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.



Bounds on Betti numbers of subvarieties?
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. 


