Examples of complex manifolds for which the logarithmic cotangent bundle is big, but the cotangent bundle is not big Let $(X,D)$ be a log pair, with $X$ a projective manifold (or quasi-projective) and $D$ a divisor with simple normal crossings.

I'd like to construct an example, or be pointed to a reference, for an
example where $\Omega_X^1(\log D)$ is big, but $\Omega_X^1$ is not
big.

There are metric characterizations of such examples due to Cadorel (in a 2016 paper), which was later developed by Guenancia. That is, $\Omega_X^1(\log D)$ is big if there is a Kähler metric $\omega$ on $X \backslash D$ with negative holomorphic sectional curvature on $X\backslash D$ and has nonpositive bisectional curvature. Then $\Omega_X^1(\log D)$ is big. If, moreover, $\omega$ is locally bounded on $X$, then $\Omega_X^1$ is big.
 A: Given any smooth projective variety $X$, it is always possible to find a normal crossing divisor $D$ such that $\Omega^1(\log D)$ is big (actually, such that it satisfies a stronger condition called almost ampleness).
This follows from the following result, see [BD18, Theorem A].

Theorem. Let $X$ be a smooth projective variety of dimension $n$ and $c \geq n$. Let $L$ be a very ample line bundle on $X$. For any $m
 \geq (4n)^{n+2}$ and for general $H_1, \ldots, H_c \in |L^m|$, writing $D=\sum_{i=1}^c H_i$, the logarithmic tangent bundle $\Omega^1(\log D)$ is almost ample.

The result is optimal in $c$: in fact, the authors prove that, if $X=\mathbb{P}^n$ and $c <n$, then the logarithmic cotangent bundle cannot be almost ample as it is not even big.
Regarding your question, you can now obtain plenty of examples starting from your preferred variety with non-big cotangent bundle ($\mathbb{P}^n$, an abelian variety, a K3 surface, etc.) and applying the construction described in the theorem above.
References.
[BD18] D. Brotbeck, Y. Deng: On the positivity of the logarithmic cotangent bundle,
Ann. Inst. Fourier 68, 7 (2018), 3001-3051.
