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A regular semisimple Hessenberg variety Hess(S,h) is a smooth subvariety of the flag variety determined by a square matrix S with distinct eigenvalues and a Hessenberg function h. The cohomology ring H^*(Hess(S,h)) is independent of the choice of S and is not explicitly described except for a few cases. In this paper, we characterize the Hessenberg function h such that H^*(Hess(S,h)) is generated in degree two as a ring. It turns out that such h is what is called a (double) lollipop.

Given a semisimple complex linear algebraic group G and a lower ideal I in positive roots of G, three objects arise: the ideal arrangement A_I, the regular nilpotent Hessenberg variety Hess(N,I), and the regular semisimple Hessenberg variety Hess(S,I). We show that a certain graded ring derived from the logarithmic derivation module of A_I is isomorphic to H^*(Hess(N,I)) and H^*(Hess(S,I))^W, the invariants in H^*(Hess(S,I)) under an action of the Weyl group W of G. This isomorphism is shown for general Lie type, and generalizes Borel's celebrated theorem showing that the coinvariant algebra of W is isomorphic to the cohomology ring of the flag variety G/B. This surprising connection between Hessenberg varieties and hyperplane arrangements enables us to produce a number of interesting consequences. For instance, the surjectivity of the restriction map H^*(G/B)to H^*(Hess(N,I)) announced by Dale Peterson and an affirmative answer to a conjecture of Sommers-Tymoczko are immediate consequences. We also give an explicit ring presentation of H^*(Hess(N,I)) in types B, C, and G. Such a presentation was already known in type A or when Hess(N,I) is the Peterson variety. Moreover, we find the volume polynomial of Hess(N,I) and see that the hard Lefschetz property and the Hodge-Riemann relations hold for Hess(N,I), despite the fact that it is a singular variety in general.

The solution of Shareshian-Wachs conjecture by Brosnan-Chow and Guay-Paquet tied the graded chromatic symmetric functions on indifference graphs (or unit interval graphs) and the cohomology of regular semisimple Hessenberg varieties with the dot action. A similar result holds between unicellular LLT polynomials and twins of regular semisimple Hessenberg varieties. A recent result by Abreu-Nigro enabled us to prove these results by showing the modular law for the geometrical objects, and this is indeed done by Precup-Sommers and Kiem-Lee. In this paper, we give elementary and simpler proofs to the modular law through GKM theory.

Fixed a polarised variety X, we can ask if it admits Ulrich bundles and, in case, what is their minimal possible rank. In this thesis, after recalling general properties of Ulrich sheaves, we show that any finite covering of P^n that embeds as a divisor in a weighted projective space with weights (1^{n+1},m) admits Ulrich sheaves, by using matrix factorisations. Among these varieties, we focus on double coverings of with nge3. Through Hartshorne--Serre correspondence, which we review along the way, we prove that the general such X admits a rank 2 Ulrich sheaf if and only if n=3 and m=2,3,4, and characterise the zero loci of their sections. Moreover, we construct generically smooth components of the expected dimension of their moduli spaces, analyse the action of the natural involution on them and the restriction of those bundles to low degree hypersurfaces. For m=2,3, we verify the existence of slope-stable Ulrich bundles of all the possible ranks.

In recent years, there has been a development in approaching rationality problems through the motivic methods (cf. [Kontsevich--Tschinkel'19], [Nicaise--Shinder'19], [Nicaise--Ottem'21]). This method requires the explicit construction of degeneration families of curves with favorable properties. While the specific construction is generally difficult, [Nicaise--Ottem'22] combines combinatorial methods to construct degeneration families of hypersurfaces in toric varieties and shows the non-stable rationality of a very general hypersurface in projective spaces. In this paper, we extend the result of [Nicaise--Ottem'22] not only for hypersurfaces in algebraic tori but also to those in sch\"{o}n affine varieties. In application, we show the irrationality of certain hypersurfaces in the complex Grassmannian variety Gr(2, n) using the motivic method, which coincides with the result obtained by the same author in the previous research.

We consider generalized homogeneous roofs, i.e. quotients of simply connected, semisimple Lie groups by a parabolic subgroup, which admit two projective bundle structures. Given a general hyperplane section on such a variety, we consider the zero loci of its pushforwards along the projective bundle structures and we discuss their properties at the level of Hodge structures. In the case of the flag variety F(1,2,n) with its projections to P^{n-1} and G(2, n), we construct a derived embedding of the relevant zero loci by methods based on the study of B-brane categories in the context of a gauged linear sigma model.