GKZ Discriminant and Multiplicities

Abstract

Let \(T=(\mathbb {C}^*)^k\) act on \(V=\mathbb {C}^N\) faithfully and preserving the volume form, i.e. \((\mathbb {C}^*)^k \hookrightarrow \text {SL}(V)\). On the B-side, we have toric stacks \(Z_W\) (see Eq. 1.1) 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 and Segal (Commun Math Phys 390:907-931, 2022), is the number of times \({{\,\textrm{Coh}\,}}(Z_{/F})\) appears in a complete SOD of \({{\,\textrm{Coh}\,}}(Z_W)\). On the A-side, we have the GKZ discriminant loci components \(\nabla _F \subset (\mathbb {C}^*)^k\), and its tropicalization \(\nabla ^{trop}_{F} \subset \mathbb {R}^k\). 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 and Segal (Commun Math Phys 390:907-931, 2022) inspired by (Aspinwall et al. in Mirror symmetry and discriminants, http://arxiv.org/abs/1702.04661, 2017). Our proof is based on the result of Horja and Katzarkov (Discriminants and toric K-theory, http://arxiv.org/abs/2205.00903, 2022) and a lemma about B-side SOD multiplicity, which allows us to reduce to lower dimension just as in A-side (Gelfand et al. in Discriminants, resultants and multidimen sional determinants, Birkahuser, Boston, 1994) [Ch 11].