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In the second half, we consider Hessenberg varieties of all types. n general, a Hessenberg variety is determined by a "good" subset of the positive root system instead of a Hessenberg function. On the other hand, a subset of the positive root system gives a hyperplane arrangement in the Lie algebra of the maximal torus. Similarly to a flag variety, the chambers of this arrangement denote a cell decomposition of the regular nilpotent Hessenberg variety. By this relation between a Hessenberg variety and a hyperplane arrangement, we describe the cohomology ring of the regular nilpotent Hessenberg variety in terms of the subset and show that its Poincaré polynomial has two expressions like the Borel's work on flag varieties. This is a joint work with Takuro Abe, Tatsuya Horiguchi, Mikiya Masuda, and Satoshi Murai.



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