18
$\begingroup$

Thom conjecture, that was originally asked in $\mathrm{CP}^2$, and is now proven for symplectic 4 manifolds, states that complex curves (symplectic surfaces) are genus minimizing in their homology class. (The $\mathrm{CP}^2$ case was proven by Kronheimer and Mrowka '94, the symplectic case by Ozsvath and Szabo '00.)

Does anybody know if there is any analogous work being done in $\mathrm{CP}^3$? More concretely, is there any reason to think that complex hypersurfaces in $\mathrm{CP}^3$ are topologically the simplest oriented 4 manifolds in their homology classes. The simplest perhaps in a sense that the sum of betti numbers is minimal.

Improve this question
$\endgroup$
1
  • $\begingroup$ I added the gt.geometric-topology tag, because that seems to fit the question better. $\endgroup$
    – Danny Ruberman
    Commented Jan 20, 2015 at 15:01

1 Answer 1

Reset to default
18
$\begingroup$

This question was addressed in higher dimensions by Mike Freedman, Surgery on codimension 2 submanifolds. Mem. Amer. Math. Soc. 12 (1977), no. 191. (I think this was his PhD thesis.) Earlier work of Thomas and Wood (On manifolds representing homology classes in codimension 2. Invent. Math. 25 (1974), 63–89) had given some bounds for the homological complexity of codimension-2 submanifolds, generalizing those given by Hsiang and Szczarba for surfaces in 4-manifolds. My recollection is that Freedman shows that the Thomas-Wood bounds are close to optimal. The technique is a very clever variation of surgery theory, rather different from the typical uses of surgery theory in constructing embeddings in higher codimensions.

I think the upshot of Freedman's work is that the analogue of the Thom conjecture in higher dimensions does not hold.

Update 9/15/21: After this MO exchange, we started discussing how to reduce $b_2$ for the degree $d$ hypersurface $V_d$ in $CP^3$. In recent preprint (On the Thom conjecture in $CP^3$), Marko (the OP), Sašo Strle, and I showed that when $d \geq 5$, there is a smooth $4$-manifold in the homology class of $V_d$ with strictly smaller $b_2$. So the analogue of the Thom conjecture doesn't hold in $CP^3$ either. (As noted in my comment below, it does actually hold for $d \leq 4$.)

It's likely that our construction doesn't produce the smallest possible $b_2$ so there is still some room for improvement.

Improve this answer
$\endgroup$
2
  • $\begingroup$ The references are very helpful. Freedman's work does provide counterexamples in higher (even?) dimensional complex projective spaces. It does seem at first glance that not much is known for CP3. Am I correct in thinking that the "simplest" submanifolds must also be taut? $\endgroup$
    – Marko Slapar
    Commented Jan 22, 2015 at 12:15
  • $\begingroup$ The degree 4 hypersurface in $CP^3$ has minimal $b_2$ in its homology class so a Thom conjecture holds in this case. The signature of an arbitrary smooth representative of a homology class in $H_4(CP^3)$ is determined by the degree, and is given by the same formula as for the hypersurface of that degree. If $d$ is even, then it is necessarily spin. By Donaldson's theorem C, the degree 4 hypersurface has the minimal $b_2$ among spin manifolds with signature $-16$. For higher degrees, there is a gap between $b_2$ of the hypersurface and the best lower bounds for $b_2$ with given signature. $\endgroup$
    – Danny Ruberman
    Commented Jan 23, 2015 at 22:14

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .