AI 中文总结
研究在KSBA模叠层上定义κ类，证明其基变换相容性等公式及性质，包括κ多项式消失、在有效圈上非负等，还表明其与陈类的关系，揭示了κ类的基本性质。
AI 中文摘要
我们在KSBA模叠层上定义κ类作为运算周环上同调中的类。它们推广了曲线模空间上的Miller-Morita-Mumford类。我们证明了基变换相容性、乘积和正规化公式以及crepant函子性，还证明了高于变化量的所有κ多项式的消失。更高的κ类在有效圈上是非负的，数值非平凡κ类的最大指标等于正规化变化量，而第一个κ类检测总变化量。随着边界系数变化，这些类在腔上是多项式且在运算壁穿越下是相容的。最后，每个正次数的κ类是自然虚拟向量丛单个陈类的有理倍数。
英文摘要
We define kappa classes on KSBA moduli stacks as classes in operational Chow cohomology. They generalize the Miller-Morita-Mumford classes on the moduli spaces of curves. We prove base-change compatibility, product and normalization formulas, and crepant functoriality, together with the vanishing of all kappa polynomials above the variation. The higher kappa classes are nonnegative on effective cycles, and the largest index of a numerically nontrivial kappa class equals the normalized variation, while the first kappa class detects the total variation. As the boundary coefficients vary, the classes are chamberwise polynomial and compatible under operational wall crossing. Finally, every positive-degree kappa class is a rational multiple of a single Chern class of a natural virtual vector bundle.
