有理椭圆曲面模空间的K-模壁跨越与自守形式
K-Moduli Wall Crossing and Automorphic Forms for the Moduli Space of Rational Elliptic Surfaces
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中文总结 AI 辅助
该研究利用对数拟映射的K-模空间构造模插值，确定其壁跨越，通过Borcherds乘积构造新自守形式，证明特定区间的K-模空间为半环面紧化。
中文摘要 AI 辅助
利用次数为12、带12个点且权重为t/12的对数拟映射的K-模空间$q_t\colon\left(\mathbb P^1,\frac{1-t}{12}D\right)\to[\mathbb A^2/\mathbb G_m]$，我们构造了Baily–Borel紧化的Heckman–Looijenga球商$X_o$与Miranda的有理椭圆曲面模空间GIT紧化之间的模插值$\{\mathcal M_t\}_{0\le t\le1}$。我们完全确定了壁跨越，且对每个有理数$t\in[0,1]$，有$\n\mathcal M_t\cong\operatorname{Proj}R\\!\left(X_o,\mathcal L+\frac t2\Delta(6)+\frac t3\Delta(9)\right)$，其中$\n\mathcal L$是自守$\n\mathbb Q$-线丛，$\n\Delta(6),\Delta(9)$是特殊Heegner除子。在自守侧，我们通过Borcherds乘积在$X_o$上构造了一个新的自守形式，其除子给出了Heegner除子间的独立关系，该关系是确定K-模壁跨越中双有理变换的关键输入。作为分析的一部分，我们证明$t\in(0,1/7)$的第一个正腔$\n\mathcal M_t$是Looijenga的半环面紧化。
英文摘要
Using K-moduli spaces for log quasimaps $q_t\colon\left(\mathbb P^1,\frac{1-t}{12}D\right)\to[\mathbb A^2/\mathbb G_m]$ of degree twelve with twelve points and weight $t/12$, we construct a modular interpolation $\{\mathcal M_t\}_{0\le t\le1}$ between the Baily--Borel compactification of the Heckman--Looijenga ball quotient $X_o$ and Miranda's GIT compactification of the moduli space of rational elliptic surfaces. We completely determine the wall-crossing and, for every rational $t\in[0,1]$, identify \[ \mathcal M_t\cong\operatorname{Proj}R\!\left(X_o,\mathcal L+\frac t2Δ(6)+\frac t3Δ(9)\right), \] where $\mathcal L$ is the automorphic $\mathbb Q$-line bundle and $Δ(6),Δ(9)$ are distinguished Heegner divisors. On the automorphic side, we construct a new automorphic form on $X_o$ via a Borcherds product, whose divisor gives an independent relation among the Heegner divisors. This relation provides a key input for determining the birational transformations in the K-moduli wall-crossing. As part of this analysis, we show that the first positive chamber $\mathcal M_t$ for $t\in (0,1/7)$ is Looijenga's semi-toroidal compactification.
发表机构
- Kyoto University（京都大学）
- Tohoku University（东北大学）
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