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Log Calabi-Yau geometry

Course Type: 
PhD Course
Academic Year: 
40 h

Calabi-Yau manifolds play a central role in algebraic/differential
geometry and mathematical physics.
Log Calabi-Yau manifolds with maximal boundary form a class of
noncompact examples with especially nice properties. The course aims
at presenting an introduction to the theory of log Calabi-Yau
surfaces, their moduli, and their mirrors following Looijenga,
Friedman and Gross-Hacking-Keel, with some applications to
mathematical physics. It could serve as a second graduate course in
algebraic geometry.

Sketch of the course:
1) Some preliminary results from the theory of algebraic surfaces.
2) Affine log Calabi-Yau surfaces and their moduli.
3) Construction of mirrors in the affine case in the sense of
4) Some connections to cluster varieties.
5) Some applications to mathematical physics (e.g. the role of
Jeffrey-Kirwan residues in mirror symmetry).

[1] Beauville, Arnaud: Complex algebraic surfaces. Cambridge University Press, Cambridge, 1996.
[2] Gross, Mark; Hacking, Paul; Keel, Sean: Moduli of surfaces with an anti-canonical cycle. Compos. Math. 151 (2015), no. 2, 265–291.
[3] Gross, Mark; Hacking, Paul; Keel, Sean: Mirror symmetry for log Calabi-Yau surfaces I. Publ. Math. Inst. Hautes Études Sci. 122 (2015), 65–168.
[4] Gross, Mark; Hacking, Paul; Keel, Sean: Birational geometry of clusteralgebras. Algebr. Geom. 2 (2015), no. 2, 137–175.

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