Flexibility of Schubert classes

Abstract

Abstract. In this note, we discuss the flexibility of Schubert classes in homogeneous varieties. We give several constructions for representing multiples of a Schubert class by irreducible subvarieties. We sharpen [R, Theorem 3.1] by proving that every positive multiple of an obstructed class in a cominuscule homogeneous variety can be represented by an irreducible subvariety.

Schur flexibility of cominuscule Schubert varieties

“…Decompose λ =μ p · · ·μ 1μ0 into blocksμ s of 'consecutive' integers, with the convention that the integers n − 1, n + 1 are considered consecutive and are placed in the sameμ sblock; likewise, the integers n, n + 2 are consecutive. For example, if n = 5, then λ = (2, 3, 4, 6, 10) has block decompositionμ 1μ0 = (2, 3, 4, 6)(10); likewise, λ = (1, 2, 5, 7, 8) has block decompositionμ 1μ0 = (1, 2) (5,7,8).…”

“…To remove the redundancies, decompose λ = µ p · · · µ 1 µ 0 into maximal blocks of consecutive integers. For example, if λ = (2,3,4,7,8,12), then µ 2 = (2, 3, 4), µ 1 = (7, 8) and µ 0 = (12). Let (A.6) j ℓ (λ) = |µ p · · · µ ℓ | be the length of the sub-partition µ p · · · µ ℓ .…”

“…As before, (A.25) j ℓ (λ) = |μ p · · ·μ ℓ | . (1, 5, 7, 9, 10, 11) 1 : 1, 3 1 (3,4,7,8,11,12) 2 : 2, 4 1 (2, 5, 7, 9, 10, 12) 2 : 1, 3, 5 2 * (1, 6, 8, 9, 10, 11) 0 : 1 0 (3, 5, 7, 9, 11, 12) 3 : 1, 3, 4 2 (2, 6, 8, 9, 10, 12) 1 : 1, 5 1 * (4, 5, 7, 10, 11, 12) 1 : 3 1 (3,6,8,9,11,12) 2 : 1, 4 1 (4, 6, 8, 10, 11, 12) 2 : 1, 3 1 (5,6,9,10,11,12) 1 : 2 1 * (7,8,9,10,11,12) As an example, Table 2 lists the partitions and corresponding a : J and r values for the Schubert varieties of S 6 = Spin 12 C/P 6 . The discussion above yields the following.…”

“…While the construction of the nontrivial solutions Y (in the proof of Theorem 4.1) is explicit, it is in terms of representation theoretic data, and as a result is not geometrically transparent. Geometric (and explicit) constructions of Y are given in [8], where a stronger (than Corollary 4.3) statement is proven: if O w = 0, then for any positive integer m there exists an irreducible variety Y representing mξ w .…”

Iitaka dimension for cycles

“…Remark 1.9. The study of the Iitaka dimension for Schubert classes is closely related to the differential-geometric notion of Schubert rigidity developed in the series of papers [Wal97], [Bry05], [Hon05], [Hon07], [Cos11], [RT12], [Rob13], [CR13], [Cos14]. A Schubert class σ is called multi rigid if the only effective cycles with class proportional to σ are sums of Schubert varieties.…”