Irrationality of Fano complete intersections with residual Serre dimension above the threshold

Determine whether a very general smooth complex Fano complete intersection of a specified multidegree with residual-category upper Serre dimension satisfying \(\sigma(Y)>\dim Y-2\), equivalently with Fano index \(r_Y\) smaller than the largest defining degree \(d_r\), is irrational, and determine which such multidegrees yield irrationality for every smooth member.

Background

For a smooth complex Fano complete intersection $Y\subset \mathbb P^n$ of dimension $m$ and multidegree $(d_1,\ldots,d_r)$, the paper recalls that the residual component in the standard semiorthogonal decomposition of $\mathrm D^{\mathrm b}(Y)$ has upper Serre dimension $\sigma(Y)=m-2r_Y/d_r$, where $r_Y$ is the Fano index and $d_r$ is the largest defining degree. The inequality $\sigma(Y)>m-2$ is equivalent to $r_Y<d_r$.

The paper develops quantum-connection invariants—ranks and regularized exponent multisets of suitable generic Hodge factors—and proves irrationality criteria in dimensions two through four. These results settle several examples, including cubic threefolds, ordinary Gushel–Mukai threefolds, and very general quartic fourfolds, but they do not resolve the stated question for all Fano complete-intersection multidegrees or establish a complete characterization of those for which every smooth member is irrational.