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Let $X$ be a smooth complex del Pezzo surface which forms non-trivial family (for example the cubic surface). Then by a Theorem of Tian every variety in the family will have a KE-metric ( Tian, G. On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), no. 1, 101–172).

Note that Tian's theorem states that all del Pezzo surfaces which are not a blow-up of $\mathbb{P}^{2}$ in 1 or 2 points have a KE-metric, but these come in trivial families.

It is proven in Example 3.9 here (https://people.math.ethz.ch/~salamon/PREPRINTS/unique.pdf), that for any two symplectic forms $\omega_{1},\omega_{2}$ on a del Pezzo surface $X$ such that $[\omega_{1}] = [\omega_{2}] \in H^{2}(X,\mathbb{R})$ there exists a diffeomorphism $f : X \rightarrow X$ such that $f^{*}(\omega_{2}) = \omega_{1}$ (see the conclusion at the bottom of page 16). Hence the underlying symplectic form of each of these KE manifolds are the same, but they cannot be identified biholomorphically, whilst simultaneously preserving the symplectic structure. This provides a negative answer to question 1.

Let $X$ be a smooth complex del Pezzo surface which forms non-trivial family (for example the cubic surface). Then by a Theorem of Tian every variety in the family will have a KE-metric ( Tian, G. On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), no. 1, 101–172).

Note that Tian's theorem states that all del Pezzo surfaces which are not a blow-up of $\mathbb{P}^{2}$ in 1 or 2 points have a KE-metric, but the 1 and 2 point blow-ups come in trivial families.

It is proven in Example 3.9 here (https://people.math.ethz.ch/~salamon/PREPRINTS/unique.pdf), that for any two symplectic forms $\omega_{1},\omega_{2}$ on a del Pezzo surface $X$ such that $[\omega_{1}] = [\omega_{2}] \in H^{2}(X,\mathbb{R})$ there exists a diffeomorphism $f : X \rightarrow X$ such that $f^{*}(\omega_{2}) = \omega_{1}$ (see the conclusion at the bottom of page 16). Hence the underlying symplectic form of each of these KE manifolds are the same, but they cannot be identified biholomorphically, whilst simultaneously preserving the symplectic structure. This provides a negative answer to question 1.

Let $X$ be a smooth complex del Pezzo surface which forms non-trivial family (for example the cubic surface). Then by a Theorem of Tian every variety in the family will have a KE-metric ( Tian, G. On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), no. 1, 101–172).

Note that Tian's theorem states that all del Pezzo surfaces which are not a blow-up of $\mathbb{P}^{2}$ in 1 or 2 points have a KE-metric, but these come in trivial families.

It is proven in Example 3.9 here (https://people.math.ethz.ch/~salamon/PREPRINTS/unique.pdf), that any two symplectic forms on a del Pezzo surface in the same cohomology class are symplectomorphic (see the conclusion at the bottom of page 16). Hence the underlying symplectic form of each of these KE manifolds are the same, but they cannot be identified biholomorphically, whilst simultaneously preserving the symplectic structure. This provides a negative answer to question 1.

Let $X$ be a smooth complex del Pezzo surface which forms non-trivial family (for example the cubic surface). Then by a Theorem of Tian every variety in the family will have a KE-metric ( Tian, G. On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), no. 1, 101–172).

Note that Tian's theorem states that all del Pezzo surfaces which are not a blow-up of $\mathbb{P}^{2}$ in 1 or 2 points have a KE-metric, but these come in trivial families.

It is proven in Example 3.9 here (https://people.math.ethz.ch/~salamon/PREPRINTS/unique.pdf), that for any two symplectic forms $\omega_{1},\omega_{2}$ on a del Pezzo surface $X$ such that $[\omega_{1}] = [\omega_{2}] \in H^{2}(X,\mathbb{R})$ there exists a diffeomorphism $f : X \rightarrow X$ such that $f^{*}(\omega_{2}) = \omega_{1}$ (see the conclusion at the bottom of page 16). Hence the underlying symplectic form of each of these KE manifolds are the same, but they cannot be identified biholomorphically, whilst simultaneously preserving the symplectic structure. This provides a negative answer to question 1.

Let $X$ be a smooth complex del Pezzo surface which forms non-trivial family (for example the cubic surface). Then by a Theorem of Tian every variety in the family will have a KE-metric ( Tian, G. On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), no. 1, 101–172).

Note that Tian's theorem states that all del Pezzo surfaces which are not a blow-up of $\mathbb{P}^{2}$ in 1 or 2 points have a KE-metric, but these come in trivial families.

By Moser's trick the Kähler forms on each variety of the family are abstractly symplectomorphic (see here symplectic form on an algebraic family), but they cannot be identified biholomorphically, whilst simultaneously preserving the symplectic structure. This provides a negative answer to question 1.

Let $X$ be a smooth complex del Pezzo surface which forms non-trivial family (for example the cubic surface). Then by a Theorem of Tian every variety in the family will have a KE-metric ( Tian, G. On Calabi’s conjecture for complex surfaces with positive first Chern class, Invent. Math. 101 (1990), no. 1, 101–172).

Note that Tian's theorem states that all del Pezzo surfaces which are not a blow-up of $\mathbb{P}^{2}$ in 1 or 2 points have a KE-metric, but these come in trivial families.

It is proven in Example 3.9 here (https://people.math.ethz.ch/~salamon/PREPRINTS/unique.pdf), that any two symplectic forms on a del Pezzo surface in the same cohomology class are symplectomorphic (see the conclusion at the bottom of page 16). Hence the underlying symplectic form of each of these KE manifolds are the same, but they cannot be identified biholomorphically, whilst simultaneously preserving the symplectic structure. This provides a negative answer to question 1.