On the algebraicity of some products of special values of Barnes' multiple gamma function

We consider partial zeta functions $\zeta (s,c)$ associated with ray classes $c$’s of a totally real field. Stark’s conjecture implies that an appropriate product of $\exp ({\zeta }^{\mathrm{\prime }}(0,c))$‘s is an algebraic number which is called a Stark unit. Shintani gave an explicit formula for $\exp ({\zeta }^{\mathrm{\prime }}(0,c))$ in terms of Barnes’ multiple gamma function. Yoshida ``decomposed’’ Shintani’s formula: he defined the symbol $X(c,\iota )$ satisfying that $\exp ({\zeta }^{\mathrm{\prime }}(0,c))={\prod }_{\iota }\exp (X(c,\iota ))$ where $\iota$ runs over all real embeddings of $F$. Hence we can decompose a Stark unit into a product of $[F:\mathbb{Q}]$ terms. The main result is to show that $([F:\mathbb{Q}]-1)$ of them are algebraic numbers. We also study a relation between Yoshida’s conjecture on CM-periods and Stark’s conjecture.