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# 2022-11-05

Canonical under blowup: see Griffiths Harris, \(K_Y=f^*\left(K_X\right)+(c-1) E\) where \(c\) is the codimension of \(V\) in \(X\) and \(E\) is the exceptional divisor.

## Looijenga Stuff

Mark Gross Smoothing cusp singularities via mirror symmetry

I will talk about joint work with Paul Hacking and Sean Keel. We use mirror symmetry to prove a conjecture of Looijenga about smoothability of cusp surface singularities. These are normal singularities whose minimal resolution has an exceptional locus given by a cycle of rational curves. It was known that these singularities come in pairs, called “dual cusps”. It turns out that this is a manifestation of mirror symmetry, and using recent results of Gross-Siebert and Gross-Pandharipande-Siebert, we are able to prove Looijenga’s conjecture, which states that a cusp singularity is smoothable if and only if the cycle of rational curves corresponding to the minimal resolution of the dual cusp can be realised as the anti-canonical class of a rational surface.

Use in string theory:

## Background

From Friedman-Miranda 82

Type \(\rm{VII}_0\)

Type III Degenerations

What is a double curve?

## Gross-Hacking-Keel

SYZ for Looijenga pairs:

Example of building an affine manifold:

## Evans-Mauri 22

## IAG of Lagrangian Fibrations , Sepe 11

## More Stuff

## More?

Symplectic toric manifolds

## Auroux

Mirror symmetry for elliptic curves:

!!!!!!!! See https://people.math.harvard.edu/~auroux/papers/slaginvol.pdf#page=11&zoom=150,-165,681

## Yet more