The Mori program and Non-Fano toric Homological Mirror Symmetry

Ballard, Matthew; Diemer, Colin; Favero, David; Katzarkov, Ludmil; Kerr, Gabriel

doi:10.1090/S0002-9947-2015-06541-6

The Mori program and Non-Fano toric Homological Mirror Symmetry
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by Matthew Ballard, Colin Diemer, David Favero, Ludmil Katzarkov and Gabriel Kerr PDF

Trans. Amer. Math. Soc. 367 (2015), 8933-8974 Request permission

Abstract:

In the case of toric varieties, we continue the pursuit of Kontsevich’s fundamental insight, Homological Mirror Symmetry, by unifying it with the Mori program. We give a refined conjectural version of Homological Mirror Symmetry relating semi-orthogonal decompositions of the $B$-model on toric varieties to semi-orthogonal decompositions on the $A$-model on the mirror Landau-Ginzburg models.

As evidence, we prove a new case of Homological Mirror Symmetry for a toric surface whose anticanonical bundle is not nef, namely a certain blow-up of $\mathbb {P}^2$ at three infinitesimally near points.

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