Hori--Vafa mirror models for complete intersections in weighted projective spaces and weak Landau--Ginzburg models

We prove that Hori--Vafa mirror models for smooth Fano complete intersections in weighted projective spaces admit an interpretation as Laurent polynomials.Comment: 5 pages; several minor changes has been mad

On birational involutions of P3P^3

Let $X$ be a rationally connected three-dimensional algebraic variety and let $\tau$ be an element of order two in the group of its birational selfmaps. Suppose that there exists a non-uniruled divisorial component of the $\tau$-fixed point locus. Using the equivariant minimal model program we give a rough classification of such elements.Comment: 24 pages, late

An example of Mirror Symmetry for Fano threefolds

In this note we illustrate the Fanosearch programme of Coates, Corti, Galkin, Golyshev, and Kasprzyk in the example of the anticanonical cone over the smooth del Pezzo surface of degree 6.Comment: 13 pages, 2 figure

Generalized Homological Mirror Symmetry and cubics

We discuss an approach to studying Fano manifolds based on Homological Mirror Symmetry. We consider some classical examples from a new point of view

Laurent polynomials in Mirror Symmetry

Mirror symmetry is one of the most exciting ideas in mathematics to emerge in the last two decades. It was introduced in physics as a duality between superconformal field theories. This book discusses this rapidly developing aspect of string theory

Phenomena

In this paper, we describe recent work towards the mirror conjecture, which relates the weight filtration on the cohomology of a log Calabi–Yau manifold to the perverse Leray filtration on the cohomology of the homological mirror dual log Calabi–Yau manifold taken with respect to the affinization map. This conjecture extends the classical relationship between Hodge numbers of mirror dual compact Calabi–Yau manifolds, incorporating tools and ideas which appear in the fascinating and groundbreaking works of de Cataldo, Hausel, and Migliorini [1] and de Cataldo and Migliorini [2]. We give a broad overview of the motivation for this conjecture, recent results towards it, and describe how this result might arise from the SYZ formulation of mirror symmetry. This interpretation of the mirror conjecture provides a possible bridge between the mirror conjecture and the well-known conjecture in non-Abelian Hodge theory