发表机构
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences; Simons Center for Geometry and Physics, Stony Brook University; International Center for Mathematical Sciences (ICMS); Instituto de Matemática, Estatística e Computação Científica (IMECC) da Universidade Estadual de Campinas (Unicamp)（保加利亚科学院数学与信息学研究所; 石溪大学西蒙斯几何与物理中心; 国际数学科学中心; 坎皮纳斯州立大学数学、统计与科学计算研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文发展根系局部化计算旗簇及齐次向量丛零轨迹的Gromov--Witten不变量，并利用量子乘法与Hodge原子理论关联，进而证明Fano四重有理性的判定条件及弱分解性质。
AI 中文摘要
我们为旗簇及整体生成齐次向量丛的光滑零轨迹的亏格零Gromov--Witten不变量发展了一种根系局部化形式体系。所得不变量表示为带装饰树上的有限和，其贡献由环境旗簇的根系和定义丛的环面权决定。我们通过第一陈类的量子小乘法矩阵将这些计算与Katzarkov--Kontsevich--Pantev--Yu的Hodge原子理论联系起来。设$X$为满足$h^{3,1}(X)=1$的Fano四重，它是合适的Fano五重的超平面截面。其余调分解为单值固定部分和中间消失上同调。该分解被第一陈类的量子乘法保持。此外，它在消失和上作用为标量。对于单值固定块，我们证明：(a)若每个特征值的代数重数至多为2且$X$是Hodge一般的，则$X$是非有理的；(b)若每个特征值的Jordan缺陷至多为1且$X$是有理的，则每个弱分解包含一个光滑曲面中心，其极小模型是射影K3曲面。我们给出了许多应用。
英文摘要
We develop a root-theoretic localization formalism for genus-zero Gromov--Witten invariants of flag varieties and of smooth zero loci of globally generated homogeneous vector bundles. The resulting invariants are expressed as finite sums over decorated trees whose contributions are determined by the root system of the ambient flag variety and by the torus weights of the defining bundle. We connect these computations with the Theory of Hodge Atoms of Katzarkov--Kontsevich--Pantev--Yu via the matrix of small quantum multiplication by first Chern class. Let $X$ be a Fano fourfold with $h^{3,1}(X)=1$ arising as a hyperplane section of a suitable Fano fivefold. Its cohomology decomposes into a monodromy-fixed part and the middle vanishing cohomology. This decomposition is preserved by quantum multiplication by the first Chern class. Moreover, it acts by a scalar on the vanishing summand. We show, for the monodromy-fixed block, that (a) if every eigenvalue has algebraic multiplicity at most two and $X$ is Hodge general, then $X$ is irrational; (b) if every eigenvalue has Jordan defect at most one and $X$ is rational, then every weak factorization contains a smooth surface center whose minimal model is a projective K3 surface. Many applications are presented.
