On sums of Betti numbers of affine varieties

TL;DR

The paper addresses explicit, uniform bounds for the sum of $\ell$-adic Betti numbers, $B(V)_{(\\ell)}$, of affine varieties $V \subset \mathbf{A}^{N}_{k}$ defined by polynomials of degree at most $d$. The main contributions are two explicit bounds: $B(V)_{(\\ell)} \le 2 r^{r}(r d+3)^{N}$ when $V$ is defined by $r$ equations, and the $r$-independent bound $B(V)_{(\\ell)} \le 2 (N+1)^{2N+1}(d+1)^{N}$. The proof strategy combines a Kronecker reduction to reduce to a complete intersection, Deligne's perverse weak Lefschetz theorem for the cohomology of perverse sheaves, and Mayer–Vietoris arguments to manage unions of hypersurfaces, yielding an inductive bound on dimension. These results extend Katz's uniform bound to all algebraically closed fields, relate to complex Milnor bounds in characteristic zero, and establish the asymptotic behavior $B(N,d) \asymp_N d^{N}$ with explicit constants.

Abstract

We show that if V is a subvariety of the affine N-space defined by polynomials of degree at most d, then the sum of its $\ell$-adic Betti numbers does not exceed $2(N + 1)^{2N +1}(d+ 1)^N$. This answers a question of Katz (FFA 2001).

Theorems & Definitions (10)

• Theorem 1
• Theorem 2
• Theorem 3
• Proposition 1
• Theorem : Deligne k1
• proof : Proof of Proposition \ref{['proposition:hyper']}
• proof : Proof of Theorem \ref{['theorem:complete-intersection']}
• Proposition 2
• proof
• proof : Proof of Theorem \ref{['theorem:with-r']}