$K_X = 0$ 的光滑射影复五维流形的 Hodge 数的导出不变性
Hirzebruch signature theorem on Hochschild homology, with applications to derived invariance of Hodge numbers
- Rectorat de Paris（巴黎学区教育局）
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AI总结：
本文证明具有平凡典范丛的导出等价光滑射影复五维流形具有相同 Hodge 数，方法结合 Hochschild 同调、Mukai 配对细化及 Hodge-Riemann 关系。
AI中文摘要：
我们证明了具有平凡典范丛的导出等价的光滑射影复五维流形具有相同的 Hodge 数。证明将 Hochschild 同调的导出不变性与对各个 Hochschild 对角线上的 Mukai 配对的细化相结合。在与复共轭复合后，该配对在奇数 Hochschild 次数上给出由 Fourier-Mukai 等价保持的 Hermite 形式。利用 Lefschetz 分解和 Hodge-Riemann 双线性关系计算它们的符号。结合对具有平凡第一陈类的五维流形特有的 Hirzebruch-Riemann-Roch 关系，这确定了所有 Hodge 数。
英文摘要：
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
