  Let $1\leq k \leq n$ and let $X_n = (x_1, \dots, x_n)$ be a list of $n$
variables. The {\em Boolean product polynomial} $B_{n,k}(X_n)$ is the product
of the linear forms $\sum_{i \in S} x_i$ where $S$ ranges over all $k$-element
subsets of $\{1, 2, \dots, n\}$. We prove that Boolean product polynomials are
Schur positive. We do this via a new method of proving Schur positivity using
vector bundles and a symmetric function operation we call {\em Chern plethysm}.
This gives a geometric method for producing a vast array of Schur positive
polynomials whose Schur positivity lacks (at present) a combinatorial or
representation theoretic proof. We relate the polynomials $B_{n,k}(X_n)$ for
certain $k$ to other combinatorial objects including derangements, positroids,
alternating sign matrices, and reverse flagged fillings of a partition shape.
We also relate $B_{n,n-1}(X_n)$ to a bigraded action of the symmetric group
$\mathfrak{S}_n$ on a divergence free quotient of superspace.
