发表机构
University of Michigan（密歇根大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究构造了满足特定条件的簇，证明了周期-指数猜想不成立，在d=3时还扩展到$\boldsymbol{\bar{Q}}$域，为相关代数几何问题提供了反例。
AI 中文摘要
对于特征为0的任意不可数代数闭域k和任意d≥3，我们在k上构造了一个维数为d的簇，其上的一个Brauer类因Hodge理论原因违反了周期-指数猜想。当d=3时，我们的构造甚至不需要k不可数的假设；特别地，周期-指数猜想在$\boldsymbol{\bar{Q}}$上不成立。
英文摘要
For any integer $d \geq 3$ and any algebraically closed field $k$ of characteristic $0$ and transcendence degree at least $d-3$, or of characteristic $p>2$ and transcendence degree at least $d-2$, we construct a $d$-dimensional variety over $k$ with a Brauer class of period $2$ and index $2^d$, violating the period-index conjecture; in particular, the period-index conjecture fails in dimension $3$ over $\overline{\mathbf{Q}}$ and $\overline{\mathbf{F}_p(t)}$, and in all dimensions $d \geq 3$ over $\mathbf{C}$ and $\overline{\mathbf{F}_p((t))}$. To bound the index of Brauer classes from below, we employ an obstruction of motivic nature, which requires proving the nonexistence of integral Hodge or Tate classes satisfying a certain equation modulo $2$. Motivated by these counterexamples, we propose a period-index conjecture with corrections at small primes and, as evidence, prove its Hodge-theoretic counterpart.
Comments41 pages, v2: added positive-characteristic counterexamples; proposed the period-index conjecture with corrections and proved its Hodge-theoretic counterpart; updated references; other minor improvements
