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Let $M$ be a Fano manifold, and $H^\star(M;\C)$ be the quantum cohomology ring of $M$ with the quantum product $\star.$ For $\sigma \in H^\star(M;\C)$, denote by $[\sigma]$ the quantum multiplication operator $\sigma\star$ on $H^\star(M;\C)$. It was conjectured several years ago \cite{GGI, GI} and has been proved for many Fano manifolds \cite{CL1, CH2, LiMiSh, Ke}, including our cases, that the operator $[c_1(M)]$ has a real valued eigenvalue $\delta_0$ which is maximal among eigenvalues of $[c_1(M)]$. Galkin's lower bound conjecture \cite{Ga} states that for a Fano manifold $M,$ $\delta_0\geq \mathrm{dim} \ M +1,$ and the equality holds if and only if $M$ is the projective space $\mathbb{P}^n.$ In this note, we show that Galkin's lower bound conjecture holds for Lagrangian and orthogonal Grassmannians, modulo some exceptions for the equality.
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참고문헌 (16)
참고문헌 신청
Daewoong Cheong / 2017 / Quantum multiplication operators for Lagrangian and orthogonal Grassmannians / Journal of Algebraic Combinatorics 45(4):1153~1171
Daewoong Cheong / 2017 / On the conjecture O of GGI for G/P / Advances in Mathematics 306:704~721
L. Evans / Galkin's lower conjecture holds for the Grassmannian
W. Fulton / 1997 / Young Tableaux / Cam-bridge University Press
W. Fulton / 2004 / On the quantum product of Schubert classes / Journal of Algebraic Geometry 13(4):641~661
Daewoong Cheong / 2017 / On the conjecture O of GGI for G/P / Advances in Mathematics 306:704~721
L. Evans / Galkin's lower conjecture holds for the Grassmannian
W. Fulton / 1997 / Young Tableaux / Cam-bridge University Press
W. Fulton / 2004 / On the quantum product of Schubert classes / Journal of Algebraic Geometry 13(4):641~661
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