A unified family of counterexamples to Batyrev's non-negativity conjecture on stringy Hodge numbers

Abstract

Recently, Huang and Satriano determined the sharp dimension threshold for Batyrev's non-negativity conjecture on stringy Hodge numbers: the conjecture holds in dimensions at most four and fails in every dimension at least five. We show that their counterexamples fit into a single broad three-parameter family $Y_{m,d}:=V({\sum }_{i=0}^m{s}_i{g}_i(x)+h(x))\subset {\mathbb{P}}^{m+d+1}$, where each $g_i$ is homogeneous of degree $d-1$ and $h$ is homogeneous of degree $d$, whose singular locus is a linear space $\Lambda \cong {\mathbb{P}}^m$. Blowing up $\Lambda$ gives a log resolution with one smooth exceptional divisor of discrepancy one. We compute the Hodge-Deligne polynomials of both the resolution and the exceptional divisor from their projective-space fibrations over ${\mathbb{P}}^d$, derive a master formula for $E_{\mathrm{st}}(Y_{m,d}\times ({\mathbb{P}}^1{)}^n)$, and prove the structure formula $h_{\mathrm{st}}^{p,q}(Y_{m,d}\times ({\mathbb{P}}^1{)}^n)=A_{p,q}+B_{p,q}-C_{p,q}$, where the three terms are non-negative. Thus the only possible source of negativity is $-C_{p,q}$, which is governed by the primitive middle cohomology of $Z_{m,d}=V(g_0,\dots ,g_m)\subset {\mathbb{P}}^d$. Off the diagonal, every negative entry equals $-C_{p,q}$, a binomially weighted sum of shifted primitive Hodge numbers of $Z_{m,d}$. On the diagonal, the behaviour is governed by the parity of $m+d$. We also obtain a complete classification for $0\le m\le d-1$, $d\ge 3$, and $n\ge 1$. The only non-counterexamples are $(m,d)=(0,3)$ for every $n\ge 1$, and $(m,d)=(2,3)$ for $n\ge 4$.

arXiv commentLaTeX, 29 pages