Pseudotoric structures and special Lagrangian torus fibrations on certain flag varieties

Abstract

We construct pseudotoric structures (à la Tyurin [26]) on the twostep flag variety F ℓ 1,n−1;n , and explain a general relation between pseudotoric structures and special Lagrangian torus fibrations, the latter of which are important in the study of SYZ mirror symmetry [25]. As an application, we speculate how our constructions can explain the number of terms in the superpotential of Rietsch's Landau-Ginzburg mirror [24,17,22,23].

W-translated Schubert divisors and transversal intersections

“…As an application of W -translated Schubert divisors, we specify an anti-canonical divisor −K X of X = F n1,...,n k ;n = SL(n, C)/P , which may have potential applications in the study of Strominger-Yau-Zaslow mirror symmetry for X. For example, for X = F 1,n−1;n , a special Lagrangian fibration for the open Calabi-Yau manifold X \ −K X was constructed in [7] with respect to such −K X .…”

“…Remark 5.3. The special paths defined above correspond to the partitions of the zero rectangle or extremal rectangles in [30], i.e., rectangles whose length equal to m or width equal to n − m. See Figure 8 for the case of Gr (4,7).…”

“…Such anti-canonical divisor degenerates to that of the Gelfand-Cetlin toric variety X 0 and is potentially useful in the study of Strominger-Yau-Zaslow (SYZ) mirror symmetry [35] for a partial flag variety X. For example, Chan et al [7] constructed a special Lagrangian fibration for the open Calabi-Yau manifold X \ −K X for the two-step flag variety F 1,n−1;n , which is one key step in the study of SYZ mirror symmetry.…”

W-translated Schubert divisors and transversal intersections

“…As an application of W -translated Schubert divisors, we specify an anti-canonical divisor −K X of X = F n1,••• ,n k ;n = SL(n, C)/P , which may have potential applications in the study of Strominger-Yau-Zaslow mirror symmetry for X. For instance for X = F 1,n−1;n , a special Lagrangian fibration for the open Calabi-Yau manifold X \ −K X was constructed in [7] with respect to such −K X .…”

“…Such anti-canonical divisor degenerates to that of the Gelfand-Cetlin toric variety X 0 and is potentially useful in the study of Strominger-Yau-Zaslow mirror symmetry [35] for a partial flag variety X. For instance, Chan, Leung and the fourth named author constructed a special Lagrangian fibration for the open Calabi-Yau manifold X \ −K X for the two-step flag variety F 1,n−1;n [7], which is one key step in the study of SYZ mirror symmetry.…”