Discriminants and Semi-orthogonal Decompositions

Abstract

The derived categories of toric varieties admit semi-orthogonal decompositions coming from wall-crossing in GIT. We prove that these decompositions satisfy a Jordan-Hölder property: the subcategories that appear, and their multiplicities, are independent of the choices made. For Calabi-Yau toric varieties wall-crossing instead gives derived equivalences and autoequivalences, and mirror symmetry relates these to monodromy around the GKZ discriminant locus. We formulate a conjecture equating intersection multiplicities in the discriminant with the multiplicities appearing in certain semi-orthogonal decompositions. We then prove this conjecture in some cases.

Fig. 1

(L) A real picture of $C_W$ as the straight line connecting the two points marked by $X_{\pm }$. (R) A complex picture of a 2-sphere near to the rational curve $C_W$, where the point $\delta$ has split into three. A loop from $X_{+}$ to $X_{-}$ and back again will factor into two loops around ${\mathrm{\Delta }}_0$ and one loop around ${\mathrm{\Delta }}_1$

Fig. 2

The phases of example 3.7