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Accueil > Agenda > Colloques et rencontres > Archives > Agenda 2010 > Mini cours de Mark Gross

Mini cours de Mark Gross

IRMA 01-06 juillet 2010

Mark Gross (San Diego)
"Deformations of surface singularities via mirror symmetry"

Résumé

I will describe recent joint work with Paul Hacking and Sean Keel
which applies the tropical vertex construction of Gross-Pandharipande-Siebert
to prove some conjectures about smoothability of surface cusp singularities.

The fundamental construction associates a flat formal family of affine
varieties X->S to a pair (Y,D), where Y is a rational surface and
D is an anti-canonical cycle of rational curves. This construction
involves the relative Gromov-Witten theory of the pair (Y,D). It is
in fact easy to write down this family in terms of this Gromov-Witten
theory, but one needs the full strength of GPS to prove it is in
fact a flat deformation.

The resulting family is a smoothing of the affine cone over
D. Depending on (Y,D), we obtain various different types of
behaviour. For example, if D supports an ample divisor, then the family
X->S is in fact algebraic, rather than formal, and can be compactified
to obtain a canonical description of universal families of rational
surfaces. On the other hand, if D is contractible on Y, the family
X->S can be extended analytically to obtain a smoothing of the so-called
"dual cusp singularity".

A cusp singularity is a normal surface singularity whose minimal resolution
has exceptional locus a cycle of rational curves. These singularities
were known to come in dual pairs, and in 1981, Looijenga conjectured that
a cusp singularity was smoothable if and only if the dual cusp is
realised as the contraction of an anti-canonical cycle on a rational
surface. He proved necessity ; the above result proves sufficiency.

I will start these lectures by discussing GPS in depth, and then
move on to its application to constructing these canonical families
associated with (Y,D).

Programme

Jeudi 1 juillet 2010 (salle de séminaires IRMA)

11h - 12h
M. Gross
"Deformations of surface singularities via mirror symmetry, I"

Vendredi 2 juillet 2010 (salle de séminaires IRMA)

11h30 - 12h30
M. Gross
"Deformations of surface singularities via mirror symmetry, II"

Mardi 6 juillet 2010 (salle de séminaires IRMA)

14h - 15h
M. Gross
"Deformations of surface singularities via mirror symmetry, III"

Dernière mise à jour le 24-06-2010