AI 中文总结
研究椭圆曲面的全纯辛几何，对非等平凡椭圆纤维化定义辛函数不变量并证明其决定辛几何，以此进行同构分类，还通过研究奇异纤维芽的辛自同构分类同构，应用中证明相关映射可扩展为K3曲面同构合成。
AI 中文摘要
当复曲面\(X\)存在处处非零的全纯2-形式时，它确定了\(X\)上的（全纯）辛结构。本文研究当\(X\)为椭圆曲面时这种辛结构的辛几何。对于非等平凡的椭圆纤维化，定义了柯达依函数不变量的一种分解，即辛函数不变量，并证明其决定了非等平凡椭圆纤维化的辛几何，进而得到具有固定源的非等平凡辛椭圆纤维化的同构分类。通过研究奇异纤维芽的辛自同构，也得到了具有固定目标的辛椭圆纤维化的同构分类。作为应用，证明了非等平凡椭圆K3曲面纤维芽之间的辛双全纯映射可扩展为K3曲面同构的合成。
英文摘要
When a complex surface $X$ admits a nowhere vanishing holomorphic 2-form, it determines a (holomorphic) symplectic structure on $X$. We study the symplectic geometry of such a symplectic structure when $X$ is an elliptic surface. When the elliptic fibration is nonisotrivial, we define a factorization of Kodaira's functional invariant, called the symplecto-functional invariant and prove that the symplecto-functional invariant determines the symplectic geometry of a nonisotrivial elliptic fibration. This leads to a classification of isogenies of nonisotrivial symplectic elliptic fibrations with a fixed source. We also classify isogenies of symplectic elliptic fibrations with a fixed target by studying symplectic automorphisms of germs of singular fibers. As an application, we prove that a symplecto-biholomorphic map between germs of fibers of nonisotrivial elliptic K3 surfaces can be extended to compositions of isogenies of K3 surfaces.
Comments32 pages, Journal of Algebraic Geometry (to appear)
