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On Landau–Ginzburg models for Fano varieties
 2008) (preprint 0707.3758). Steklov Mathematical Institute, 8 Gubkina
"... Abstract. We observe a method for finding weak LandauGinzburg models for Fano varieties and find them for smooth Fano threefolds of genera 9, 10, and 12. In the late 1980’s physicists discovered a phenomenon of Mirror Symmetry. They found that given a Calabi–Yau variety one can construct the so cal ..."
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Abstract. We observe a method for finding weak LandauGinzburg models for Fano varieties and find them for smooth Fano threefolds of genera 9, 10, and 12. In the late 1980’s physicists discovered a phenomenon of Mirror Symmetry. They found that given a Calabi–Yau variety one can construct the so called Superconformal Field Theory. This can be done in two ways: “algebrogeometric ” and “symplectic”. Based on this, they suggested that for each Calabi–Yau X there is another one Y (not necessary uniquely determined), whose algebrogeometric properties “correspond ” to symplectic ones of X and symplectic ones “corresponds ” to algebro–geometric ones of X. In particular, the Hodge diamond of Y is a reflection (a mirror image) of the Hodge diamond of X about a 45 ◦ line. That’s why X and Y are called a mirror pair. Later, in order to formalize this empiric approach, mathematicians formulated a series of Mirror Symmetry Conjectures. They generalize the correspondence to Fano varieties (Batyrev, Givental, Hori, Vafa, etc.). The pair to a Fano variety X is conjecturally a Landau–Ginzburg model, that is, a (noncompact) manifold M with complexvalued function f on it. The dual model M have a series of properties that correspond to properties of X. Homological Mirror Symmetry Conjecture (Kontsevich, [Ko94]), for instance, states that
Weak Landau–Ginzburg models for smooth Fano threefolds
, 902
"... To the memory of my teacher V.A.Iskovskikh. Abstract. We prove that Landau–Ginzburg models for all 17 smooth Fano threefolds with Picard rank 1 can be represented as Laurent polynomials in 3 variables exhibiting them case by case. We prove that general elements of all the pencils we found are birati ..."
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Cited by 11 (5 self)
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To the memory of my teacher V.A.Iskovskikh. Abstract. We prove that Landau–Ginzburg models for all 17 smooth Fano threefolds with Picard rank 1 can be represented as Laurent polynomials in 3 variables exhibiting them case by case. We prove that general elements of all the pencils we found are birational to K3 surfaces. We state most of known methods of finding Landau–Ginzburg models in terms of Laurent polynomials. We state some problems concerning Laurent polynomial representations of Landau–Ginzburg models of Fano varieties. 1.
Toric Degenerations of Fano Threefolds Giving Weak Landau–Ginzburg Models
 Journal of Algebra
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