Tom Coates

Wall-crossings in toric Gromov-Witten theory I: crepant examples

Let X be a Gorenstein orbifold with projective coarse moduli space X and let Y be a crepant resolution of X . We state a conjecture relating the genus-zero Gromov‐ Witten invariants of X to those of Y , which differs in general from the Crepant Resolution Conjectures of Ruan and Bryan‐Graber, and prove our conjecture when XD P.1;1;2/ and XD P.1;1;1;3/. As a consequence, we see that the original form of the Bryan‐Graber Conjecture holds for P.1;1;2/ but is probably false for P.1;1;1;3/. Our methods are based on mirror symmetry for toric orbifolds. 53D45; 14N35, 83E30

A mirror theorem for toric stacks

We prove a Givental-style mirror theorem for toric Deligne–Mumford stacks

Mirror Symmetry and Fano Manifolds

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Quantum periods for 3-dimensional Fano manifolds

. This determines the genus-zero Gromov–Witten invariants of

Minkowski Polynomials and Mutations

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The Crepant Transformation Conjecture for toric complete intersections

in terms of an explicit hypergeometric function, called the

On the Crepant Resolution Conjecture in the Local Case

I

A Fock sheaf for Givental quantization

-function, that takes values in the Chen–Ruan orbifold cohomology of

Four-dimensional Fano toric complete intersections.

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The quantum Lefschetz hyperplane principle can fail for positive orbifold hypersurfaces

.