Complete intersections
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[edit] 1 Construction and examples
Let be complex, homogeneous polynomials of degree in complex variables . Define
The set is an algebraic variety. It is non-singular if for all with the affine function
where , has as a regular value. In this case is a smooth, complex variety and in particular defines a closed, smooth, oriented dimensional manifold of real dimension . This is called a complete intersection.
By a theorem of Thom [???] the diffeomorphism type of depends only upon the multi-degree, , and we write for .
For example:
- ,
- the oriented surface of genus ,
- is a complex K3 surface,
- is a Calabi-Yau 3-fold.
[edit] 2 Invariants
By the Lefschetz hyperplane theorem the inclusion is an n-connected map. Hence:
- if n > 1 then ,
- if ,
- for all unless .
- where is the Euler characteristic of which we discuss further below.
- When is even generates a summand of .
Note that here and throughout integer coefficients are use for (co)homology.
[edit] 2.1 Cohomology ring
Let be the canonical line bundle over and let be defined by where is the first Chern class of . Let denote the fundamental class of and let be the product of all degrees, called the total degree. We have the following useful identity [???]
Let or and consider the graded ring
where the dimensions of and are and respectively. Let , considered as a graded ring in dimension
- If is odd:
- the ring is determined by , and ,
- there is a short exact sequence
- the intersection form is of course skew hyperbolic.
- If is even:
- the ring is determined by , and the pair :
- there is a short exact sequence .
- Some properties of are described below.
Proposition 2.1 [Libgober&Wood1981]. If is even, then is indefinite unless or .
[edit] 2.2 Characteristic classes
The stable tangent bundle of is isomorphic to , [Milnor&Stasheff1974], and the normal bundle of the inclusion is given by the identity [???]
where denotes the -fold tensor product of with itself. From this one deduces that the stable tangent bundle of , , satisfies the equation
It follows immediately that the total Chern class and the total Pontrjagin class of are given by
Moreover, the Euler class and Euler characteristic of is given by
[edit] 3 Classification
The smooth classification of complete intersections for is organised by the following conjecture, often called the Sullivan Conjecture after Dennis Sullivan.
Conjecture 3.1. For , complete intersections and are diffeomorphic if and only if all of the following conditions hold
- ,
- ,
- .
Note that we regard as multiplies of in order to view as an element of .
[edit] 3.1 Classification in low dimensions
- For the Sullivan Conjecture holds by the classification of surfaces.
- For the topological Sullivan Conjecture holds for by applying [Freedman1982]. The Sullivan Conjecture fails smoothly by [Ebeling1990] and [Libgober&Wood1990].
- For the Sullivan Conjecture holds by [Wall1966] and [Jupp1973]. See the page 6-manifolds: 1-connected.
- For the topological Sullivan Conjecture is true [Fang&Klaus1996]. Indeed, Fang and Klaus prove that the smooth conjecture, which is still open in general, holds up to connected sum with a homotopy 8-sphere.
- For and the topological Sullivan Conjecture is true [Fang&Wang2009].
[edit] 3.2 Further classification theorems
We now discuss further classification results for complete intersections. Let where is prime.
Theorem 3.2 [Traving1985]. If and for all such that then the Sullivan Conjecture holds.
Traving's proof uses modified surgery: see [Kreck1999, Section 8] for a summary.
Theorem 3.3 [Fang1997]. If and for all such that then the topological Sullivan Conjecture holds.
Fang's proof proceeds by extending results of [Libgober&Wood1982] on the homotopy classification of complete intersections where is even to the case of odd. He then solves the homeomorphism classification by exhibiting characteristic varieties for complete intersections and using them to calculate classical surgery obstructions. In particular he proves
Theorem 3.4 [Fang1997]. Let and be homotopy equivalent complete intersections. If is odd and for all then and are homeomorphic to each other if and only if their Pontrjagin classes agree.
[edit] 3.3 Classification up to homotopy
Recall that the signature of, , of is the signature of its intersection form. If is odd and if is even can be computed from the via Hirzebruch's signature theorem.
Theorem 3.5 [Libgober&Wood1982] and [Fang1997]. Let and be complete intersections with the same total degree . Suppose that if . If then and are homotopy equivalent if and only if they have the same signature and Euler characteristic.
Finally we summarise theorems of [Astey&Gitler&Micha&Pastor2003]. Define to be the unordered sequence
obtained by removing entries where and write if for all primes .
Theorem 3.6 [Astey&Gitler&Micha&Pastor2003]. If and then is homotopy equivalent to if and only if they have the same Euler characteristic and signature.
In contrast to the results of Fang and Ligober and Wood which hold when is small relative to the primes dividing , the above theorem leads to a homotopy classification which holds when is large relative to . If for all p then define otherwise let
Theorem 3.7 [Astey&Gitler&Micha&Pastor2003]. If and have the same multidegree and then and are homotopy equivalent if and only if they have the same signature and Euler characteritic.
[edit] 4 Further discussion
Conjecture 4.1 Hartshorne. Every smooth algebraic variety of dimension n which is embedded in is isomorphic to a complete intersection, if .
[edit] 4.1 Splitting theorems
[edit] 5 References
- [???] Template:???
- [Astey&Gitler&Micha&Pastor2003] L. Astey, S. Gitler, E. Micha and G. Pastor, On the homotopy type of complete intersections, Topology 44 (2005), no.1, 249–260. MR2104011 (2005i:14058) Zbl 1072.14063
- [Ebeling1990] W. Ebeling, An example of two homeomorphic, nondiffeomorphic complete intersection surfaces, Invent. Math. 99 (1990), no.3, 651–654. MR1032884 (91c:57038) Zbl 0707.14045
- [Fang&Klaus1996] F. Fang and S. Klaus, Topological classification of -dimensional complete intersections, Manuscripta Math. 90 (1996), no.2, 139–147. MR1391206 (97g:57046) Zbl 0866.57015
- [Fang&Wang2009] F. Fang and J. Wang, Homeomorphism classification of complex projective complete intersections of dimensions 5, 6 and 7, to appear in Math. Zeit. (2009).
- [Fang1997] F. Fang, Topology of complete intersections, Comment. Math. Helv. 72 (1997), no.3, 466–480. MR1476060 (98k:14071) Zbl 0896.14028
- [Freedman1982] M. H. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), no.3, 357–453. MR679066 (84b:57006) Zbl 0528.57011
- [Jupp1973] P. E. Jupp, Classification of certain -manifolds, Proc. Cambridge Philos. Soc. 73 (1973), 293–300. MR0314074 (47 #2626) Zbl 0249.57005
- [Kreck1999] M. Kreck, Surgery and duality, Ann. of Math. (2) 149 (1999), no.3, 707–754. MR1709301 (2001a:57051) Zbl 0935.57039
- [Libgober&Wood1981] A. S. Libgober and J. W. Wood, On the topological structure of even-dimensional complete intersections, Trans. Amer. Math. Soc. 267 (1981), no.2, 637–660. MR626495 (83e:57017) Zbl 0475.57013
- [Libgober&Wood1982] A. S. Libgober and J. W. Wood, Differentiable structures on complete intersections. I, Topology 21 (1982), no.4, 469–482. MR670748 (84a:57037) Zbl 0504.57015
- [Libgober&Wood1990] A. S. Libgober and J. W. Wood, Uniqueness of the complex structure on Kähler manifolds of certain homotopy types, J. Differential Geom. 32 (1990), no.1, 139–154. MR1064869 (91g:32039) Zbl 0711.53052
- [Milnor&Stasheff1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Princeton University Press, Princeton, N. J., 1974. MR0440554 (55 #13428) Zbl 1079.57504
- [Traving1985] C. Traving, Zur Diffeomorphieklassifikation vollständiger Durchschnitte, Diplomarbeit, Mainz (1985).
- [Wall1966] C. T. C. Wall, Classification problems in differential topology. V. On certain -manifolds, Invent. Math. 1 (1966), 355-374; corrigendum, ibid 2 (1966), 306. MR0215313 (35 #6154) Zbl 0149.20601
[edit] 6 External links
- The Wikipedia page about complete intersections