Pfaffian Calabi-Yau threefolds and mirror symmetry.

*(English)*Zbl 1274.14047This 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.

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.

Reviewer: Noriko Yui (Kingston)

##### MSC:

14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |

14J33 | Mirror symmetry (algebro-geometric aspects) |