The Sullivan Conjecture for complex projective complete intersections of dimension at least five: diffeomorphism classification beyond the homeomorphism theorem of Fang and Wang

A smooth complex projective complete intersection $X_n(\underline{d})$ is the transverse intersection of hypersurfaces of degrees $d_1,\dots,d_r$ in $\mathbb{CP}^{n+r}$. By an observation of Thom its diffeomorphism type depends only on the multidegree, but the multidegree is not a topological invariant, so classification must proceed through the invariants topology can see: the total degree $d=d_1\cdots d_r$, the Pontryagin classes (which for complete intersections are integer multiples of powers of the hyperplane class), and the Euler characteristic — together the Sullivan data. Sullivan's conjecture predicts that for complex dimension $n\ge3$, complete intersections with equal Sullivan data are orientation-preserving diffeomorphic; the converse implication is known. Dimensions $n=3$ and $n=4$ are settled, and Fang and Wang proved that for $n=5,6,7$ equal Sullivan data classify complete intersections up to homeomorphism. The remaining problem — the Sullivan Conjecture in its open range $n\ge5$, recorded as open by Crowley and Nagy — is the diffeomorphism classification for all $n\ge5$: upgrading the known homeomorphism classification in dimensions 5–7, and establishing any unconditional classification in dimensions at least 8, where even the homeomorphism level is open.

Background

A complex projective complete intersection $X_n(\underline{d})\subset\mathbb{CP}^{n+r}$ is the transverse intersection of $r$ complex hypersurfaces of degrees $d_1,\dots,d_r$; the tuple $\underline{d}=(d_1,\dots,d_r)$ is the multidegree and $d=d_1\cdots d_r$ the total degree. As a complex manifold, $X_n(\underline{d})$ carries a canonical orientation. An observation of Thom shows that the diffeomorphism type of $X_n(\underline{d})$ depends only on $n$ and the multidegree, but the multidegree itself is not a topological invariant, so one seeks to classify complete intersections by invariants that homeomorphisms and diffeomorphisms can see. By the Lefschetz hyperplane theorem (which makes the pair $(\mathbb{CP}^{n+r},X_n(\underline{d}))$ $n$-connected), the cohomology is concentrated in even degrees, and each Pontryagin class has the form $p_i(X_n(\underline{d}))=p_i(n,\underline{d})\,x^{2i}$ for an integer $p_i(n,\underline{d})$, where $x\in H^2$ is the restriction of the hyperplane class; the Euler characteristic $\chi$ is likewise a polynomial in the degrees. The tuple $SD_n(\underline{d})=(d,\,p_1(n,\underline{d}),\dots,p_{\lfloor n/2\rfloor}(n,\underline{d}),\,\chi)$ is called the Sullivan data of $X_n(\underline{d})$.

In his 1970 MIT lecture notes (published in 2005), Dennis Sullivan conjectured that for $n\ge3$ the Sullivan data are a complete set of invariants up to diffeomorphism: if $SD_n(\underline{d})=SD_n(\underline{d}')$, then $X_n(\underline{d})$ and $X_n(\underline{d}')$ are orientation-preserving diffeomorphic. The converse — diffeomorphic complete intersections have equal Sullivan data — is known for $n\ge3$ (Libgober and Wood; see Crowley and Nagy, Proposition 2.10), so the conjecture is equivalent to the assertion that Sullivan data classify complete intersections of dimension at least three up to diffeomorphism. The conjecture organizes a long classification program: for $n=1$ it is classical; for $n=2$ (complex surfaces) the smooth classification is out of reach and even the homeomorphism classification belongs to the subtler topology of 4-manifolds; in complex dimension 3 the conjecture follows from the classification theorems for simply connected 6-manifolds of Wall and Jupp; in dimension 4, Fang and Klaus proved that equal Sullivan data imply a diffeomorphism after connected sum (gluing along a disk) with a homotopy 8-sphere — a sphere carrying an exotic smooth structure, from the group $\Theta_8\cong\mathbb{Z}/2$ of such structures — and Crowley and Nagy settled dimension 4 completely. In all dimensions $n>2$, a theorem recorded by Kreck (Theorem A of his modified surgery paper) gives the diffeomorphism classification under a divisibility hypothesis on the total degree: writing $\nu_p(d)$ for the exponent of a prime $p$ in $d$, it suffices that $\nu_p(d)\ge\frac{2n+1}{2(p-1)}+1$ for every prime $p$ with $p(p-1)\le n+1$.

For complex dimensions 5, 6 and 7, Fang and Wang [1] proved that two complete intersections are homeomorphic if and only if their Sullivan data agree, extending Fang and Klaus's dimension-4 homeomorphism classification. The standard attack on such problems is Kreck's modified surgery theory: one shows that $X_n(\underline{d})$ and $X_n(\underline{d}')$ with equal Sullivan data admit bordant normal $(n-1)$-smoothings — bundle-covered reference maps into the normal type $(\mathbb{CP}^\infty\times BO\langle n+1\rangle,\,\xi_n(\underline{d})\times\gamma)$, where $BO\langle n+1\rangle$ is the stage of the Postnikov tower of $BO$ retaining only homotopy in degrees above $n$ and $\xi_n(\underline{d})$ encodes the stable normal bundle determined by the multidegree. The obstruction to diffeomorphism then lies in twisted bordism groups $\Omega^{O\langle n+1\rangle}_{2n}(\mathbb{CP}^\infty;\xi_n(\underline{d}))$, equivalently in stable homotopy groups of a Thom spectrum (a spectrum built from the Thom spaces of a vector bundle, whose homotopy groups generalize framed bordism), which are attacked with the Adams spectral sequence (a spectral sequence computing stable homotopy groups from Ext-groups over the mod-2 Steenrod algebra, the algebra of stable cohomology operations on mod-2 cohomology). What survives at the homeomorphism level but may obstruct diffeomorphism is smooth torsion: the action of the groups $\Theta_{2n}$ of homotopy spheres (smooth structures on the $2n$-sphere under connected sum, computed by Kervaire and Milnor; e.g. $\Theta_{10}\cong\mathbb{Z}/2\times\mathbb{Z}/3$) and Kervaire-type invariants — $\mathbb{Z}/2$-valued quadratic-form invariants of framed manifolds, generalized by Brown and Kervaire, which detect precisely 2-torsion invisible to topological arguments — together with the $\alpha$-invariant, a stable-cohomotopy refinement of the $\hat{A}$-genus defined for spin manifolds (those with $w_2=0$) and sensitive to the smooth structure. Fang and Wang's vanishing arguments control this torsion only in the topological category, which is why their theorem stops at homeomorphism.

Problem Statement

Let $X_n(\underline{d})$ and $X_n(\underline{d}')$ be smooth complex projective complete intersections of complex dimension $n\ge5$ with equal Sullivan data $SD_n(\underline{d})=SD_n(\underline{d}')$ (same total degree, same integers $p_i(n,\underline{d})$, same Euler characteristic). Since the converse implication is known for $n\ge3$, the open cases of the Sullivan Conjecture assert the following, and the problem is to prove or refute it:

$SD_n(\underline{d})=SD_n(\underline{d}')\quad\Longrightarrow\quad X_n(\underline{d})\ \text{and}\ X_n(\underline{d}')\ \text{are orientation-preserving diffeomorphic, for every}\ n\ge5\ \text{and all multidegrees.}$

The regime $n=5,6,7$ is the immediate one left open by Fang and Wang: there, equal Sullivan data are already known to give a homeomorphism, so the question is whether homeomorphic complete intersections in these dimensions are diffeomorphic — equivalently, whether every Sullivan class in dimensions 5-7 carries a unique smooth structure, with the actions of the homotopy-sphere groups $\Theta_{10},\Theta_{12},\Theta_{14}$ and all Kervaire-type 2-torsion obstructions vanishing on complete intersections. For $n\ge8$ the statement subsumes the still-open homeomorphism-level classification, since no unconditional classification theorem is known there. Refuting the implication would require a dimension $n\ge5$ and two multidegrees $\underline{d}\ne\underline{d}'$ with $SD_n(\underline{d})=SD_n(\underline{d}')$ whose complete intersections are not diffeomorphic.

References

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