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Register a new userPeriods of double octic Calabi--Yau manifolds
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We compute numerical approximations of the period integrals for eleven rigid double octic Calabi--Yau threefolds and compare them with the periods of corresponding weight our cusp forms and find, as to be expected, commensurabilities. These give information on character of the correspondences of these varieties with the associated Kuga-Sato modular threefolds.
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In the present paper we propose a combinatorial approach to study the so called double octic Clabi--Yau threefolds. We use this description to give a complete classification of double octics with $h^{1,2}le1$ and to derive their geometric properties (Kummer surface fibrations, automorphisms, special elements in families).
We prove that the categorical entropy of the autoequivalence $T_{mathcal{O}}circ(-otimesmathcal{O}(-1))$ on a Calabi-Yau manifold is the unique positive real number $lambda$ satisfying $$ sum_{kgeq 1}frac{chi(mathcal{O}(k))}{e^{klambda}}=e^{(d-1)t}. $$ We then use this result to construct the first counterexamples of a conjecture on categorical entropy by Kikuta and Takahashi.
In this paper we study Higgs and co-Higgs $G$-bundles on compact Kahler manifolds $X$. Our main results are: (1) If $X$ is Calabi-Yau, and $(E,,theta)$ is a semistable Higgs or co-Higgs $G$-bundle on $X$, then the principal $G$-bundle $E$ is semistable. In particular, there is a deformation retract of ${mathcal M}_H(G)$ onto $mathcal M(G)$, where $mathcal M(G)$ is the moduli space of semistable principal $G$-bundles with vanishing rational Chern classes on $X$, and analogously, ${mathcal M}_H(G)$ is the moduli space of semistable principal Higgs $G$-bundles with vanishing rational Chern classes. (2) Calabi-Yau manifolds are characterized as those compact Kahler manifolds whose tangent bundle is semistable for every Kahler class, and have the following property: if $(E,,theta)$ is a semistable Higgs or co-Higgs vector bundle, then $E$ is semistable.
Motivated by mirror symmetry of one-parameter models, an interesting class of Fuchsian differential operators can be singled out, the so-called Calabi--Yau operators, introduced by Almkvist and Zudilin. They conjecturally determine $Sp(4)$-local systems that underly a $mathbb{Q}$-VHS with Hodge numbers [h^{3 0}=h^{2 1}=h^{1 2}=h^{0 3}=1] and in the best cases they make their appearance as Picard--Fuchs operators of families of Calabi--Yau threefolds with $h^{12}=1$ and encode the numbers of rational curves on a mirror manifold with $h^{11}=1$. We review some of the striking properties of this rich class of operators.
For complete intersection Calabi-Yau manifolds in toric varieties, Gross and Haase-Zharkov have given a conjectural combinatorial description of the special Lagrangian torus fibrations whose existence was predicted by Strominger, Yau and Zaslow. We present a geometric version of this construction, generalizing an earlier conjecture of the first author.
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