Related papers: Mirror symmetry for concavex vector bundles on pro…

The construction introduced by Gross, Hacking and Keel allows one to construct a formal mirror family to a pair $(S,D)$ where $S$ is a smooth rational projective surface and $D$ a certain type of Weil divisor supporting an ample or…

We prove a generalized mirror conjecture for non-negative complete intersections in symplectic toric manifolds. Namely, we express solutions of the PDE system describing quantum cohomology of such a manifold in terms of suitable…

Let G be a Lie goup, let M and N be smooth connected G-manifolds, let f be a smooth G-map from M to N, and let P denote the fiber of f. Given a closed and equivariantly closed relative 2-form for f with integral periods, we construct the…

To a finite quiver equipped with a positive integer on each of its vertices, we associate a holomorphic symplectic manifold having some parameters. This coincides with Nakajima's quiver variety with no stability parameter/framing if the…

We generalize our theorems in "Mirror Principle I" to a class of balloon manifolds. Many of the results are proved for convex projective manifolds. In a subsequent paper, Mirror Principle III, we will extend the results to projective…

We construct decompositions of: (1) the cohomology of smooth stacks, (2) the Borel--Moore homology of $0$-shifted symplectic stacks, and (3) the vanishing cycle cohomology of $(-1)$-shifted symplectic stacks, assuming a good moduli space…

We introduce the concept of Bergman bundle attached to a hermitian manifold X, assuming the manifold X to be compact - although the results are local for a large part. The Bergman bundle is some sort of infinite dimensional very ample…

We construct a global B-model for weighted homogeneous polynomials based on K. Saito's theory of primitive forms. Our main motivation is to give a rigorous statement of the so called global mirror symmetry conjecture relating Gromov-Witten…

We define the notion of mirror of a Calabi-Yau manifold with a stable bundle in the context of type II strings in terms of supersymmetric cycles on the mirror. This allows us to relate the variation of Hodge structure for cohomologies…

Let $X$ be a smooth projective manifold with $\dim_\mathbb{C} X=n$. We show that if a line bundle $L$ is $(n-1)$-ample, then it is $(n-1)$-positive. This is a partial converse to the Andreotti-Grauert theorem. As an application, we show…

Let X be a smooth projective variety and let K be the canonical divisor of X. In this paper, we study embeddings of X given by adjoint line bundles of the form K+L, where L is an ample line bundle. When X is a regular surface (i.e. H^1(X,…

We prove a conjecture of Artur Elezi in a generalized form suggested by Givental. Namely, our main result relates genus-0 Gromov--Witten invariants of a bundle space with such invariants of the base, provided that the fiber is a toric…

We produce an equality between the Gromov-Witten invariants of the moduli space M of rank two odd degree stable vector bundles over a Riemann surface $\Sigma$ and the Donaldson invariants of the algebraic surface $\Sigma \times P^1$. We…

Given a finite, flat morphism between embeddable noetherian schemes of pure dimension 1, we define the notion of direct and inverse image for generalized divisors and generalized line bundles. In the case when we deal with (possibly…

Given a quasi-projective scheme M over complex numbers equipped with a perfect obstruction theory and a morphism to a nonsingular quasi-projective variety B, we show it is possible to find an affine bundle M'/ M that admits a perfect…

We construct a family of points on the Lagrangian cone of a partial flag bundle associated to a (possibly non-split) vector bundle from any Weyl-invariant $I$-function of a prequotient. This result can be seen as the nonabelian analogue of…

Let $X$ be a smooth complex projective algebraic variety. Given a line bundle $\mathcal{L}$ over $X$ and an integer $r>1$ one defines the stack $\sqrt[r]{\mathcal{L}/X}$ of $r$-th roots of $\mathcal{L}$. Motivated by Gromov-Witten theoretic…

We prove that open Gromov-Witten invariants for semi-Fano toric manifolds of the form $X=\mathbb{P}(K_Y\oplus\mathcal{O}_Y)$, where $Y$ is a toric Fano manifold, are equal to certain 1-pointed closed Gromov-Witten invariants of $X$. As…

The Conifold Gap Conjecture asserts that the polar part of the Gromov-Witten potential of a Calabi-Yau threefold near its conifold locus has a universal expression described by the logarithm of the Barnes $G$-function. In this paper, I…

In this paper, we study the structure of the quantum cohomology ring of a projective hypersurface with non-positive 1st Chern class. We prove a theorem which suggests that the mirror transformation of the quantum cohomology of a projective…