论文标题
GKZ判别和多重性
GKZ discriminant and Multiplicities
论文作者
论文摘要
令$ t =（\ c^*）^k $在$ v = \ c^n $上忠实地保存卷形式，即$（\ c^*）^k \ tove \ text {sl}（v）$。在B面上，我们有折叠件$ z_w $（请参阅gkz粉丝中的walls $ w $标记的等式\ ref {eq：zw}），以及$ z _ {/f} $标记为由与最小半循环分解（Sod）组合（SOD）组合的polytope相应的面孔所标记的。 b端多重性$ n^b_ {w，f} $，由风筝segal \ cite {kite-segal}的结果很好地定义，是$ \ coh（z _ {/f}）$的次数。在A侧，我们有GKZ判别基因座组件$ \ nabla_f \ in（\ c^*）^k $及其热带化$ \ nabla^{trop} _ {f} _ {f} \ in \ r^k $。 a端多重性$ n^a_ {w，f} $定义为壁$ w $上的热带复杂$ \ nabla^{trop} _ {f} $的多样性。我们证明$ n^a_ {w，f} = n^b_ {w，f} $，在\ cite {aspinwall2017mirror}启发的Kite-Segal \ cite {kite-segal}中确认一个猜想。我们的证明是基于Horja-katzarkov \ cite {Horja2022discriminans}的结果和关于B侧SOD多样性的引理，这使我们能够像A端\ cite \ cite \ cite {gkz-book} [gkz-book} [ch ch 11]一样降低较低的维度。
Let $T=(\C^*)^k$ act on $V=\C^N$ faithfully and preserving the volume form, i.e. $(\C^*)^k \into \text{SL}(V)$. On the B-side, we have toric stacks $Z_W$ (see Eq. \ref{eq:ZW})labelled by walls $W$ in the GKZ fan, and $Z_{/F}$ labelled by faces of a polytope corresponding to minimal semi-orthogonal decomposition (SOD) components. The B-side multiplicity $n^B_{W,F}$, well-defined by a result of Kite-Segal \cite{kite-segal}, is the number of times $\Coh(Z_{/F})$ appears in a complete SOD of $\Coh(Z_W)$. On the A-side, we have the GKZ discriminant loci components $\nabla_F \In (\C^*)^k$, and its tropicalization $\nabla^{trop}_{F} \In \R^k$. The A-side multiplicity $n^A_{W, F}$ is defined as the multiplicity of the tropical complex $\nabla^{trop}_{F}$ on wall $W$. We prove that $n^A_{W,F} = n^B_{W,F}$, confirming a conjecture in Kite-Segal \cite{kite-segal} inspired by \cite{aspinwall2017mirror}. Our proof is based on the result of Horja-Katzarkov \cite{horja2022discriminants} and a lemma about B-side SOD multiplicity, which allows us to reduce to lower dimension just as in A-side \cite{GKZ-book}[Ch 11].
