Shadow-complexity and trisection genus

The shadow-complexity is an invariant of closed $4$-manifolds defined by using $2$-dimensional polyhedra called Turaev’s shadows, which, roughly speaking, measures how complicated a $2$-skeleton of the $4$-manifold is. In this paper, we define a new version ${\mathrm{s}\mathrm{c}}_r$ of shadow-complexity depending on an extra parameter $r\ge 0$, and we investigate the relationship between this complexity and the trisection genus $g$. More explicitly, we prove an inequality $g(W)\le 2+2{\mathrm{s}\mathrm{c}}_r(W)$ for any closed $4$-manifold $W$ and any $r\ge 1\mathrm{/}2$. Moreover, we determine the exact values of ${\mathrm{s}\mathrm{c}}_{1\mathrm{/}2}$ for infinitely many $4$-manifolds, and also we classify all the closed $4$-manifolds with ${\mathrm{s}\mathrm{c}}_{1\mathrm{/}2}\le 1\mathrm{/}2$.