Classification of smooth complete intersections by total degree, Pontrjagin classes, and Euler characteristic in complex dimension at least 8

A smooth complete intersection $X_n(d_1,\dots,d_r)\subset\mathbb{CP}^{n+r}$ is determined up to diffeomorphism by its multidegree (Thom), but the multidegree is not a topological invariant. The organizing question of the field — the Sullivan conjecture — asks whether, for complex dimension $n\ge 3$, the total degree $d=d_1\cdots d_r$, the Pontrjagin classes, and the Euler characteristic (together, the Sullivan data) determine the diffeomorphism type, with a parallel homeomorphism question. Homeomorphism classification by these invariants is known for $n\le 7$, diffeomorphism classification for $n=3$ and $n=4$ (the latter only in 2025), and both conditionally for every $n$ under strong divisibility hypotheses on the total degree (Kreck–Traving). The source (Wang–Du, 2014/2016) explicitly leaves open the classification in complex dimensions $n\ge 8$ (Remark 3.12): their equal-invariant multidegree pairs extend to $n=8,\dots,11$, but they state that no classification theorem allows one to conclude homeomorphism or diffeomorphism there. The source's own dimension $8$–$11$ examples already satisfy the Kreck–Traving divisibility hypotheses and are therefore conditionally diffeomorphic, so the surviving open core is the unconditional homeomorphism and diffeomorphism classification by the Sullivan data in complex dimension at least 8. This classification is open.

Background

A smooth complete intersection is the transverse intersection $X_n(d_1,\dots,d_r)$ of $r$ complex hypersurfaces in complex projective space $\mathbb{CP}^{n+r}$; the unordered tuple $(d_1,\dots,d_r)$ is the multidegree and $d=d_1\cdots d_r$ the total degree; entries equal to $1$ may be inserted into or deleted from a multidegree without changing the complete intersection (a degree-$1$ hypersurface does not cut the variety), so multidegrees are compared up to this convention and a codimension is the length of the multidegree after deleting $1$s. By a theorem of Thom, the complex dimension $n$ and the multidegree determine the diffeomorphism type, and all complete intersections of a fixed multidegree are diffeomorphic. The multidegree itself, however, is not a topological invariant, and classifying complete intersections by classical numerical invariants has been a central problem of the topology of projective varieties since the 1980s.

The relevant invariants are expressed through the power sums $s_k=\sum_{j=1}^r d_j^k$. Writing $x\in H^2(X_n(d_1,\dots,d_r);\mathbb{Z})$ for the pullback of the hyperplane class, each Pontrjagin class (the characteristic class $p_k\in H^{4k}$ controlling the real vector bundle underlying the tangent bundle) is an integral multiple of $x^{2k}$, and the Euler characteristic is $d$ times a universal polynomial in $n+r+1-s_1,\dots,n+r+1-s_n$ given by Newton's formulas. The Sullivan data of $X_n(d_1,\dots,d_r)$ is the tuple $\mathrm{SD}_n=(d,(p_i)_{i=1}^{\lfloor n/2\rfloor},e)$ of total degree, Pontrjagin classes viewed as integers, and Euler characteristic. For $n\ge 3$ the Sullivan data are invariants of the diffeomorphism type, and the Sullivan conjecture predicts that they are complete: if $\mathrm{SD}_n(d_1,\dots,d_r)=\mathrm{SD}_n(d'_1,\dots,d'_{r'})$ and $n\ge 3$, then the two complete intersections are diffeomorphic.

The confirmed cases are: $n=1$ (classical theory of Riemann surfaces); $n=2$ topologically, via Freedman's classification of simply connected 4-manifolds, where Ebeling and Libgober–Wood famously produced homeomorphic but non-diffeomorphic complete intersection surfaces; $n=3$, by classification theorems of Wall and Jupp for certain 6-manifolds; $n=4$, up to connected sum with a homotopy 8-sphere by Fang–Klaus (a homotopy sphere is a manifold homeomorphic but possibly not diffeomorphic to the standard sphere) and fully, in 2025, by Crowley–Nagy; and $n=5,6,7$, up to homeomorphism, by Fang–Wang. In addition, a theorem obtained through Kreck's modified surgery theory (a classification technique for manifolds with a fixed normal $k$-type, recorded as Theorem A of Kreck's paper following Traving's thesis) settles the diffeomorphism classification for every $n>2$ under an arithmetic hypothesis on the total degree: writing $\nu_p(d)$ for the exponent of a prime $p$ in $d$, it suffices that $\nu_p(d)>\frac{2n+1}{2(p-1)}+1$ for every prime $p$ with $p(p-1)\le n+1$. Kreck states explicitly that it is open whether the conclusion holds for arbitrary complete intersections.

Two further results shape the problem. First, Libgober–Wood proved (Proposition 7.3 of their 1982 paper) that if $n>2$ and $2r\le n+2$ (small codimension), then the total degree and Pontrjagin classes already determine the multidegree; so different-multidegree pairs with identical invariants can exist only once the codimension exceeds roughly $n/2$, and whether this threshold is sharp is unknown. Second, Libgober–Wood showed by a counting argument, valid in all dimensions, that for sufficiently large codimension there are arbitrarily many distinct multidegrees with the same Sullivan data, and that diffeomorphic complete intersections with different multidegrees lie in different connected components of the moduli space of complex structures on the underlying smooth manifold (the space parametrizing all complex structures) — the geometric payoff of the classification program.

The source, a 2014 preprint of Wang and Du (published 2016), realizes the known classification theorems by explicit examples in complex dimensions $n\le 7$ — confirming, in particular, the Libgober–Wood expectation that diffeomorphic complex 3-dimensional complete intersections with different first Chern classes exist — and acknowledges the classification in complex dimensions $n\ge 8$ as unsettled: their equal-invariant multidegree pairs extend to $n=8,\dots,11$, but they state that no classification theorem allows one to conclude homeomorphism or diffeomorphism there (Remark 3.12). The Libgober–Wood bound $2r\le n+2$ enters the source only as the small-codimension obstruction that forces their different-multidegree examples to have large codimension, and their examples were found partly by computer search.

Problem Statement

Determine what the total degree, Pontrjagin classes, and Euler characteristic reveal about the topology and smooth structure of a complete intersection in the regime not covered by the existing classification theorems, namely complex dimension at least 8.

Core classification question. Prove or disprove: for every complex dimension $n\ge 8$, two smooth complete intersections $X_n(d_1,\dots,d_r)$ and $X_n(d'_1,\dots,d'_{r'})$ with the same total degree $d=d_1\cdots d_r=d'_1\cdots d'_{r'}$, the same Pontrjagin classes $p_i$ (viewed as the integer multiples of $x^{2i}$), and the same Euler characteristic are homeomorphic. Establish the corresponding diffeomorphism statement — the Sullivan conjecture for dimensions $n\ge 8$ — for arbitrary total degrees, that is, without the Kreck–Traving divisibility hypothesis $\nu_p(d)>\frac{2n+1}{2(p-1)}+1$ for all primes $p$ with $p(p-1)\le n+1$; or else exhibit two complete intersections of some dimension $n\ge 8$ with identical Sullivan data that are not homeomorphic, or not diffeomorphic. These two statements are nested: the diffeomorphism statement is the stronger claim, because diffeomorphic complete intersections are homeomorphic. A proof of the diffeomorphism statement therefore settles the homeomorphism statement as well, and a pair of dimension $n\ge 8$ with identical Sullivan data that is not homeomorphic refutes both statements at once; a pair with identical Sullivan data that is homeomorphic but not diffeomorphic refutes only the diffeomorphism statement, leaving the homeomorphism statement open.

The objective is to complete the picture of which numerical invariants determine the homeomorphism and diffeomorphism type of a complete intersection in the dimensions where this is currently unknown. Explicit different-multidegree pairs with identical Sullivan data make the classification visible — including through the connected-component structure of the moduli spaces of complex structures on the underlying smooth manifolds, which by Libgober–Wood is governed by different-multidegree diffeomorphic pairs — but the open problem is the unconditional classification itself.

References

1. J. Wang and J. Du, Geometrical realization of diffeomorphic (homeomorphic) low dimensional complete intersections, arXiv:1411.6117 (2014); published as Geometrical realization of low-dimensional complete intersections, Chinese Annals of Mathematics, Series B 37(4) (2016) 523–532. DOI: 10.1007/s11401-016-1022-0, https://doi.org/10.1007/s11401-016-1022-0
2. A. S. Libgober and J. W. Wood, Differentiable structures on complete intersections. I, Topology 21(4) (1982) 469–482. DOI: 10.1016/0040-9383(82)90024-6, https://doi.org/10.1016/0040-9383(82)90024-6
3. A. S. Libgober and J. W. Wood, Diffeomorphic complete intersections with different multidegrees, Bulletin of the American Mathematical Society (new series) 2(3) (1980) 459–461. DOI: 10.1090/s0273-0979-1980-14767-9, https://doi.org/10.1090/s0273-0979-1980-14767-9
4. A. S. Libgober and J. W. Wood, Uniqueness of the complex structure on Kähler manifolds of certain homotopy types, Journal of Differential Geometry 32(1) (1990) 139–154. DOI: 10.4310/jdg/1214445041, https://doi.org/10.4310/jdg/1214445041
5. W. Ebeling, An example of two homeomorphic, nondiffeomorphic complete intersection surfaces, Inventiones Mathematicae 99 (1990) 651–654. DOI: 10.1007/BF01234435, https://doi.org/10.1007/BF01234435
6. F. Fang and S. Klaus, Topological classification of 4-dimensional complete intersections, Manuscripta Mathematica 90(1) (1996) 139–147. DOI: 10.1007/BF02568299, https://doi.org/10.1007/BF02568299
7. F. Fang and J. Wang, Homeomorphism classification of complex projective complete intersections of dimensions 5, 6 and 7, Mathematische Zeitschrift 266(3) (2010) 719–746. DOI: 10.1007/s00209-009-0597-5, https://doi.org/10.1007/s00209-009-0597-5
8. M. Kreck, Surgery and duality, Annals of Mathematics (2) 149(3) (1999) 707–754. DOI: 10.2307/121071, arXiv:math/9905211, https://arxiv.org/abs/math/9905211
9. D. Crowley and Cs. Nagy, The smooth classification of 4-dimensional complete intersections, Geometry & Topology 29(1) (2025) 269–311. DOI: 10.2140/gt.2025.29.269, arXiv:2003.09216, https://arxiv.org/abs/2003.09216
10. J. Wang, Z. Yu and Y. Wang, Examples of diffeomorphic complete intersections with different Hodge numbers, Topology and its Applications 339 (2023) 108690. DOI: 10.1016/j.topol.2023.108690, arXiv:2004.07142, https://arxiv.org/abs/2004.07142
11. J. Wang, Remarks on 5-dimensional complete intersections, Electronic Research Announcements in Mathematical Sciences 21 (2014) 28–40. DOI: 10.3934/era.2014.21.28, https://doi.org/10.3934/era.2014.21.28
12. J. Wang and Y. Wang, Moduli spaces of 6 and 7-dimensional complete intersections, Topology and its Applications 199 (2016) 63–69. DOI: 10.1016/j.topol.2015.12.007, https://doi.org/10.1016/j.topol.2015.12.007