Hirzebruch signature theorem on Hochschild homology, with applications to derived invariance of Hodge numbers
Abstract
We prove a refinement of Hirzebruch's signature formula on the individual Hochschild diagonals of a smooth projective complex variety. Natural Hermitian forms, obtained by symmetrizing a Todd-corrected pairing, have explicitly computable positive and negative indices and are preserved by derived equivalences. On diagonals of the same parity as the dimension, their signatures separate the even and odd contributions to the Hochschild-Kostant-Rosenberg decomposition; on the other diagonals, their radicals and signatures are governed by the first Chern class. We deduce the derived invariance of hn−2,0, hn−1,1 and hn−3,1 in any dimension, and of h2,0 when the canonical bundle is trivial. Combining the invariance of h2,0 with the multiplicative structure of Hochschild cohomology and Verbitsky's orthogonal action, we give a new short proof of the theorem of Huybrechts and Nieper-Wißkirchen that a derived partner of an irreducible holomorphic symplectic variety is again irreducible holomorphic symplectic. In dimension five with trivial canonical bundle, the signature invariants and the Libgober-Wood identity together establish the derived invariance of all Hodge numbers
Related
- The Hochschild (co)homology of gentle algebras
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BibTeX
@misc{abuaf2026hirzebruch,
title = {{Hirzebruch signature theorem on Hochschild homology, with applications to derived invariance of Hodge numbers}},
author = {Roland Abuaf},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08917},
eprint = {2609.08917},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08917},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hirzebruch signature theorem on Hochschild homology, with applications to derived invariance of Hodge numbers
AU - Roland Abuaf
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.08917
UR - https://arxiv.org/abs/2609.08917
AB - We prove a refinement of Hirzebruch's signature formula on the individual Hochschild diagonals of a smooth projective complex variety. Natural Hermitian forms, obtained by symmetrizing a Todd-corrected pairing, have explicitly computable positive and negative indices and are preserved by derived equivalences. On diagonals of the same parity as the dimension, their signatures separate the even and odd contributions to the Hochschild-Kostant-Rosenberg decomposition; on the other diagonals, their radicals and signatures are governed by the first Chern class. We deduce the derived invariance of h^n−2,0, h^n−1,1 and h^n−3,1 in any dimension, and of h^2,0 when the canonical bundle is trivial. Combining the invariance of h^2,0 with the multiplicative structure of Hochschild cohomology and Verbitsky's orthogonal action, we give a new short proof of the theorem of Huybrechts and Nieper-Wißkirchen that a derived partner of an irreducible holomorphic symplectic variety is again irreducible holomorphic symplectic. In dimension five with trivial canonical bundle, the signature invariants and the Libgober-Wood identity together establish the derived invariance of all Hodge numbers
ER -