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Consider a map $$ f: \Sigma \to X/\sigma,$$ where $\Sigma=\Sigma_g/\Omega$ is a quotient of a Riemann surface by an antiholomorphic involution, $\sigma:X\to X$ is an antiholomorphic involution of some CY threefold $X$ (with pointwise fixed set $L$) and the holomorphic map $\hat f: \Sigma_g \to \Sigma_g$ is equivariant, $\hat f \circ \Omega = \sigma \circ \hat f$, and descends to $f$.

How can we get a nice intuition about the degree $$d := f_* ([\Sigma]) \in H_2(X,L;\mathbb{Z}),$$ namely is it true e.g. that $d=\sum d_i$ where each $d_i$ comes either from a boundary or a crosscap (i.e. an $\mathbb{RP}^2$) of $\Sigma$, and thus it is somehow 'localized' on $L$?

(this question has a double on math.SE)

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