Let M be the period domain for elliptic K3 surfaces $(X,\Omega)$ with a holomorphic two-form. Denote the fiber class $f$. Then $$M=\{\Omega\in f^\perp\otimes \mathbb{C}\,:\, \Omega\cdot \Omega=0, \,\Omega\cdot \overline{\Omega}>0\}$$ where $\cdot$ is the extension of the intersection product on the K3 lattice to $\mathbb{C}$. There is an elliptic fibration $X\rightarrow \mathbb{P}^1$ whose fiber class is $f$. Now consider a path connecting two singular fibers of $X\rightarrow \mathbb{P}^1$ and let $C$ be a cylinder lying over this path. Then the integral $$\int_C\Omega$$ is locally a well-defined function $f_C\,:\,M\rightarrow \mathbb{C}$ because any two choices of cylinder differ by a homology lying within a fiber of the elliptic fibration, and $\Omega$ restricts to zero on such a fiber. My question is the following: Is $f_C$ a linear function (in the sense that it is a restriction of a linear function on the vector space containing the quadric $M$)? This would make sense, because whenever a collection of cylinders join to form an actual homology class, the sum of these integral is linear by definition.


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