Imperial College London
Alternate

ProfessorAlessioCorti

Faculty of Natural Sciences, Department of Mathematics

Chair in Pure Mathematics
 
 
 
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Contact

 

+44 (0)20 7594 1870a.corti Website

 
 
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Location

 

673Huxley BuildingSouth Kensington Campus

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Summary

 

Publications

Citation

BibTex format

@article{Coates:2016:10.2140/gt.2016.20.103,
author = {Coates, T and Corti, A and Galkin, S and Kasprzyk, A},
doi = {10.2140/gt.2016.20.103},
journal = {Geometry & Topology},
pages = {103--256},
title = {Quantum Periods for 3-Dimensional Fano Manifolds},
url = {http://dx.doi.org/10.2140/gt.2016.20.103},
volume = {20},
year = {2016}
}
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RIS format (EndNote, RefMan)

TY  - JOUR
AB  - The quantum period of a variety X is a generating function for certainGromov-Witten invariants of X which plays an important role in mirror symmetry.In this paper we compute the quantum periods of all 3-dimensional Fanomanifolds. In particular we show that 3-dimensional Fano manifolds with veryample anticanonical bundle have mirrors given by a collection of Laurentpolynomials called Minkowski polynomials. This was conjectured in joint workwith Golyshev. It suggests a new approach to the classification of Fanomanifolds: by proving an appropriate mirror theorem and then classifying Fanomirrors.  Our methods are likely to be of independent interest. We rework theMori-Mukai classification of 3-dimensional Fano manifolds, showing that each ofthem can be expressed as the zero locus of a section of a homogeneous vectorbundle over a GIT quotient V/G, where G is a product of groups of the formGL_n(C) and V is a representation of G. When G=GL_1(C)^r, this expresses theFano 3-fold as a toric complete intersection; in the remaining cases, itexpresses the Fano 3-fold as a tautological subvariety of a Grassmannian,partial flag manifold, or projective bundle thereon. We then compute thequantum periods using the Quantum Lefschetz Hyperplane Theorem ofCoates-Givental and the Abelian/non-Abelian correspondence ofBertram-Ciocan-Fontanine-Kim-Sabbah.
AU  - Coates,T
AU  - Corti,A
AU  - Galkin,S
AU  - Kasprzyk,A
DO  - 10.2140/gt.2016.20.103
EP  - 256
PY  - 2016///
SN  - 1465-3060
SP  - 103
TI  - Quantum Periods for 3-Dimensional Fano Manifolds
T2  - Geometry & Topology
UR  - http://dx.doi.org/10.2140/gt.2016.20.103
UR  - http://arxiv.org/abs/1303.3288v3
UR  - http://hdl.handle.net/10044/1/24559
VL  - 20
ER  -
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