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Pfaffian Calabi-Yau threefolds and mirror symmetry. (English) Zbl 1274.14047
This paper constructs four new examples of Calabi-Yau threefolds with $$h^{1,1}=1$$. These Calabi-Yau threefolds are Pfaffian Calabi-Yau threefolds in weighted projective spaces, which are non-complete intersections. The existence of non-complete intersection Pfaffian Calabi-Yau threefolds was conjectured by C. van Enckevort and D. van Straten [AMS/IP Studies in Advanced Mathematics 38, 539–559 (2006; Zbl 1117.14043)]. The main result of the paper is the following:
Theorem. There exist four Pfaffian threefolds $$X_5, X_7, X_{10}$$ and $$X_{25}$$, which are smooth Calabi-Yau threefolds with $$h^{1,1}=1$$. All of these are non-complete intersections. The fundamental topological invariants $$\int_X H^3$$, and $$\int_X c_2(X)\cdot H$$ and $$c_3(X)$$ are also computed for these Calabi-Yau threefolds.
Complete intersections of Pfaffian varieties and hypersurfaces in weighted projective spaces are studied focusing on examples. For instance, $$X_{25}$$ is such an example, and it turns out that the Calabi-Yau equation has two maximally unipotent monodromy points of the same type.
Next mirror families for degree $$5,7,10$$ Pfaffian Calabi-Yau threefolds are constructed, following a detailed discussion on mirror construction of the degree $$13$$ Pfaffian Calabi-Yau threefold, which is not a complete intersection. $$X_{13}$$ was constructed by F. Tonoli [J. Algebr. Geom. 13, No. 2, 209–232 (2004; Zbl 1060.14060)], and a candidate mirror partner was proposed by J. Böhm [“Mirror symmetry and tropical geometry”, arXiv:0708.4402].
This paper computes the fundamental period integrals and Picard-Fuchs differential equations for the degree $$13$$ Calabi-Yau threefold first and then for the degree $$5,7$$ and $$10$$ new Pfaffian Calabi-Yau threefolds. It is verified that the Picard-Fuchs differential equations coincide with the predicted Calabi-Yau equations in the aforementioned article of van Enkevort and van Straten.

##### MSC:
 14J32 Calabi-Yau manifolds (algebro-geometric aspects) 14J33 Mirror symmetry (algebro-geometric aspects)
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