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We bound from below the complexity of the top Chern class of the Hodge bundlein the Chow ring of the moduli space of curves: no formulas in terms of classesof degrees 1 and 2 can exist. As a consequence of the Torelli map, the0-section over the second Voronoi compactification of the moduli of principallypolarized abelian varieties also can not be expressed in terms of classes ofdegree 1 and 2. Along the way, we establish new cases of Pixton's conjecturefor tautological relations.  In the log Chow ring of the moduli space of curves, however, we prove thatthe top Chern class of the Hodge bundle lies in the subalgebra generated bylogarithmic boundary divisors. The proof is effective and uses Pixton's doubleramification cycle formula together with a foundational study of thetautological ring defined by a normal crossings divisor. The results open thedoor to the search for simpler formulas for the top Chern class of the Hodgebundle on the moduli of curves after log blow-ups.

The Hodge bundle, the universal 0-section, and the log Chow ring of the moduli space of curves