Molcho, Pandharipande, Schmitt — The Hodge Bundle, The Universal 0-Section, And The Log Chow Ring Of The Moduli Space Of Curves (2023)

We bound from below the complexity of the top Chern class {\(\lambdag\backslash\)}lambda _g of the Hodge bundle in the Chow ring of the moduli space of curves: no formulas for {\(\lambdag\backslash\)}lambda _g in terms of classes of degrees 1 and 2 can exist. As a consequence of the Torelli map, the 0-section over the second Voronoi compactification of the moduli of principally polarized abelian varieties also cannot be expressed in terms of classes of degree 1 and 2. Along the way, we establish new cases of Pixton’s conjecture for tautological relations. In the log Chow ring of the moduli space of curves, however, we prove {\(\lambdag\backslash\)}lambda _g lies in the subalgebra generated by logarithmic boundary divisors. The proof is effective and uses Pixton’s double ramification cycle formula together with a foundational study of the tautological ring defined by a normal crossings divisor. The results open the door to the search for simpler formulas for {\(\lambdag\backslash\)}lambda _g on the moduli of curves after log blow-ups.