发表机构
University of Thessaly（色萨利大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一个统一的三参数反例族，证明Batyrev非负猜想在至少五维失败，并给出完整分类，指出唯一例外情况。
AI 中文摘要
最近，Huang和Satriano确定了Batyrev关于弦性Hodge数的非负猜想的尖锐维度阈值：该猜想在至多四维时成立，而在每个至少五维时失败。我们证明他们的反例属于一个单一的三参数族$Y_{m,d}:=V(\sum_{i=0}^m s_i g_i(x)+h(x))\subset\mathbb{P}^{m+d+1}$，其中每个$g_i$是$d-1$次齐次多项式，$h$是$d$次齐次多项式，其奇点轨迹是线性空间$\Lambda\cong\mathbb{P}^m$。爆破$\Lambda$得到一个对数解消，其有一个光滑例外除子，差异为一。我们从它们在$\mathbb{P}^d$上的射影空间纤维化计算了解消和例外除子的Hodge-Deligne多项式，推导出$E_{\mathrm{st}}(Y_{m,d}\times(\mathbb{P}^1)^n)$的主公式，并证明了结构公式$h_{\mathrm{st}}^{p,q}(Y_{m,d}\times(\mathbb{P}^1)^n)=A_{p,q}+B_{p,q}-C_{p,q}$，其中三项均为非负。因此，唯一可能的负性来源是$-C_{p,q}$，它由$Z_{m,d}=V(g_0,\ldots,g_m)\subset\mathbb{P}^d$的原始中间上同调控制。在对角线之外，每个负项等于$-C_{p,q}$，这是$Z_{m,d}$的移位原始Hodge数的二项式加权和。在对角线上，行为由$m+d$的奇偶性决定。我们还获得了$0\le m\le d-1$，$d\ge3$，$n\ge1$的完整分类。唯一的非反例是$(m,d)=(0,3)$对所有$n\ge1$，以及$(m,d)=(2,3)$对$n\ge4$。
英文摘要
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 $Λ\cong\mathbb{P}^m$. Blowing up $Λ$ 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 $Z_{m,d}=V(g_0,\ldots,g_m)\subset\mathbb{P}^d$. 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 $0\le m\le d-1$, $d\ge3$, and $n\ge1$. The only non-counterexamples are $(m,d)=(0,3)$ for every $n\ge1$, and $(m,d)=(2,3)$ for $n\ge4$.
CommentsLaTeX, 29 pages
