五维中的Bondal猜想
Bondal's conjecture in dimension five
AI总结：
本文证明了五维Fano流形上的Bondal猜想，并给出了所有奇数维Fano流形的部分结果，证明使用了弱Fano流形上余维一叶状结构的代数可积性准则、Poisson结构的模留数以及Pfaff域不变子簇的上同调约束。
摘要：
Poisson几何中的Bondal猜想给出了Poisson Fano流形退化轨迹的下界，其中Poisson结构的秩下降。通过多位作者的工作，先前已知该猜想对维数不超过四的Fano流形成立。我们首次给出了五维Fano流形上该猜想的证明，以及所有奇数维Fano流形的部分结果。证明使用了：(i) 弱Fano流形上余维一叶状结构的代数可积性准则，推广了第一作者先前的结果；(ii) Gualtieri和第三作者引入的Poisson结构的“模留数”；以及(iii) Pfaff域不变子簇的上同调约束，将Esteves--Kleiman的早期结果推广到子簇上的Pfaff分布允许闭强有向正电流的情形。
英文摘要：
Bondal's conjecture in Poisson geometry gives lower bounds on the degeneracy loci of Poisson Fano manifolds, where the rank of the Poisson structure drops. By work of several authors, it was previously known to hold for Fano manifolds of dimension at most four. We give the first proof of this conjecture for Fano manifolds of dimension five, and partial results for Fano manifolds of all odd dimensions. The proof uses: (i) an algebraic integrability criterion for codimension-one foliations on weak Fano manifolds, extending a previous result of the first author; (ii) the "modular residues" of Poisson structures introduced by Gualtieri and the third author; and (iii) a cohomological constraint on invariant subvarieties for Pfaff fields, extending earlier results of Esteves--Kleiman to the case in which the Pfaff distribution on the subvariety admits a closed strongly directed positive current.
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