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Mohammad Farajzadeh-Tehrani
Mohammad Farajzadeh-Tehrani
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It is possible to construct (in many ways) a family of Calabi-Yau quintics $\mathcal{X}\rightarrow \Delta$, over disk, such that the fiber over $0$ has $A_2$a singularities (locallylocally given by the equation $x_1^2+x_2^2+x_3^2+x_4^2=0$) and the others are smooth quintics. In such families there is vanishing cycle in generic fiber which is a Lagrangian isomorphic to $S^3$ and collapses to the node at central fiber. This Lagrangian $S^3$ is obtained by the deformation of the local equation: $$x_1^2+x_2^2+x_3^2+x_4^2=0 \rightarrow x_1^2+x_2^2+x_3^2+x_4^2=\epsilon$$

Here is my questions: Consider the Fermat quintic $x_1^5+\cdots+x_5^5=0$$z_1^5+\cdots+z_5^5=0$ and its real locus which is a Lagrangian isomorphic to $S^3 /\mathbb{Z}_2$. Is there any one parameter family of C.Y 3-folds with one fiber isomorphic to Fermat quintic, such that real locus of quintic apears as vanishing cycle for this family ?

Comment: I think if there is such family then the singular fiber has to have orbifold singularities given by local equation: ${x_1^2+x_2^2+x_3^2+x_4^2=0}/\mathbb{Z}_2$ and I think its impossible to write a family of degree 5 equations with this type local singularities!

It is possible to construct (in many ways) a family of Calabi-Yau quintics $\mathcal{X}\rightarrow \Delta$, over disk, such that the fiber over $0$ has $A_2$ singularities (locally given by $x_1^2+x_2^2+x_3^2+x_4^2=0$) and the others are smooth quintics. In such families there is vanishing cycle in generic fiber which is a Lagrangian isomorphic to $S^3$ and collapses to the node at central fiber. This Lagrangian $S^3$ is obtained by the deformation of the local equation: $$x_1^2+x_2^2+x_3^2+x_4^2=0 \rightarrow x_1^2+x_2^2+x_3^2+x_4^2=\epsilon$$

Here is my questions: Consider the Fermat quintic $x_1^5+\cdots+x_5^5=0$ and its real locus which is a Lagrangian isomorphic to $S^3 /\mathbb{Z}_2$. Is there any one parameter family of C.Y 3-folds with one fiber isomorphic to Fermat quintic, such that real locus of quintic apears as vanishing cycle for this family ?

Comment: I think if there is such family then the singular fiber has to have orbifold singularities given by local equation: ${x_1^2+x_2^2+x_3^2+x_4^2=0}/\mathbb{Z}_2$ and I think its impossible to write a family of degree 5 equations with this type local singularities!

It is possible to construct (in many ways) a family of Calabi-Yau quintics $\mathcal{X}\rightarrow \Delta$, over disk, such that the fiber over $0$ has a singularities locally given by the equation $x_1^2+x_2^2+x_3^2+x_4^2=0$ and the others are smooth quintics. In such families there is vanishing cycle in generic fiber which is a Lagrangian isomorphic to $S^3$ and collapses to the node at central fiber. This Lagrangian $S^3$ is obtained by the deformation of the local equation: $$x_1^2+x_2^2+x_3^2+x_4^2=0 \rightarrow x_1^2+x_2^2+x_3^2+x_4^2=\epsilon$$

Here is my questions: Consider the Fermat quintic $z_1^5+\cdots+z_5^5=0$ and its real locus which is a Lagrangian isomorphic to $S^3 /\mathbb{Z}_2$. Is there any one parameter family of C.Y 3-folds with one fiber isomorphic to Fermat quintic, such that real locus of quintic apears as vanishing cycle for this family ?

Comment: I think if there is such family then the singular fiber has to have orbifold singularities given by local equation: ${x_1^2+x_2^2+x_3^2+x_4^2=0}/\mathbb{Z}_2$ and I think its impossible to write a family of degree 5 equations with this type local singularities!

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Mohammad Farajzadeh-Tehrani
Mohammad Farajzadeh-Tehrani
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Looking for a particular family of C.Y quintics

It is possible to construct (in many ways) a family of Calabi-Yau quintics $\mathcal{X}\rightarrow \Delta$, over disk, such that the fiber over $0$ has $A_2$ singularities (locally given by $x_1^2+x_2^2+x_3^2+x_4^2=0$) and the others are smooth quintics. In such families there is vanishing cycle in generic fiber which is a Lagrangian isomorphic to $S^3$ and collapses to the node at central fiber. This Lagrangian $S^3$ is obtained by the deformation of the local equation: $$x_1^2+x_2^2+x_3^2+x_4^2=0 \rightarrow x_1^2+x_2^2+x_3^2+x_4^2=\epsilon$$

Here is my questions: Consider the Fermat quintic $x_1^5+\cdots+x_5^5=0$ and its real locus which is a Lagrangian isomorphic to $S^3 /\mathbb{Z}_2$. Is there any one parameter family of C.Y 3-folds with one fiber isomorphic to Fermat quintic, such that real locus of quintic apears as vanishing cycle for this family ?

Comment: I think if there is such family then the singular fiber has to have orbifold singularities given by local equation: ${x_1^2+x_2^2+x_3^2+x_4^2=0}/\mathbb{Z}_2$ and I think its impossible to write a family of degree 5 equations with this type local singularities!