• Title, Summary, Keyword: Grassmannians

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CLASSIFICATION OF SMOOTH SCHUBERT VARIETIES IN THE SYMPLECTIC GRASSMANNIANS

  • HONG, JAEHYUN
    • Journal of the Korean Mathematical Society
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    • v.52 no.5
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    • pp.1109-1122
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    • 2015
  • A Schubert variety in a rational homogeneous variety G/P is defined by the closure of an orbit of a Borel subgroup B of G. In general, Schubert varieties are singular, and it is an old problem to determine which Schubert varieties are smooth. In this paper, we classify all smooth Schubert varieties in the symplectic Grassmannians.

COMMUTING STRUCTURE JACOBI OPERATOR FOR HOPF HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS

  • Jeong, Im-Soon;Suh, Young-Jin;Yang, Hae-Young
    • Bulletin of the Korean Mathematical Society
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    • v.46 no.3
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    • pp.447-461
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    • 2009
  • In this paper we give a non-existence theorem for Hopf real hypersurfaces in complex two-plane Grassmannians $G_2(\mathbb{C}^{m+2})$ satisfying the condition that the structure Jacobi operator $R_{\xi}$ commutes with the 3-structure tensors ${\phi}_i$, i = 1, 2, 3.

REAL HYPERSURFACES IN COMPLEX TWO-PLANE GRASSMANNIANS WHOSE SHAPE OPERATOR IS OF CODAZZI TYPE IN GENERALIZED TANAKA-WEBSTER CONNECTION

  • Cho, Kyusuk;Lee, Hyunjin;Pak, Eunmi
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.1
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    • pp.57-68
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    • 2015
  • In this paper, we give a non-existence theorem of Hopf hypersurfaces in complex two-plane Grassmannians $G_2(\mathbb{C}^{m+2})$, $m{\geq}3$, whose shape operator is of Codazzi type in generalized Tanaka-Webster connection $\hat{\nabla}^{(k)}$.

DEFORMATION RIGIDITY OF ODD LAGRANGIAN GRASSMANNIANS

  • Park, Kyeong-Dong
    • Journal of the Korean Mathematical Society
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    • v.53 no.3
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    • pp.489-501
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    • 2016
  • In this paper, we study the rigidity under $K{\ddot{a}}hler$ deformation of the complex structure of odd Lagrangian Grassmannians, i.e., the Lagrangian case $Gr_{\omega}$(n, 2n+1) of odd symplectic Grassmannians. To obtain the global deformation rigidity of the odd Lagrangian Grassmannian, we use results about the automorphism group of this manifold, the Lie algebra of infinitesimal automorphisms of the affine cone of the variety of minimal rational tangents and its prolongations.

REAL HYPERSURFACES WITH ∗-RICCI TENSORS IN COMPLEX TWO-PLANE GRASSMANNIANS

  • Chen, Xiaomin
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.3
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    • pp.975-992
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    • 2017
  • In this article, we consider a real hypersurface of complex two-plane Grassmannians $G_2({\mathbb{C}}^{m+2})$, $m{\geq}3$, admitting commuting ${\ast}$-Ricci and pseudo anti-commuting ${\ast}$-Ricci tensor, respectively. As the applications, we prove that there do not exist ${\ast}$-Einstein metrics on Hopf hypersurfaces as well as ${\ast}$-Ricci solitons whose potential vector field is the Reeb vector field on any real hypersurfaces.

LAURENT PHENOMENON FOR LANDAU-GINZBURG MODELS OF COMPLETE INTERSECTIONS IN GRASSMANNIANS OF PLANES

  • Przyjalkowski, Victor;Shramov, Constantin
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.5
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    • pp.1527-1575
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    • 2017
  • In a spirit of Givental's constructions Batyrev, Ciocan-Fontanine, Kim, and van Straten suggested Landau-Ginzburg models for smooth Fano complete intersections in Grassmannians and partial flag varieties as certain complete intersections in complex tori equipped with special functions called superpotentials. We provide a particular algorithm for constructing birational isomorphisms of these models for complete intersections in Grassmannians of planes with complex tori. In this case the superpotentials are given by Laurent polynomials. We study Givental's integrals for Landau-Ginzburg models suggested by Batyrev, Ciocan-Fontanine, Kim, and van Straten and show that they are periods for pencils of fibers of maps provided by Laurent polynomials we obtain. The algorithm we provide after minor modifications can be applied in a more general context.

ONE-POINTED GRAVITATIONAL GROMOV-WITTEN INVARIANTS FOR GRASSMANNIANS

  • Kim, Bum-Sig
    • Journal of the Korean Mathematical Society
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    • v.38 no.5
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    • pp.1061-1068
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    • 2001
  • We write down explicity a recursive formula of one-pointed gravitational Gromov-Witten invariants and reduce the computation of them to a combinatoric problem which is not solved yet. The one-pointed invariants were played important role in Givental’s program in mirror symmetry. In section 3, we describe the combinatoric problem which can be read independently.

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