[Paper Review] Mirror symmetry and projective geometry of Reye congruences I

Shinobu Hosono, Hiromichi Takagi|arXiv (Cornell University)|Jan 14, 2011

Algebraic Geometry and Number Theory34 references3 citations

TL;DR

This paper investigates mirror symmetry for Calabi-Yau threefolds arising from Reye congruences in $\mathbb{P}^4$, conjecturing and constructing a non-trivial Fourier-Mukai partner $Y$ for the Reye congruence Calabi-Yau threefold $X$. Using toric mirror symmetry and period integral analysis, the authors identify two large complex structure limit points, one corresponding to $X$ and the other to $Y$, which is realized as a $\mathbb{Z}_2$-cover of a determinantal quintic in $\mathbb{P}^4$ branched over a genus 26 curve. The key contribution is the explicit construction of $Y$ and the computation of BPS numbers for $X$, $Y$, and a related complete intersection $\tilde{X}_0$ via mirror symmetry.

ABSTRACT

Studying the mirror symmetry of a Calabi-Yau threefold $X$ of the Reye congruence in $\mP^4$, we conjecture that $X$ has a non-trivial Fourier-Mukai partner $Y$. We construct $Y$ as the double cover of a determinantal quintic in $\mP^4$ branched over a curve. We also calculate BPS numbers of both $X$ and $Y$ (and also a related Calabi-Yau complete intersection $ ilde X_0$) using mirror symmetry.

Motivation & Objective

• To investigate mirror symmetry for Calabi-Yau threefolds defined as Reye congruences in $\mathbb{P}^4$ via the geometry of five symmetric $(1,1)$-divisors in $\mathbb{P}^4 \times \mathbb{P}^4$.
• To conjecture the existence of a non-trivial Fourier-Mukai partner $Y$ for the Reye congruence Calabi-Yau threefold $X$, analogous to the $G(2,7)$ and $Pf(7)$ duality.
• To construct $Y$ explicitly as a $\mathbb{Z}_2$-cover of a determinantal quintic in $\mathbb{P}^4$ branched over a smooth curve of genus 26 and degree 20.
• To compute Gromov-Witten invariants and BPS numbers for $X$, $Y$, and the related complete intersection $\tilde{X}_0$ using mirror symmetry.

Proposed method

• Construct the Reye congruence $X$ as the $\mathbb{Z}_2$-quotient of a generic complete intersection $\tilde{X}_0$ of five symmetric $(1,1)$-divisors in $\mathbb{P}^4 \times \mathbb{P}^4$, using the involution $\sigma: (z,w) \mapsto (w,z)$.
• Apply the toric mirror construction of Batyrev and Borisov to build a mirror family $\mathcal{X}_0^\vee$ over $\mathbb{P}^2$, then reduce it to a diagonal family $\mathcal{X}^\vee$ over $\mathbb{P}^1$ to obtain the mirror of $X$.
• Analyze period integrals of the families and identify two distinct large complex structure limit points with maximal unipotent monodromy on $\mathcal{X}^\vee$, one corresponding to $X$ and the other to the predicted $Y$.
• Use mirror symmetry to compute Gromov-Witten invariants and BPS numbers at each boundary point, verifying predictions via the period map and monodromy analysis.
• Construct $Y$ explicitly as a double cover of a determinantal quintic threefold in $\mathbb{P}^4$, ramified along a smooth curve of genus 26 and degree 20, using the projective geometry of the Reye congruence.
• Verify the BPS numbers for $X$, $Y$, and $\tilde{X}_0$ via mirror symmetry, with explicit tables provided for $g \leq 14$ and degrees up to $d=16$.

Experimental results

Research questions

• RQ1 Does the Reye congruence Calabi-Yau threefold $X$ admit a non-trivial Fourier-Mukai partner $Y$ arising from its projective geometry?
• RQ2 Can the mirror family of $X$ be reduced to a diagonal family over $\mathbb{P}^1$, and what are the implications for the large complex structure limit points?
• RQ3 Is the predicted Calabi-Yau threefold $Y$ constructible as a $\mathbb{Z}_2$-cover of a determinantal quintic in $\mathbb{P}^4$ branched over a curve of genus 26 and degree 20?
• RQ4 Do the BPS numbers computed via mirror symmetry for $X$, $Y$, and $\tilde{X}_0$ match expected invariants from Gromov-Witten theory?
• RQ5 Is the derived category equivalence $D^b(Coh(X)) \cong D^b(Coh(Y))$ expected to hold in this setting, as in the $G(2,7)$ and $Pf(7)$ case?

Key findings

• The Reye congruence Calabi-Yau threefold $X$ is constructed as the $\mathbb{Z}_2$-quotient of a complete intersection $\tilde{X}_0$ of five symmetric $(1,1)$-divisors in $\mathbb{P}^4 \times \mathbb{P}^4$, with Hodge numbers $h^{1,1}(X) = 1$, $h^{2,1}(X) = 26$.
• The mirror family $\mathcal{X}_0^\vee$ is constructed over $\mathbb{P}^2$ via the Batyrev-Borisov toric method, and reduced to a diagonal family $\mathcal{X}^\vee$ over $\mathbb{P}^1$, yielding two large complex structure limit points with maximal unipotent monodromy.
• One of the two boundary points corresponds to the mirror of $X$, and the other predicts the existence of a new Calabi-Yau threefold $Y$, which is constructed as a $\mathbb{Z}_2$-cover of a determinantal quintic in $\mathbb{P}^4$ branched over a smooth curve of genus 26 and degree 20.
• The BPS numbers for $X$, $Y$, and $\tilde{X}_0$ are computed via mirror symmetry and tabulated up to genus $g=14$ and degree $d=16$, with explicit values provided in Tables 1–6 of the appendix.
• For $X$, the BPS number $n_1^X(1,1) = 650$, $n_2^X(1,1) = 148525$, and $n_3^X(1,1) = 3270050$; for $Y$, $n_1^Y(1,1) = 1475$, $n_2^Y(1,1) = 29350$, and $n_3^Y(1,1) = 148525$, showing a duality pattern.
• The BPS numbers for $\tilde{X}_0$ are computed in terms of bidegrees $(i,j)$, with $n_0^{\tilde{X}_0}(1,1) = 650$, $n_1^{\tilde{X}_0}(1,1) = 29350$, and $n_2^{\tilde{X}_0}(1,1) = 148525$, confirming consistency with mirror symmetry.