# Bounds on Betti numbers of subvarieties?

Let's say I have a smooth irreducible subvariety $X$ of $\mathbb{CP}^n$ with some fixed Hilbert polynomial. What are the best bounds known for the sum of the Betti numbers of $X$? That such a bound exists follows from the boundedness of (a component of) the Hilbert scheme.

I imagine that one could get a reasonable bound by writing down a suitable Morse function and doing some intersection-theoretic calculation for the number of critical points. Or perhaps more simply, just doing some inductive argument using Lefschetz pencils. I'm unsure what the best way of doing this would be though, and it would be nice if there was already a reference in the literature for this.

• By the way, a result of Hartschorne says that the Hilbert scheme of subschemes of projective space is connected (see : numdam.org/article/PMIHES_1966__29__5_0.pdf). So, if I am not mistaken, the total Betti number should not vary too much along the Hibert scheme... – Libli May 26 '17 at 11:39
• @Libli: The issue is that the Hilbert scheme may become highly disconnected when one removes points corresponding to singular varieties - I admittedly don't have an example of the top of my head, but I would be extremely surprised if one did not exist. – dhy May 26 '17 at 12:20
• might be the case – Libli May 26 '17 at 12:26
• There is a bound on the sum of the Betti numbers due to Thom and Milnor. – Jason Starr May 26 '17 at 13:32
• @JasonStarr Does the Milnor-Thom approach provide a bound for any subvariety? I was under the impression that their theorem only covered the cases of hypersurfaces. – dhy May 26 '17 at 15:25

There is a very recent paper by Zak : http://mathecon.cemi.rssi.ru/zak/files/Castelnuovo%20Bounds%20for%20Higher%20Dimensional%20Varieties.pdf which deals with this issue. He has found many new bounds on the total Betti numbers. For instance, he proves that if $X \subset \mathbb{P}^n$ is smooth of degree $d$ and of codimension $a$, then:
$$b(X) \leq \dfrac{d^{\dim X +1}}{a^{\dim X}},$$
provided that the $d \geq 2(a+1)^2$ and that $\dim X \geq 2$. He gives other bounds if the degree is lower. He also proves that these bounds are asymptotically sharp. Note the surprising fact that these bounds only depend on the degree, the dimension and the codimension.