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Lets see why the study of moduli space of complex structures is closely related to mirror pair.

From mirror map, we can heuristically identify the moduli space of symplectic structures $\mathcal M_{sym}$ and moduli space of complex structures $\mathcal M_{cpx}$ as follows $$\mathcal M_{sym}(M)\cong\mathcal M_{cpx}(M^\vee)$$ and $$\mathcal M_{sym}(M^\vee)\cong\mathcal M_{cpx}(M)$$

where $M$ and $M^\vee$ are mirror to each other.

We have the following relation for tangent space of moduli space of symplectic structures

$$T_\omega\mathcal M_{sym}(M)\cong H^2(M)$$

and from BTT Theorem we have $$T_J\mathcal M_{cpx}(M)\cong H^1(M,TM)$$ for tangent space of moduli of complex structures.

So from mirror map  , we have $$H^2(M)=H^1(M^\vee, TM^\vee)$$

More generally if we take extended moduli space of complex and symplectic structures then

$$T_\omega\mathcal M_{sym}^{ext}(M)\cong H^*(M)$$

and

$$T_J\mathcal M_{cpx}^{ext}(M)\cong \oplus_{p,q}H^q(M,\wedge^pTM)$$

and hence from extended mirror map we can identify

$$H^q(M,\Omega^pM)\cong H^q(M^\vee, \wedge^pTM^\vee)$$ and in the case when the mirror $M^\vee$ is itself Calabi-Yau variety then

$$H^q(M,\Omega^pM)\cong H^q(M^\vee, \wedge^pTM^\vee)\cong H^q(M^\vee, \Omega^{n-p}M^\vee)$$

See Kontsevich paper

Let's see why the study of moduli space of complex structures is closely related to mirror pairs.

From the mirror map, we can heuristically identify the moduli space of symplectic structures $\mathcal M_{sym}$ and moduli space of complex structures $\mathcal M_{cpx}$ as follows $$\mathcal M_{sym}(M)\cong\mathcal M_{cpx}(M^\vee)$$ and $$\mathcal M_{sym}(M^\vee)\cong\mathcal M_{cpx}(M)$$

where $M$ and $M^\vee$ are mirror to each other.

We have the following relation for tangent space of moduli space of symplectic structures

$$T_\omega\mathcal M_{sym}(M)\cong H^2(M)$$

and from the BTT Theorem we have $$T_J\mathcal M_{cpx}(M)\cong H^1(M,TM)$$ for the tangent space of moduli of complex structures.

So from the mirror map, we have $$H^2(M)=H^1(M^\vee, TM^\vee)$$

More generally if we take extended moduli space of complex and symplectic structures then

$$T_\omega\mathcal M_{sym}^{ext}(M)\cong H^*(M)$$

and

$$T_J\mathcal M_{cpx}^{ext}(M)\cong \oplus_{p,q}H^q(M,\wedge^pTM)$$

and hence from the extended mirror map we can identify

$$H^q(M,\Omega^pM)\cong H^q(M^\vee, \wedge^pTM^\vee)$$ and in the case when the mirror $M^\vee$ is itself a Calabi-Yau variety then

$$H^q(M,\Omega^pM)\cong H^q(M^\vee, \wedge^pTM^\vee)\cong H^q(M^\vee, \Omega^{n-p}M^\vee)$$

See this Kontsevich paper.

Lets see why the study of moduli space of complex structures is closely related to mirror pair.

From mirror map we can heuristically identify the moduli space of symplectic structures $\mathcal M_{sym}$ and moduli space of complex structures $\mathcal M_{cpx}$ as follows $$\mathcal M_{sym}(M)\cong\mathcal M_{cpx}(M^\vee)$$ and $$\mathcal M_{sym}(M^\vee)\cong\mathcal M_{cpx}(M)$$

where $M$ and $M^\vee$ are mirror to each other.

We have the following relation for tangent space of moduli space of symplectic structures

$$T_\omega\mathcal M_{sym}(M)\cong H^2(M)$$

and from BTT theorem we have $$T_J\mathcal M_{cpx}(M)\cong H^1(M,TM)$$ for tangent space of moduli of complex structures.

So from mirror map , we have $$H^2(M)=H^1(M^\vee, TM^\vee)$$

More generally if we take extended moduli space of complex and symplectic structures then

$$T_\omega\mathcal M_{sym}^{ext}(M)\cong H^*(M)$$

and

$$T_J\mathcal M_{cpx}^{ext}(M)\cong \oplus_{p,q}H^q(M,\wedge^pTM)$$

and hence from extended mirror map we can identify

$$H^q(M,\Omega^pM)\cong H^q(M^\vee, \wedge^pTM^\vee)$$ and in the case when the mirror $M^\vee$ is itself Calabi-Yau variety then

$$H^q(M,\Omega^pM)\cong H^q(M^\vee, \wedge^pTM^\vee)\cong H^q(M^\vee, \Omega^{n-p}M^\vee)$$

See Kontsevich paper

Lets see why the study of moduli space of complex structures is closely related to mirror pair.

From mirror map, we can heuristically identify the moduli space of symplectic structures $\mathcal M_{sym}$ and moduli space of complex structures $\mathcal M_{cpx}$ as follows $$\mathcal M_{sym}(M)\cong\mathcal M_{cpx}(M^\vee)$$ and $$\mathcal M_{sym}(M^\vee)\cong\mathcal M_{cpx}(M)$$

where $M$ and $M^\vee$ are mirror to each other.

We have the following relation for tangent space of moduli space of symplectic structures

$$T_\omega\mathcal M_{sym}(M)\cong H^2(M)$$

and from BTT Theorem we have $$T_J\mathcal M_{cpx}(M)\cong H^1(M,TM)$$ for tangent space of moduli of complex structures.

So from mirror map , we have $$H^2(M)=H^1(M^\vee, TM^\vee)$$

More generally if we take extended moduli space of complex and symplectic structures then

$$T_\omega\mathcal M_{sym}^{ext}(M)\cong H^*(M)$$

and

$$T_J\mathcal M_{cpx}^{ext}(M)\cong \oplus_{p,q}H^q(M,\wedge^pTM)$$

and hence from extended mirror map we can identify

$$H^q(M,\Omega^pM)\cong H^q(M^\vee, \wedge^pTM^\vee)$$ and in the case when the mirror $M^\vee$ is itself Calabi-Yau variety then

$$H^q(M,\Omega^pM)\cong H^q(M^\vee, \wedge^pTM^\vee)\cong H^q(M^\vee, \Omega^{n-p}M^\vee)$$

See Kontsevich paper

Lets see why the study of moduli space of complex structure is closely related to mirror pair.

From mirror map we can heuristically identify the moduli space of symplectic structures $\mathcal M_{sym}$ and moduli space of complex structures $\mathcal M_{cpx}$ as follows $$\mathcal M_{sym}(M)\cong\mathcal M_{cpx}(M^\vee)$$ and $$\mathcal M_{sym}(M^\vee)\cong\mathcal M_{cpx}(M)$$

where $M$ and $M^\vee$ are mirror to each other.

We have the following relation for tangent space of moduli space of symplectic structures

$$T_\omega\mathcal M_{sym}(M)\cong H^2(M)$$

and from BTT theorem we have $$T_J\mathcal M_{cpx}(M)\cong H^1(M,TM)$$ for tangent space of moduli of complex structures.

So from mirror map , we have $$H^2(M)=H^1(M^\vee, TM^\vee)$$

More generally if we take extended moduli space of complex and symplectic structures then

$$T_\omega\mathcal M_{sym}^{ext}(M)\cong H^*(M)$$

and

$$T_J\mathcal M_{cpx}^{ext}(M)\cong \oplus_{p,q}H^q(M,\wedge^pTM)$$

and hence from extended mirror map we can identify

$$H^q(M,\Omega^pM)\cong H^q(M^\vee, \wedge^pTM^\vee)$$ and in the case when the mirror $M^\vee$ is itself Calabi-Yau variety then

$$H^q(M,\Omega^pM)\cong H^q(M^\vee, \wedge^pTM^\vee)\cong H^q(M^\vee, \Omega^{n-p}M^\vee)$$

See Kontsevich paper

Lets see why the study of moduli space of complex structures is closely related to mirror pair.

From mirror map we can heuristically identify the moduli space of symplectic structures $\mathcal M_{sym}$ and moduli space of complex structures $\mathcal M_{cpx}$ as follows $$\mathcal M_{sym}(M)\cong\mathcal M_{cpx}(M^\vee)$$ and $$\mathcal M_{sym}(M^\vee)\cong\mathcal M_{cpx}(M)$$

where $M$ and $M^\vee$ are mirror to each other.

We have the following relation for tangent space of moduli space of symplectic structures

$$T_\omega\mathcal M_{sym}(M)\cong H^2(M)$$

and from BTT theorem we have $$T_J\mathcal M_{cpx}(M)\cong H^1(M,TM)$$ for tangent space of moduli of complex structures.

So from mirror map , we have $$H^2(M)=H^1(M^\vee, TM^\vee)$$

More generally if we take extended moduli space of complex and symplectic structures then

$$T_\omega\mathcal M_{sym}^{ext}(M)\cong H^*(M)$$

and

$$T_J\mathcal M_{cpx}^{ext}(M)\cong \oplus_{p,q}H^q(M,\wedge^pTM)$$

and hence from extended mirror map we can identify

$$H^q(M,\Omega^pM)\cong H^q(M^\vee, \wedge^pTM^\vee)$$ and in the case when the mirror $M^\vee$ is itself Calabi-Yau variety then

$$H^q(M,\Omega^pM)\cong H^q(M^\vee, \wedge^pTM^\vee)\cong H^q(M^\vee, \Omega^{n-p}M^\vee)$$

See Kontsevich paper