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# Mirror symmetry for K3surfacehyperkahlermanifold

Hi there,

I have some questions about the mirror symmetry of hyperkahler manifold and K3 surface.

The well-known result said: the mirror symmetry for K3 surface is just given by its hyperkahler rotation.

1) In what sense, the rotation gives the mirror map?

2) Does this means:

if we start from $(M,\omega_I,I)$, here the k3 surface $M$ has a special lagrangian fibration structure with respect to $\omega_I$ and $I$, and also a special lagrangian section. Denote its SYZ mirror by $(M^{mirror},J_{mirror})$. Then exist a (fiberwised) diffeomorphism $\phi: M \rightarrow M^{mirror}$, s.t. $\phi^{*} (J_{mirror}) = K$?

(Here $I,K$ are the standard base of the $S^2$-family of compatible complex structure of the hyperkahler metric on $M$.)

Thanks!

3 added 4 characters in body

Hi there,

I have some questions about the mirror symmetry of K3 surface.

The well-known result said: the mirror symmetry for K3 surface is just given by its hyperkahler rotation.

1) In what sense, the rotation gives the mirror map?

2) Does this means:

if we start from $(M,\omega_I,I)$, here the k3 surface $M$ has a special lagrangian fibration structure with respect to $\omega_I$ and $I$, and also a special lagrangian section. Denote its SYZ mirror by $(M^{mirror},J_{mirror})$. Then exist a (fiberwised) diffeomorphism $\phi: M \rightarrow M^{mirror}$, s.t. $\phi^{*} (J_{mirror}) = K$?

(Here $I,K$ are the standard base of the $S^2$-family of compatible complex structure of the hyperkahler metric on $M$.)

Thanks!

2 added 11 characters in body

Hi there,

I have some questions about the mirror symmetry of K3 surface.

The well-known result said: the mirror symmetry for K3 surface is just given by its hyperkahler rotation.

1) In what sense, the rotation gives the mirror map?

2) Does this means:

if we start from $(M,\omega_I,I)$, here the k3 surface $M$ has a special lagrangian fibration structure with respect to $\omega_I$ and $I$, and also a special lagrangian section. Denote its mirror by $(M^{mirror},J_{mirror})$. Then exist a (fiberwised) diffeomorphism $\phi: M \rightarrow M^{mirror}$, s.t. $\phi^{*} (J_{mirror}) = K$?

(Here $I,K$ are the standard base of the $S^2$-family of compatible complex structure of the hyperkahler metric on $M$.)

Thanks!

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