Quiver flag zero loci are subvarieties of quiver flag varieties cut out by sections of representation theoretic vector bundles. We prove the Abelian/non-Abelian correspondence in this context: this allows us to compute genus zero Gromov–Witten invariants of quiver flag zero loci. We determine the ample cone of a quiver flag variety, and disprove a conjecture of Craw. In the appendices (which can be found in the electronic supplementary material), which are joint work with Tom Coates and Alexander Kasprzyk, we use these results to find four-dimensional Fano manifolds that occur as quiver flag zero loci in ambient spaces of dimension up to 8, and compute their quantum periods. In this way, we find at least 141 new four-dimensional Fano manifolds.

We generalize the cohomological mirror duality of Borcea and Voisin in any dimension and for any number of factors. Our proof applies to all examples which can be constructed through Berglund-Hübsch duality. Our method is a variant of the so-called Landau-Ginzburg/Calabi-Yau correspondence of Calabi-Yau orbifolds with an involution that does not preserve the volume form. We deduce a version of mirror duality for the fixed loci of the involution, which are beyond the Calabi-Yau category and feature hypersurfaces of general type.The above theorem provides many examples of Calabi-Yau mirror pairs unknown before. These statements turn into ordinary cohomology statement whenever crepant resolutions on the two sides exist.Crepant resolution conjecture [28]. Finally, very recently, Hull, Israel and Sarti used mirror symmetry for K3 surfaces to form "non-geometric backgrounds" in the physics paper [21].1.10. Contents. In §2 we recall terminology briefly. In §3 we recall some basic definitions about Berglund-Hübsch invertible polynomials. In §4 we treat orbifold cohomology, its σorbifold variant, and we prove the compatibility result (5) stated above. In §5 we prove all the relevant statements at the level of Landau-Ginzburg state spaces. In §6 we derive the corresponding geometric versions stated above, see in particular §6.3 with some examples. Relation to K3 surfaces is treated in §6.4; we compare to the approach of [2] in Example 6.4.3.Higher dimensional Borcea-Voisin mirror theorem is deduced in §6.5.Aknowledgements. We are grateful to Alfio Ragusa, Francesco Russo and Giuseppe Zappalà for organising Pragmatic 2015, where this work started. We are grateful to London 2. Terminology 2.1. Conventions. We work with schemes and stacks over the complex numbers. All schemes are Noetherian and separated. By linear algebraic group we mean a closed subgroup of GL m (C) for some m. We often use strict Henselizations in order to describe a stack or a morphism between stacks locally at a closed point: by "local picture of X at the geometric point x ∈ X" we mean the strict Henselization of X at x.2.2. Notation. We list here notation that occurs throughout the entire paper.V K the invariant subspace of a vector space V linearized by a finite group K; P(w w w) the quotient stack [(C n \ 0 0 0)/G m ] with w w w-weighted G m -action; Z(f ) the variety defined as zero locus of f ∈ C[x 1 , . . . , x n ].Remark 2.2.1 (zero loci). We add the subscript P(w w w) when we refer to the zero locus in P(w w w) of a polynomial f which is w w w-weighted homogeneous. In this way we haveRemark 2.2.2 (graphs and maps). Given an automorphism α of X we write Γ α for the graph X → X × X. However, to simplify formulae, we often abuse notation and use α for the graph Γ α as well as the automorphism. In this way, the diagonal ∆ : X → X × X will be often written as id X or simply id. Berglund-Hübsch polynomialsThe setup presented here is due to Berglund-Hübsch [3]. We also refer to [4,15, 16,24,22]. It can be motivated as the simplest generalization of Gree...

The classification of Fano varieties is an important open question, motivated in part by the MMP. Smooth Fano varieties have been classified up to dimension three: one interesting feature of this classification is that they can all be described as certain subvarieties in GIT quotients; in particular, they are all either toric complete intersections (subvarieties of toric varieties) or quiver flag zero loci (subvarieties of quiver flag varieties). There is a program to use mirror symmetry to classify Fano varieties in higher dimensions. Fano varieties are expected to correspond to certain Laurent polynomials under mirror symmetry; given such a Fano toric complete intersections, one can produce a Laurent polynomial via the Hori-Vafa mirror. In this paper, we give a method to find Laurent polynomial mirrors to Fano quiver flag zero loci in Y -shaped quiver flag varieties. To do this, we generalise the Gelfand-Cetlin degeneration of type A flag varieties to Fano Y -shaped quiver flag varieties, and give a new description of these degenerations as canonical toric quiver varieties. We find conjectural mirrors to 99 four dimensional Fano quiver flag zero loci, and check them up to 20 terms of the period sequence.

First, I will discuss Golay sequences as Golay himself defined them, presenting his results on their lengths and his direct and recursive constructions. I will then discuss the broadest generalization yet defined, Golay array pairs. It is fruitful to understand complementary sequences as a special case of Golay array pairs, rather than Golay array pairs simply as an extension of the more fundamental complementary sequences. First, I will discuss their basic properties, and then the ways in which they can be transformed into higher or lower dimensional arrays. Secondly, within the context of Golay array pairs, I will discuss the other main generalizations and developments, beginning with Jedwab and Davis’ non-recursive structure, which led to the division of standard and non-standard Golay sequences, both of which I will discuss. Thirdly, I will present the basic extensions of Golay sequences: Golay sets, and multiple L-shift complementary sequences.

We show that there is an extra grading in the mirror duality discovered in the early nineties by Greene–Plesser and Berglund–Hübsch. Their duality matches cohomology classes of two Calabi–Yau orbifolds. When both orbifolds are equipped with an automorphism s of the same order, our mirror duality involves the weight of the action of $s^*$ on cohomology. In particular it matches the respective s-fixed loci, which are not Calabi–Yau in general. When applied to K3 surfaces with nonsymplectic automorphism s of odd prime order, this provides a proof that Berglund–Hübsch mirror symmetry implies K3 lattice mirror symmetry replacing earlier case-by-case treatments.