半正度量延拓在光滑射影族中的失败
Failure of semipositive metric extension in a smooth projective family
- Chern Institute of Mathematics and LPMC, Nankai University（南开大学陈省身数学研究所和LPMC）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
构造了一个光滑射影有理曲面族，其中中心纤维上的半正奇异 Hermite 度量无法延拓到任何邻域，即使仅要求奇点类型相同也失败，从而否定了 Păun 的度量延拓问题。
AI中文摘要：
我们构造了一个光滑射影有理曲面族、其全空间上的一个有效线丛，以及在中心纤维上预先指定的一个半正曲率奇异 Hermite 度量，该度量无法延拓到该纤维的任何邻域。该度量具有解析奇点且为最小奇点类型。即使仅要求限制具有与指定度量相同的奇点类型，延拓失败仍然存在。该族通过 blowing up 一个乘积的四个不相交截面得到，其中三个点在中心纤维上共线。该纤维上的一个可积伴随截面无法延拓，因为相应的伴随系统在每个邻近纤维上消失。这给出了 Păun 度量延拓问题（由 Dinew、Guedj 和 Zeriahi 列为问题 38）的否定回答。
英文摘要：
We construct a smooth projective family of rational surfaces, an effective line bundle on its total space, and a prescribed semipositively curved singular Hermitian metric on the central fibre that has no semipositive extension to any neighbourhood of that fibre. It has analytic singularities and is of minimal singularity type. The failure persists even when the restriction is only required to have the same singularity type as the prescribed metric. The family is obtained by blowing up four disjoint sections of a product, with three points becoming collinear on the central fibre. An integrable adjoint section on that fibre cannot extend because the corresponding adjoint systems vanish on every nearby fibre. This gives a negative answer to Păun's metric-extension question, listed as Question 38 by Dinew, Guedj, and Zeriahi.
