The minimal-class characterization by strong theta-regularity

Let $(A,\Theta)$ be an irreducible principally polarized abelian variety, and let $Y\subset A$ be a non-degenerate subvariety. Say that $Y$ represents a minimal class when its codimension is $d$ and its class is $[\Theta]^d/d!$. Say that the ideal sheaf $\mathcal I_Y$ is strongly $3$-$\Theta$-regular when $\mathcal I_Y(2\Theta_\alpha)$ satisfies the Index Theorem with index $0$, namely

$$H^i\bigl(\mathcal I_Y((2\Theta)_\alpha)\bigr)=0,\qquad \forall i>0,\quad \forall \alpha\in \operatorname{Pic}^0(A).$$

The minimal-class characterization by strong theta-regularity. The subvariety $Y$ represents a minimal class if and only if its ideal sheaf is strongly $3$-$\Theta$-regular. This proposes a Castelnuovo-type characterization of the special minimal-class subvarieties of principally polarized abelian varieties. Strong $3$-$\Theta$-regularity is motivated by the known examples of the special subvarieties $W_d$ in Jacobians and, for a general cubic threefold, the Fano surface of lines; the stated equivalence is not resolved in the supplied text.