Are there two non-isomorphic compact Riemann surfaces with isomorphic integral Hodge structure on $H^1(-,\mathbb{Z})$?
Recall Torelli theorem for Riemann surface: For two compact Riemann surfaces $X,Y$, if there is an isomorphism $H^1(X,\mathbb{Z})\to H^1(Y,\mathbb{Z})$ preserving the intersection pairing, then $X,Y$ are biholomorphic.
I'm wondering what if we ignore the intersection pairing.