Bounds on the individual Betti numbers of complex varieties, stability and algorithms

TL;DR

This paper establishes explicit, dimension-independent bounds on the Betti numbers of complex varieties defined by polynomials, and explores stability and computational complexity aspects.

Contribution

It provides the first explicit bounds on Betti numbers independent of ambient space dimension and introduces stability results for sequences of complex varieties.

Findings

Betti number bounds are independent of ambient space dimension

Homological and representational stability results for complex projective varieties

Differences in computational complexity between complex and real varieties

Abstract

We prove graded bounds on the individual Betti numbers of affine and projective complex varieties. In particular, we give for each $p,d,r$, explicit bounds on the $p$-th Betti numbers of affine and projective subvarieties of $\mathrm{C}^k$, $\mathbb{P}^k_{\mathrm{C}}$, as well as products of projective spaces, defined by $r$ polynomials of degrees at most $d$ as a function of $p,d$ and $r$. Unlike previous bounds these bounds are independent of $k$, the dimension of the ambient space. We also prove as consequences of our technique certain homological and representational stability results for sequences of complex projective varieties which could be of independent interest. Finally, we highlight differences in computational complexities of the problem of computing Betti numbers of complex as opposed to real projective varieties.