= Solution
The <canonical bundle> is $K_M=\bigwedge^m(T^{1,0}M)^*$, the <holomorphic line bundle> of holomorphic top forms on a complex $m$-manifold. For an embedded complex submanifold $V$ of dimension $n$, the holomorphic <normal bundle> is the quotient
$$
0\longrightarrow T^{1,0}V\longrightarrow T^{1,0}M|_V\longrightarrow N_{V/M}\longrightarrow0.
$$
Taking determinants and duals gives <adjunction for a smooth submanifold>:
$$
\boxed{K_V\cong K_M|_V\otimes\det N_{V/M}.}
$$
For a closed smooth hypersurface, $[V]=\mathcal O_M(V)$ is its <divisor line bundle>, equipped with a holomorphic section whose zero divisor is $V$ with multiplicity one. If $f_\alpha$ locally defines $V$, local meromorphic generators $1/f_\alpha$ transform by the holomorphic units $f_\alpha/f_\beta$. The stated normal-bundle relation is
$$
\boxed{[V]|_V\cong N_{V/M},\qquad K_V\cong(K_M\otimes[V])|_V.}
$$
Equivalently the conormal is $\mathcal O_M(-V)|_V$. The closed-divisor interpretation is needed to define $[V]$; an arbitrary nonclosed local submanifold does not define that global divisor bundle.
For a holomorphic subbundle $F\subset E$, define $D_Fs=\pi D_Es$ on sections of $F$, with $\pi$ acting on the bundle factor of a bundle-valued form. It obeys the <Leibniz rule>. Since $F$ is holomorphic, $D_E^{0,1}s=\bar\partial_Es$ lies in $F$, so $D_F^{0,1}=\bar\partial_F$. For $s,t\in F$, their inner products with the normal components of $D_Es,D_Et$ vanish; hence projecting preserves <metric compatibility>. The uniqueness proved for the <Chern connection> identifies
$$
\boxed{D_F=\pi\circ D_E|_F.}
$$
This is the <projected Chern connection>.
Choose the adapted smooth unitary frame and retain Question 2's input-index convention. The <connection matrix> is skew-Hermitian. Its upper-left block is $\theta_1$ for $D_F$; writing the lower-left block as $-A$ forces the upper-right block to be $\overline A^t$:
$$
\boxed{\theta_E=\begin{pmatrix}\theta_1&\overline A^t\\-A&\theta_2\end{pmatrix}.}
$$
Holomorphicity of $F$ makes the normal part of $D_E^{0,1}s$ vanish, so $B=\overline A^t$ has type $(1,0)$, and $A$ has type $(0,1)$. The matrix $B$ represents the <second fundamental form of a holomorphic subbundle> in this input-index convention. These types would appear in the opposite blocks with coefficient-column indices; they must not be swapped while retaining the printed minus-sign Cartan equation.
Take the upper-left block of the <Cartan curvature equation with input indices>. It is
$$
(\Theta_E)_{11}=d\theta_1-\theta_1\wedge\theta_1+\overline A^t\wedge A=\Theta_F+\overline A^t\wedge A.
$$
Therefore the <curvature formula for a holomorphic subbundle> is
$$
\boxed{\Theta_F=(\Theta_E)_{11}-\overline A^t\wedge A=(\Theta_E)_{11}-B\wedge\overline B^t.}
$$
The sign and the order of the factors both follow from the displayed block multiplication.
Now let $M$ be a <complex torus> and $V$ a closed smooth hypersurface. The torus has a translation-invariant flat <Kähler metric>, and $T^{1,0}M$ and $K_M$ are holomorphically trivial. Give $E=T^{1,0}M|_V$ its flat metric and apply the preceding formula to $F=T^{1,0}V$. Since $\Theta_E=0$,
$$
\Theta_{TV}=-B\wedge\overline B^t,\qquad\Theta_{K_V}=-\operatorname{tr}\Theta_{TV}=\operatorname{tr}(B\wedge\overline B^t).
$$
For a $(1,0)$ tangent vector $X$, the latter coefficient is $\operatorname{tr}(B(X)B(X)^\dagger)\geq0$, so $i\Theta_{K_V}$ is a semipositive real $(1,1)$ form. Thus $K_V$ is a <semipositive holomorphic line bundle>. This proves the <semipositive canonical bundle of a submanifold of a complex torus> assertion directly from curvature. By <adjunction for a smooth submanifold>, $K_V\cong[V]|_V$.
Restrict a positive metric on $L$ to $V$. The tensor-product metric on
$$
P_a=(L\otimes[V]^{\otimes a})|_V
$$
has curvature $i\Theta_{L|_V}+a\,i\Theta_{K_V}$, which is positive for every nonnegative integer $a$. Hence
$$
\boxed{(L\otimes[V]^{\otimes a})|_V\text{ is a positive holomorphic line bundle for every }a\in\mathbb Z_{\geq0}.}
$$
The integer condition is implicit in a tensor power. No assertion of strict positivity of $[V]|_V$ is required.
For vanishing, write $E_a=L\otimes[V]^{\otimes a}$. The allowed <Kodaira vanishing theorem> says that on a compact <Kähler manifold> $X$, $H^i(X,K_X\otimes P)=0$ for $i>0$ when $P$ is a <positive holomorphic line bundle>. Since $K_M$ is trivial, its application on the torus gives $H^i(M,E_0)=H^i(M,L)=0$ for $i>0$.
For every $a\geq1$, the divisor section gives the exact sequence
$$
0\longrightarrow E_{a-1}\longrightarrow E_a\longrightarrow i_*(E_a|_V)\longrightarrow0,
$$
where $i:V\hookrightarrow M$. Crucially,
$$
E_a|_V\cong K_V\otimes P_{a-1},
$$
and $P_{a-1}$ is positive by the proved restriction statement. The induced <Kähler metric> makes $V$ compact Kähler, so <Kodaira vanishing theorem> on $V$ gives $H^i(V,E_a|_V)=0$ for all $i>0$. In the <long exact sequence in sheaf cohomology>, the segment
$$
H^i(M,E_{a-1})\longrightarrow H^i(M,E_a)\longrightarrow H^i(V,E_a|_V)
$$
then has zero outside terms by induction. Therefore
$$
\boxed{H^i(M,L\otimes[V]^{\otimes a})=0\qquad(i>0,\ a\in\mathbb Z_{\geq0}).}
$$
This proves <vanishing for divisor twists on a complex torus> using restriction positivity and the divisor exact sequence, rather than presuming ambient positivity of every divisor twist.
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