The Sullivan Conjecture for complete intersections of complex dimension at least 8

A smooth complete intersection $X_n(\mathbf d)$ in complex projective space is the transverse intersection of hypersurfaces of degrees $d_1,\dots,d_r$; by an observation of Thom its diffeomorphism type is determined by the multidegree $\mathbf d$, but the multidegree is not itself a topological invariant. The organizing open problem is the Sullivan Conjecture: for complex dimension $n \ge 3$, two complete intersections with the same Sullivan data — total degree $d_1\cdots d_r$, integral Pontryagin classes, and Euler characteristic — must be diffeomorphic. The conjecture is proved for $n = 3$ and $n = 4$ (the smooth case for $n=4$ completed by Crowley–Nagy in 2025), holds up to homeomorphism for $n=5,6,7$ (Fang–Wang), and holds in all dimensions $\ge 3$ under strong prime-divisibility assumptions on the total degree (Kreck's Theorem A). For $n \ge 8$ no classification theorem covering the general case (homeomorphism or diffeomorphism, beyond the divisible case and the small-codimension range $r \le (n+2)/2$, in which the multidegree is determined by the total degree and Pontryagin classes) is known; the dimension-$n \ge 8$ segment of the conjecture remains an open problem.

Background

A smooth complete intersection $X_n(d_1,\dots,d_r) \subset \mathbb{CP}^{n+r}$ is the common zero locus of $r$ homogeneous polynomials in $\mathbb{CP}^{n+r}$, of degrees $d_1,\dots,d_r$, meeting transversely; it is a smooth complex projective variety of complex dimension $n$. The unordered tuple $\mathbf d=(d_1,\dots,d_r)$ is the multidegree (counted up to permutation and insertion or deletion of entries equal to $1$, which do not change the variety), and $d = d_1 d_2 \cdots d_r$ is the total degree. By an observation of Thom the diffeomorphism type of $X_n(\mathbf d)$ depends only on the multidegree. The multidegree, however, is not a topological invariant — for instance $X_2(4)$, $X_2(3,2)$ and $X_2(2,2,2)$ are all diffeomorphic, each being a K3 surface (a simply connected complex surface with second Betti number 22) — so the natural classification question is by numerical topological invariants.

By the Lefschetz hyperplane section theorem, for $n \ge 3$ the manifold $X_n(\mathbf d)$ is simply connected with $H^2 \cong \mathbb{Z}$ generated by the hyperplane class $x$. Writing power sums $s_k = \sum_{j=1}^r d_j^{\,k}$, Libgober and Wood expressed all Chern classes, Pontryagin classes and the Euler characteristic as explicit universal polynomials in $n$, $r$, $d$ and the $s_k$; in particular each Pontryagin class is an integral multiple of $x^{2k}$, written $p_k = p_k(n,\mathbf d)\,x^{2k}$, so these invariants can be compared across complete intersections as integers. Following the formulation of Crowley and Nagy, the Sullivan data of $X_n(\mathbf d)$ is the tuple

$$SD_n(\mathbf d) = \bigl(d,\; (p_k(n,\mathbf d))_{1 \le k \le \lfloor n/2 \rfloor},\; \chi(X_n(\mathbf d))\bigr) \in \mathbb{Z}_{>0} \times \mathbb{Z}^{\lfloor n/2 \rfloor} \times \mathbb{Z},$$

consisting of the total degree, the Pontryagin integers, and the Euler characteristic. Each entry is a diffeomorphism invariant (for $n \ge 3$ even the total degree is, by results of Libgober–Wood), and conversely the Sullivan Conjecture asserts completeness: for $n \ge 3$, $SD_n(\mathbf d) = SD_n(\mathbf d')$ implies $X_n(\mathbf d) \cong X_n(\mathbf d')$ (orientation-preserving diffeomorphism).

The conjecture is known in low dimensions. For $n=1$ it is the classical classification of surfaces; for $n=2$ the homeomorphism classification follows from topological 4-manifold theory (same $p_1$ and Euler characteristic), while smooth classification is out of reach and homeomorphic non-diffeomorphic complete-intersection surfaces are known. For $n=3$ the conjecture follows from classical classification theorems for simply connected 6-manifolds. For $n=4$, Fang and Klaus proved it up to connected sum with a homotopy 8-sphere (a smooth manifold homeomorphic, but not known a priori to be diffeomorphic, to the 8-sphere), and Crowley and Nagy (2025) removed that ambiguity and proved the full 4-dimensional Sullivan Conjecture. For $n=5,6,7$, Fang and Wang proved the homeomorphism version: equality of Sullivan data implies homeomorphism. In general dimension $n \ge 3$, Kreck's Theorem A, obtained from his modified surgery theory (a variant of surgery theory that classifies manifolds by fixing a normal type rather than a full homotopy type), proves the diffeomorphism classification under a strong arithmetic hypothesis: for every prime $p$ with $p(p-1) \le n+1$, the exponent of $p$ in the total degree must satisfy $\nu_p(d) \ge \frac{2n+1}{2(p-1)}+1$. Kreck states explicitly that it is open whether the theorem holds for arbitrary complete intersections of complex dimension $> 2$. Outside the small-codimension range $r \le (n+2)/2$ in which the multidegree is determined, nothing is known for $n \ge 8$ without such divisibility assumptions — even the homeomorphism classification, available for $n = 5,6,7$, has no analogue there.

Two further results frame the question. First, Libgober and Wood proved that for $n > 2$ and small codimension $r \le (n+2)/2$, the total degree and Pontryagin classes already determine the multidegree, so in that range equal Sullivan data forces equal multidegrees and the conjecture holds trivially by Thom's observation; the substance of the problem lies at larger codimension, where the power sums $s_1,\dots,s_r$ are no longer recoverable from the invariants. Second, Libgober–Wood showed that diffeomorphic complete intersections with different multidegrees carry complex structures lying in different connected components of the moduli space of complex structures on the underlying smooth manifold, and a counting argument shows that the classes of multidegrees sharing the same Sullivan data can be arbitrarily large; the Sullivan Conjecture is thus the gateway to controlling how disconnected these moduli spaces are. In this spirit, Wang and Du exhibited explicit multidegree pairs (two length-30 multidegrees with equal power sums $s_1,\dots,s_{11}$) whose complete intersections have equal Sullivan data in complex dimensions $n = 8,9,10,11$. The total degree of these pairs is highly divisible (by $2^{51}$ and $3^{18}$), so they lie in the divisible case already covered by Kreck's Theorem A and are in fact diffeomorphic. It is the non-divisible, large-codimension regime, where the power sums are no longer recoverable from the Sullivan data, that furnishes a concrete incarnation of the open problem stated below.

Problem Statement

Let $X_n(\mathbf d)$ and $X_n(\mathbf d')$ be smooth complete intersections of complex dimension $n \ge 8$ (multidegrees $\mathbf d, \mathbf d'$ counted up to permutation and insertion/deletion of entries equal to $1$), and suppose their Sullivan data coincide: same total degree $d_1 \cdots d_r$, same Pontryagin integers $p_k(n,\mathbf d) = p_k(n,\mathbf d')$ for $1 \le k \le \lfloor n/2 \rfloor$, and same Euler characteristic. As in the standard formulation, homeomorphisms and diffeomorphisms are required to preserve the complex orientations. (i) Homeomorphism version: must $X_n(\mathbf d)$ and $X_n(\mathbf d')$ be homeomorphic? (ii) Diffeomorphism version (the Sullivan Conjecture restricted to $n \ge 8$): must they be diffeomorphic? The diffeomorphism version (ii) is the target of the problem: it implies the homeomorphism version (i), and a positive answer to (i) alone does not settle (ii).

References

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