• Title/Summary/Keyword: Fano manifolds

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A SUFFICIENT CONDITION FOR A TORIC WEAK FANO 4-FOLD TO BE DEFORMED TO A FANO MANIFOLD

  • Sato, Hiroshi
    • Journal of the Korean Mathematical Society
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    • v.58 no.5
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    • pp.1081-1107
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    • 2021
  • In this paper, we introduce the notion of toric special weak Fano manifolds, which have only special primitive crepant contractions. We study their structure, and in particular completely classify smooth toric special weak Fano 4-folds. As a result, we can confirm that almost every smooth toric special weak Fano 4-fold is a weakened Fano manifold, that is, a weak Fano manifold which can be deformed to a Fano manifold.

FANO MANIFOLDS AND BLOW-UPS OF LOW-DIMENSIONAL SUBVARIETIES

  • Chierici, Elena;Occhetta, Gianluca
    • Journal of the Korean Mathematical Society
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    • v.47 no.1
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    • pp.189-213
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    • 2010
  • We study Fano manifolds of pseudoindex greater than one and dimension greater than five, which are blow-ups of smooth varieties along smooth centers of dimension equal to the pseudoindex of the manifold. We obtain a classification of the possible cones of curves of these manifolds, and we prove that there is only one such manifold without a fiber type elementary contraction.

MORPHISMS BETWEEN FANO MANIFOLDS GIVEN BY COMPLETE INTERSECTIONS

  • Choe, Insong
    • Journal of the Chungcheong Mathematical Society
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    • v.22 no.4
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    • pp.689-697
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    • 2009
  • We study the existence of surjective morphisms between Fano manifolds of Picard number 1, when the source is given by the intersection of a cubic hypersurface and either a quadric or another cubic hypersurface in a projective space.

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SASAKIAN 3-METRIC AS A *-CONFORMAL RICCI SOLITON REPRESENTS A BERGER SPHERE

  • Dey, Dibakar
    • Bulletin of the Korean Mathematical Society
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    • v.59 no.1
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    • pp.101-110
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    • 2022
  • In this article, the notion of *-conformal Ricci soliton is defined as a self similar solution of the *-conformal Ricci flow. A Sasakian 3-metric satisfying the *-conformal Ricci soliton is completely classified under certain conditions on the soliton vector field. We establish a relation with Fano manifolds and proves a homothety between the Sasakian 3-metric and the Berger Sphere. Also, the potential vector field V is a harmonic infinitesimal automorphism of the contact metric structure.

GALKIN'S LOWER BOUND CONJECURE FOR LAGRANGIAN AND ORTHOGONAL GRASSMANNIANS

  • Cheong, Daewoong;Han, Manwook
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.4
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    • pp.933-943
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    • 2020
  • 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.