On real Calabi–Yau threefolds twisted by a section

Abstract

AbstractWe study the mod 2 cohomology of real Calabi–Yau threefolds given by real structures that preserve the torus fibrations constructed by Gross. We extend the results of Castaño–Bernard–Matessi and Arguz–Prince to the case of real structures twisted by a Lagrangian section. In particular, we find exact sequences linking the cohomology of the real Calabi–Yau with the cohomology of the complex one. Applying Strominger–Yau–Zaslow mirror symmetry, we show that the connecting homomorphism is determined by a “twisted squaring of divisors” in the mirror Calabi–Yau, that is, by where is a divisor in the mirror and is the divisor mirror to the twisting section. We use this to find an example of a connected ‐real quintic threefold.