Narrow quantum D-modules and quantum Serre duality
[D-modules et dualité de Serre quantiques dans le secteur troit]
Annales de l'Institut Fourier, Online first, 49 p.

Étant donnée une variété ou une orbifold non compacte 𝒴, on définit un sous-espace naturel de la cohomologie de 𝒴 que nous appelons cohomologie étroite. On montre qu’en dépit du fait que 𝒴 est non compacte, il existe un couplage non-dégénéré sur ce sous-espace. Cette cohomologie étroite s’avère utile pour l’étude de la théorie de Gromov–Witten en genre 0. Lorsque 𝒴 est une variété complexe lisse ou un champ de Deligne–Mumford lisse, on peut définir un D-module quantique sur sa cohomologie étroite. Ceci nous amène à une nouvelle formulation de la dualité de Serre quantique.

Given 𝒴 a non-compact manifold or orbifold, we define a natural subspace of the cohomology of 𝒴 called the narrow cohomology. We show that despite 𝒴 being non-compact, there is a well-defined and non-degenerate pairing on this subspace. The narrow cohomology proves useful for the study of genus zero Gromov–Witten theory. When 𝒴 is a smooth complex variety or Deligne–Mumford stack, one can define a quantum D-module on the narrow cohomology of 𝒴. This yields a new formulation of quantum Serre duality.

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DOI : https://doi.org/10.5802/aif.3419
Classification : 14N35,  53D45,  14E16
Mots clés : théorie de Gromov–Witten, D-modules quantiques, dualité de Serre quantique, symétrie miroir
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Shoemaker, Mark. Narrow quantum $D$-modules and quantum Serre duality. Annales de l'Institut Fourier, Online first, 49 p.

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