Let
X
X
be a smooth projective variety over the complex numbers, and let
D
⊂
X
D\subset X
be an ample divisor. For which spaces
Y
Y
is the restriction map
r
:
H
o
m
(
X
,
Y
)
→
H
o
m
(
D
,
Y
)
\begin{equation*}r: \mathrm {Hom}(X, Y)\to \mathrm {Hom}(D, Y) \end{equation*}
an isomorphism?
Using positive characteristic methods, we give a fairly exhaustive answer to this question. An example application of our techniques is: if
dim
(
X
)
≥
3
\dim (X)\geq 3
,
Y
Y
is smooth,
Ω
Y
1
\Omega ^1_Y
is nef, and
dim
(
Y
)
>
dim
(
D
)
,
\dim (Y)> \dim (D),
the restriction map
r
r
is an isomorphism. Taking
Y
Y
to be the classifying space of a finite group
B
G
BG
, the moduli space of pointed curves
M
g
,
n
\mathscr {M}_{g,n}
, the moduli space of principally polarized Abelian varieties
A
g
\mathscr {A}_g
, certain period domains, and various other moduli spaces, one obtains many new and classical Lefschetz hyperplane theorems.