• Title/Summary/Keyword: Calabi-Yau

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CALABI-YAU THREEFOLDS FROM BUILDING BLOCKS OF G2-MANIFOLDS

  • Lee, Nam-Hoon
    • Journal of the Chungcheong Mathematical Society
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    • v.30 no.3
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    • pp.331-335
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    • 2017
  • We construct Calabi-Yau threefolds by smoothing normal crossing varieties, which are made from the building blocks of $G_2-manifolds$. We compute the Hodge numbers of those Calabi-Yau threefolds. Some of those Hodge number pairs ($h^{1,1}$, $h^{1,2}$) do not overlap with those of Calabi-Yau threefolds constructed in the toric setting.

Some Siegel Threefolds with a Calabi-Yau Model II

  • Freitag, Eberhard;Manni, Riccardo Salvati
    • Kyungpook Mathematical Journal
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    • v.53 no.2
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    • pp.149-174
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    • 2013
  • In a previous paper, we described some Siegel modular threefolds which admit a Calabi-Yau model. Using a different method we give in this paper an enlarged list of such varieties. Basic for this method is a paper of van Geemen and Nygaard. They study a variety $\mathcal{X}$ that is the complete intersection of four quadrics in $\mathbb{P}^7(\mathbb{C})$. This is biholomorphic equivalent to the Satake compactification of $\mathcal{H}_2/{\Gamma}^{\prime}$ for a certain subgroup ${\Gamma}^{\prime}{\subset}Sp(2,\mathbb{Z})$ and it will be the starting point of our investigation. It has been pointed out that a (projective) small resolution of this variety is a rigid Calabi-Yau manifold $\tilde{\mathcal{X}}$. Then we will consider the action of quotients of modular groups on $\mathcal{X}$ and study possible resolutions that admits a Calabi-Yau model in the category of complex spaces.

SMOOTH, ISOLATED CURVES IN FAMILIES OF CALABI-YAU THREEFOLDS IN HOMOGENEOUS SPACES

  • Knutsen, Andreas Leopold
    • Journal of the Korean Mathematical Society
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    • v.50 no.5
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    • pp.1033-1050
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    • 2013
  • We show the existence of smooth isolated curves of different degrees and genera in Calabi-Yau threefolds that are complete intersections in homogeneous spaces. Along the way, we classify all degrees and genera of smooth curves on BN general K3 surfaces of genus ${\mu}$, where $5{\leq}{\mu}{\leq}10$. By results of Mukai, these are the K3 surfaces that can be realised as complete intersections in certain homogeneous spaces.

HOMOGENEOUS POLYNOMIAL HYPERSURFACE ISOLATED SINGULARITIES

  • Akahori, Takao
    • Journal of the Korean Mathematical Society
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    • v.40 no.4
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    • pp.667-680
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    • 2003
  • The mirror conjecture means originally the deep relation between complex and symplectic geometry in Calabi-Yau manifolds. Recently, this conjecture is posed beyond Calabi-Yau, and even to, open manifolds (e.g. $A_{n}$ singularities and its resolution) is discussed. While if we treat open manifolds, we can't avoid the boundary (in our case, CR manifolds). Therefore we pose the more precise conjecture (mirror symmetry with boundaries). Namely, in mirror symmetry, for boundaries, what kind of structure should correspond\ulcorner For this problem, the $A_{n}$ case is studied.

On Semiparallel and Weyl-semiparallel Hypersurfaces of Kaehler Manifolds

  • Ozgur, Cihan;Murathan, Cengizhan;Arslan, Kadri
    • Kyungpook Mathematical Journal
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    • v.49 no.1
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    • pp.133-141
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    • 2009
  • We study on semiparallel and Weyl semiparallel Sasakian hypersurfaces of Kaehler manifolds. We prove that a (2n + 1)-dimensional Sasakian hypersurface M of a (2n+2)-dimensional Kaehler manifold $\widetilde{M}^{2n+2}$ is semiparallel if and only if it is totally umbilical with unit mean curvature, if dimM = 3 and $\widetilde{M}^4$ is a Calabi-Yau manifold, then $\widetilde{M}$ is flat at each point of M. We also prove that such a hypersurface M is Weyl-semiparallel if and only if it is either an ${\eta}$-Einstein manifold or semiparallel. We also investigate the extended classes of semiparallel and Weyl semiparallel Sasakian hypersurfaces of Kaehler manifolds.

ON VOISIN'S CONJECTURE FOR ZERO-CYCLES ON HYPERKÄHLER VARIETIES

  • Laterveer, Robert
    • Journal of the Korean Mathematical Society
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    • v.54 no.6
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    • pp.1841-1851
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    • 2017
  • Motivated by the Bloch-Beilinson conjectures, Voisin has made a conjecture concerning zero-cycles on self-products of Calabi-Yau varieties. We reformulate Voisin's conjecture in the setting of $hyperk{\ddot{a}}hler$ varieties, and we prove this reformulated conjecture for one family of $hyperk{\ddot{a}}hler$ fourfolds.

SPACE-LIKE COMPLEX HYPERSURFACES OF A COMPLEX LORENTZ MANIFOLD

  • Kwon, Jung-Hwan;Nakagawa, Hisao
    • Bulletin of the Korean Mathematical Society
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    • v.26 no.1
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    • pp.75-82
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    • 1989
  • It is recently proved by Aiyama and the authors [2] that a complete space-like complex submanifold of a complex space form $M^{n+p}$$_{p}$ (c') (c'.geq.0) is to totally geodesic. This is a complex version of the Bernstein-type theorem in the Minkowski space due to Calabi [4] and Cheng and Yau [5], which is generalized by Nishikawa[7] in the Lorentz manifold satisfying the strong energy condition. The purpose of this paper is to consider his result in the complex Lorentz manifold and is to prove the following.e following.

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