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An abundance of $$K3$$ fibrations from polyhedra with interchangeable parts. (English) Zbl 1284.14051
Three-dimensional Calabi-Yau manifolds are characterized by two independent topological invariants, the Hodge numbers $$h^{1,1}$$ and $$h^{1,2}$$. The largest class of such manifolds that have been constructed explicitly so far are the hypersurfaces in toric varieties associated with 4-dimensional reflexive polytopes. The point plot of $$y=h_{11}+h_{21}$$ vs. $$\chi=2(h^{1,1}-h^{1,2})$$ for these manifolds reveals a symmetry about the vertical axis at $$\chi=0$$, as well as further more intricate substructure. For instance, there are certain shift vectors $$\Delta(h^{1,1},h^{1,2})$$ that leave large parts of the plot invariant. Namely, if there exists a Calabi-Yau hypersurface in a toric variety with Hodge numbers $$(h^{1,1},h^{1,2})$$, then there frequently also exists such a hypersurface with Hodge numbers $$(h^{1,1},h^{1,2})+ \Delta(h^{1,1},h^{1,2})$$. The presence of these shift vectors gives the Hodge plot an intriguing “fractal”, or “self-similar” appearance.
The symmetry about the $$\chi=0$$ axis is known as mirror symmetry and was explained long time ago by V. V. Batyrev [J. Algebr. Geom. 3, No. 3, 493–535 (1994; Zbl 0829.14023)]: At the level of the toric data, the symmetry corresponds simply to an exchange of a reflexive polytope with its dual.
The purpose of this paper is to elucidate the origin of the finer substructure. The authors observe that manifolds in extremal regions of the Hodge plot admit the structure of elliptic-$$K3$$ fibrations. The associated 4-dimensional polytope can be bisected into “top” and “bottom” along a 3-dimensional interface that corresponds to the $$K3$$ fiber. Any given $$K3$$-polytope arises from many different Calabi-Yau threefold polytopes, and trade of top and bottom amongst those threefolds gives rise, at the level of the Hodge numbers, precisely to shifts of the type described above.

##### MSC:
 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) 81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
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