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GALKIN'S LOWER BOUND CONJECURE FOR LAGRANGIAN AND ORTHOGONAL GRASSMANNIANS

  • Cheong, Daewoong (Department of Mathematics Chungbuk National University) ;
  • Han, Manwook (Department of Mathematics Chungbuk National University)
  • Received : 2019.06.26
  • Accepted : 2019.11.06
  • Published : 2020.07.31

Abstract

Let M be a Fano manifold, and H🟉(M; ℂ) be the quantum cohomology ring of M with the quantum product 🟉. For 𝜎 ∈ H🟉(M; ℂ), denote by [𝜎] the quantum multiplication operator 𝜎🟉 on H🟉(M; ℂ). It was conjectured several years ago [7,8] and has been proved for many Fano manifolds [1,2,10,14], including our cases, that the operator [c1(M)] has a real valued eigenvalue 𝛿0 which is maximal among eigenvalues of [c1(M)]. Galkin's lower bound conjecture [6] states that for a Fano manifold M, 𝛿0 ≥ dim M + 1, and the equality holds if and only if M is the projective space ℙn. In this note, we show that Galkin's lower bound conjecture holds for Lagrangian and orthogonal Grassmannians, modulo some exceptions for the equality.

Keywords

References

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