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Register a new userGeometry of Hessenberg varieties with applications to Newton-Okounkov bodies
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In this paper, we study the geometry of various Hessenberg varieties in type A, as well as families thereof, with the additional goal of laying the groundwork for future computations of Newton-Okounkov bodies of Hessenberg varieties. Our main results are as follows. We find explicit and computationally convenient generators for the local defining ideals of indecomposable regular nilpotent Hessenberg varieties, and then show that all regular nilpotent Hessenberg varieties are local complete intersections. We also show that certain families of Hessenberg varieties, whose generic fibers are regular semisimple Hessenberg varieties and the special fiber is a regular nilpotent Hessenberg variety, are flat and have reduced fibres. This result further allows us to give a computationally effective formula for the degree of a regular nilpotent Hessenberg variety with respect to a Plucker embedding. Furthermore, we construct certain flags of subvarieties of a regular nilpotent Hessenberg variety, obtained by intersecting with Schubert varieties, which are suitable for computing Newton-Okounkov bodies. As an application of our results, we explicitly compute many Newton-Okounkov bodies of the two-dimensional Peterson variety with respect to Plucker embeddings.
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Tropical geometry and the theory of Newton-Okounkov bodies are two methods which produce toric degenerations of an irreducible complex projective variety. Kaveh-Manon showed that the two are related. We give geometric maps between the Newton-Okounkov bodies corresponding to two adjacent maximal-dimensional prime cones in the tropicalization of $X$. Under a technical condition, we produce a natural algebraic wall-crossing map on the underlying value semigroups (of the corresponding valuations). In the case of the tropical Grassmannian $Gr(2,m)$, we prove that the algebraic wall-crossing map is the restriction of a geometric map. In an Appendix by Nathan Ilten, he explains how the geometric wall-crossing phenomenon can also be derived from the perspective of complexity-one $T$-varieties; Ilten also explains the connection to the combinatorial mutations studied by Akhtar-Coates-Galkin-Kasprzyk.
The main result of this note is that the toric degenerations of flag varieties associated to string polytopes and certain Bott-Samelson resolutions of flag varieties fit into a commutative diagram which gives a resolution of singularities of singular toric varieties corresponding to string polytopes. Our main tool is a result of Anderson which shows that the toric degenerations arising from Newton-Okounkov bodies are functorial in an appropriate sense. We also use results of Fujita which show that Newton-Okounkov bodies of Bott-Samelson varieties with respect to a certain valuation $ u_{max}$ coincide with generalized string polytopes, as well as previous results by the authors which explicitly describe the Newton-Okounkov bodies of Bott-Samelson varieties with respect to a different valuation $ u_{min}$ in terms of Grossberg-Karshon twisted cubes. A key step in our argument is that, under a technical condition, these Newton-Okounkov bodies coincide.
We show that quite universally the holonomicity of the complexity function of a big divisor on a projective variety does not predict the polyhedrality of the Newton-Okounkov body associated to every flag.
We compute the Newton--Okounkov bodies of line bundles on the complete flag variety of GL_n for a geometric valuation coming from a flag of translated Schubert subvarieties. The Schubert subvarieties correspond to the terminal subwords in the decomposition (s_1)(s_2s_1)(s_3s_2s_1)(...)(s_{n-1}...s_1) of the longest element in the Weyl group. The resulting Newton--Okounkov bodies coincide with the Feigin--Fourier--Littelmann--Vinberg polytopes in type A.
This paper studies the geometry and combinatorics of three interrelated varieties: Springer fibers, Steinberg varieties, and parabolic Hessenberg varieties. We prove that each parabolic Hessenberg variety is the pullback of a Steinberg variety under the projection of the flag variety to an appropriate partial flag variety and we give three applications of this result. The first application constructs an explicit paving of all Steinberg varieties in Lie type $A$ in terms of semistandard tableaux. As a result, we obtain an elementary proof of a theorem of Steinberg and Shimomura that the well-known Kostka numbers count the maximal-dimensional irreducible components of Steinberg varieties. The second application proves an open conjecture for certain parabolic Hessenberg varieties in Lie type A by showing that their Betti numbers equal those of a specific union of Schubert varieties. The third application proves that the irreducible components of parabolic Hessenberg varieties are in bijection with the irreducible components of the Steinberg variety. All three of these applications extend our geometric understanding of the three varieties at the heart of this paper, a full understanding of which is unknown even for Springer varieties, despite over forty years worth of work.
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