# Calibrated submanifolds in Spin(7) and Calabi-Yau threefold

Suppose I have a Cayley cycle in a $Spin(7)$ holonomy manifold $M$, i.e. a calibrated submanifold. In the special case that $M=CY_3\times T^2$, is it possible that the Cayley cycle reduces to a Kahler cycle, i.e. a calibrated submanifold of $CY_3$?

• Please see "how to ask question". – BS. Jun 10 '17 at 7:05
• The literal answer to your question is 'yes, it is possible that it does (though it does not necessarily do so)', but you may have meant to ask something like 'is it possible that all the Cayley cycles in $M$ are Kahler cycles in $M$ as an SU(4)-holonomy manifold' or you may have meant something else entirely. As BS suggested, you need to formulate your question more carefully if you want an answer. Also, you should not assume that everyone knows what you mean by '$CY_3$'; giving a little background or a reference helps. – Robert Bryant Jun 10 '17 at 12:00