The Sullivan Conjecture for smooth complete intersections

A complete intersection $X_n(\underline d)\subseteq\mathbb{C}P^{n+k}$ is the transverse intersection of $k$ complex hypersurfaces of multidegree $\underline d=(d_1,\ldots,d_k)$. Let $d=d_1\cdots d_k$ be its total degree. Its Sullivan data is

$$SD_n(\underline d):=\bigl(d,(p_i(n,\underline d))_{i=1}^{\lfloor n/2\rfloor},\chi(X_n(\underline d))\bigr),$$

where $p_i(n,\underline d)$ are defined by $p_i(X_n(\underline d))=p_i(n,\underline d)x^{2i}$ and $x\in H^2(X_n(\underline d))$ is the pullback of a generator of $H^2(\mathbb{C}P^{n+k})$. The Sullivan Conjecture. Suppose that $X_n(\underline d)$ and $X_n(\underline d')$ are complete intersections with $SD_n(\underline d)=SD_n(\underline d')$. If $n\geq 3$, then $X_n(\underline d)$ is diffeomorphic to $X_n(\underline d')$. This conjecture organises the smooth classification of complete intersections; prior results established it under divisibility hypotheses and up to the action of the group of homotopy $8$-spheres, while the paper studies the remaining diffeomorphism and inertia-group issues in dimension $4$.