Brasselet-Schürmann-Yokura关于奇异簇$L$-类的猜想
The Brasselet-Schürmann-Yokura conjecture on $L$-classes of singular varieties
AI总结：
本文基于立方超解、分解定理和Hodge理论，证明了Brasselet-Schürmann-Yokura关于奇异簇$L$-类的猜想在射影簇情形下成立，并给出了有理同调流形的新刻画。
摘要：
2010年，Brasselet、Schürmann和Yokura猜想：对于紧致复代数簇$X$（其为有理同调流形），Goresky-MacPherson $L$-类$L_*(X)$与Hirzebruch同调类$T_{1*}(X)$之间存在特征类等式。本文基于立方超解、分解定理和Hodge理论，给出了该猜想在射影簇情形下的证明。证明的关键步骤是给出有理同调流形的一个新的立方超解刻画，我们认为该刻画具有独立的意义。
英文摘要：
In 2010, Brasselet, Schürmann and Yokura conjectured an equality of characteristic classes of singular varieties between the Goresky-MacPherson $L$-class $L_*(X)$ and the Hirzebruch homology class $T_{1*}(X)$ for a compact complex algebraic variety $X$ that is a rational homology manifold. In this note we give a proof of this conjecture for projective varieties based on cubical hyperresolutions, the Decomposition Theorem, and Hodge theory. The crucial step of the proof is a new characterization of rational homology manifolds in terms of cubical hyperresolutions which we find of independent interest.
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