中文总结
针对固定库默尔K3曲面的单参数类，给出卡拉比-丘定理保证的里奇平坦凯勒度量的收敛显式表示，构造了具O(a⁴)归一化蒙日-安培残差的显式径向背景，证明其在小参数下绝对收敛，得到对应卡拉比-丘势并给出退化椭圆K3曲面的递归分析条件。
AI 中文摘要
我们针对固定库默尔K3曲面的单参数类，给出卡拉比-丘定理保证的里奇平坦凯勒度量的收敛显式表示。一个显式径向背景具有归一化蒙日-安培残差\boldsymbol{O(a^4)}。其标量格林算子\boldsymbol{G_a}由周期埃瓦尔德核、分离径向核、有限维舒尔补及收敛诺伊曼级数表示。系数定义为\boldsymbol{U_{a,1}=-G_af_a}，\boldsymbol{U_{a,n}=G_a\boldsymbol{\textstyle\bigcup_{j=1}^{n-1}}Q_a(U_{a,j},U_{a,n-j})}，其中\boldsymbol{Q_a}为极化二次蒙日-安培项。阶为\boldsymbol{a^{-1}}的格林估计给出阶为\boldsymbol{a^3}的一阶修正和阶为\boldsymbol{a^2}的收敛参数。卡特兰优函数证明，对每个足够小的固定\boldsymbol{a}，该级数在整个光滑曲面上绝对收敛。该和定义了解体积方程的正形式；通过比较可将该和与光滑归一化卡拉比-丘势等同。我们还记录了退化椭圆K3曲面上同一递归的充分分析条件。
英文摘要
We give a convergent explicit representation of the Ricci-flat Kähler metrics supplied by the Calabi--Yau theorem in a one-parameter family of classes on a fixed Kummer K3 surface. An explicit radial background has normalized Monge--Ampère residual \(O(a^4)\). Its scalar Green operator \(G_a\) is represented by periodic Ewald kernels, separated radial kernels, finite-dimensional Schur complements, and convergent Neumann series. The coefficients are defined by \(U_{a,1}=-G_af_a\) and \(U_{a,n}=G_a\sum_{j=1}^{n-1}Q_a(U_{a,j},U_{a,n-j})\), where \(Q_a\) is the polarized quadratic Monge--Ampère term. A Green estimate of order \(a^{-1}\) gives a first correction of order \(a^3\) and a convergence parameter of order \(a^2\). A Catalan majorant proves absolute convergence on the entire smooth surface for every sufficiently small fixed \(a\). The sum defines a positive form solving the volume equation; comparison identifies the sum with the smooth normalized Calabi--Yau potential. We also record sufficient analytic conditions for the same recursion on collapsing elliptic K3 surfaces.
Comments35 pages
