Hirzebruch signature theorem on Hochschild homology, with applications to derived invariance of Hodge numbers

RESUMO

We prove a refinement of Hirzebruch's signature formula on the individual Hochschild diagonals of a smooth projective complex variety. We construct natural non-degenerate Hermitian forms on the diagonals whose degree has the same parity as the dimension, and compute their positive and negative indices explicitly in terms of Hodge numbers. When the canonical bundle is trivial, these forms are preserved by derived equivalences. Their signatures yield the derived invariance of $h^{2,0}$ and $h^{1,n-1}$ in every dimension $n\geq2$. In dimension five, combining these invariants with Hochschild homology and the Libgober-Wood identity proves that derived-equivalent smooth projective complex varieties with trivial canonical bundle have identical Hodge numbers.