I'm trying to understand a part of N.J.Hitchin's paper "The moduli space of complex Lagrangian submanifolds" https://arxiv.org/pdf/math/9901069.pdf . In particular I would like to understand how any given holomorphic function $\mathcal{F}$ will naturally give a special Kahler manifold.

In the following I will try my best to summarize his Remark 1 (page 11). Let $V$ be a real symplectic vector space with coordinates $\{x_1,...,x_{2n}\}$ and a symplectic form $\omega = \sum_{i = 1}^n dx_{i} \wedge dx_{n+i}$. Consider $V \times V$, let's denote a coordinates on the second factor by $\{\xi_1,...,\xi_{2n}\}$ then we introduce a complex structure $I$ on $V\times V$ by \begin{equation} I\frac{\partial}{\partial x_i} = \frac{\partial}{\partial \xi_{n+i}}, \qquad I\frac{\partial}{\partial \xi_i} = \frac{\partial}{\partial x_{n+i}}. \end{equation} So we can define a holomorphic complex coordinates $\{v_i = x_i + i\xi_{n+i}, w_i = \xi_i + ix_{n+i}|i = 1,...,n\}$ and the canonical complex symplectic form $\omega^c = \sum_{i = 1}^n dv_i \wedge dw_i = \omega_1 + i\omega_2$ where \begin{equation} \omega_1 = \sum_{i = 1}^{2n}dx_i\wedge d\xi_i, \qquad \omega_2 = \sum_{i=1}^n (dx_i \wedge dx_{n+i} - d\xi_i \wedge d\xi_{n+i}). \end{equation}

The claim is that given any holomorphic function $\mathcal{F} = \mathcal{F}(w_1,...,w_n)$ a complex Lagrangian submanifold $M = \{z_i = \partial\mathcal{F}/\partial w_i\} \subset V\times V$ is special Kahler with complex structure $I|_M$.

According to the paper this follows since $\omega^c|_M = 0 \implies \omega_i|_M = 0, i = 1,2$ hence $M$ is bi-Lagrangian, then

"From Theorem 2 this is all we need for a special Kahler manifold" (page 12).

I'm having a difficult time understanding this because Theorem 2 (page 7) also required that $M\subset V\times V$ is transversal to two projections onto $V$ in addition to the bi-Lagrangian condition. I interpreted the transversal condition as the two projections of $M$ are surjective to $V$, which seems to be quite an important part in proving Theorem 2. However I don't see why \begin{equation} M = \Big\{v_i = \frac{\partial \mathcal{F}}{\partial w_i}\Big\} = \Big\{x_i = \frac{\partial \Re \mathcal{F}}{\partial \xi_i} = \frac{\partial \Im \mathcal{F}}{\partial x_{n+i}},\ \xi_{n+i} = \frac{\partial \Im \mathcal{F}}{\partial \xi_i} = -\frac{\partial \Re \mathcal{F}}{\partial x_{n+i}}\Big\} \end{equation} would satisfies such condition for an arbitrary $\mathcal{F}$. For example I could take $\mathcal{F} = 0$ then obviously $x_i$ and $\xi_{n+i}$ would be fixed and transversality condition would failed.

Question : what exactly did I miss in trying to follow his arguments above? Or, if not every holomorphic functions $\mathcal{F}$ would yield a special Kahler Lagrangian then what is a criterion to determine which $\mathcal{F}$ would works?

Thank you!


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