  We present new anomalies in two-dimensional ${\mathcal N} =(2, 2)$
superconformal theories. They obstruct the shortening conditions of chiral and
twisted chiral multiplets at coincident points. This implies that marginal
couplings cannot be promoted to background superfields in short
representations. Therefore, standard results that follow from ${\mathcal N}
=(2, 2)$ spurion analysis are invalidated. These anomalies appear only if
supersymmetry is enhanced beyond ${\mathcal N} =(2, 2)$. These anomalies
explain why the conformal manifolds of the K3 and $T^4$ sigma models are not
K\"ahler and do not factorize into chiral and twisted chiral moduli spaces and
why there are no ${\mathcal N} =(2, 2)$ gauged linear sigma models that cover
these conformal manifolds. We also present these results from the point of view
of the Riemann curvature of conformal manifolds.
