Control of the domino number by relative Frobenius geometry

Investigate whether the relative Frobenius geometry of a generically ordinary semistable fibration with nonzero Kodaira–Spencer map controls the domino number T^{0,2}(S), specifically by proving the estimate 2T^{0,2}(S)≤G_f−r_f, in order to derive the classical Arakelov inequality without assuming that the total surface is Hodge Witt.

Background

The refined Arakelov inequality established in the paper includes the correction term T^{0,2}(S), which measures the failure of the slope spectral sequence to degenerate integrally. The Hodge Witt hypothesis eliminates this term, yielding the classical Arakelov inequality, but the paper gives an example of a semistable fibration over P^1 for which T^{0,2}(S)=1.

The authors observe that, for a generically ordinary semistable fibration with nonzero Kodaira–Spencer map, an inequality bounding twice the domino number by the difference G_f−r_f would remove the correction term through the refined estimate. Establishing this bound would therefore extend the classical Arakelov inequality to settings where the total surface is not assumed to be Hodge Witt.