AI 中文总结
该研究发展射影簇拟映射的K-模壁跨越理论，构造多 regime 模空间，关联稳定映射、拟映射与紧化空间，用于椭圆曲面紧化间的模插值。
AI 中文摘要
我们发展了射影簇上拟映射的模壁跨越理论，允许独立变动边界系数与拟映射权重。基于Hashizume与第一作者引入的拟映射K-稳定性，我们在稳定、Calabi-Yau及对数Fano regime中构造了射影模空间，同时给出壁跨越态射。核心构造是边界极化Calabi-Yau拟映射的模理论，其在数值平凡轨迹上保有丰富极化，可与向稳定及对数Fano区域的适当扰动作比较。稳定理论适用于任意亏格，而通过Calabi-Yau轨迹的比较仅涉及零亏格。对于次数为1的边界除子，所得框架将加权稳定映射与拟映射关联到Hassett空间及射影直线上加权点的GIT商。在配套论文中，我们将该框架用于在有理椭圆曲面的Miranda GIT紧化与8维球商的Baily-Borel紧化之间建立模插值。
英文摘要
We develop a modular wall crossing theory for quasimaps to a projective variety, allowing independent variation of the boundary coefficients and the quasimap weight. Building on the K-stability of quasimaps introduced by Hashizume and the first author, we construct projective moduli spaces in the stable, Calabi--Yau, and log Fano regimes, together with wall crossing morphisms. A central construction is the moduli theory of boundary polarized Calabi--Yau quasimaps, which retains an ample polarization at the numerically trivial locus and allows comparison with suitable perturbations toward the stable and log Fano regions. The stable theory applies in arbitrary genus, while the comparisons through the Calabi--Yau locus concern genus zero. For degree-one boundary divisors, the resulting framework relates weighted stable maps and quasimaps to Hassett spaces and GIT quotients of weighted points on $\mathbb P^1$. In a companion paper, we apply this framework to give a modular interpolation between Miranda's GIT compactification of rational elliptic surfaces and the Baily--Borel compactification of an eight-dimensional ball quotient.
Comments51 pages, comments are welcome!
