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

Gross-Hacking-Keel.

4) Some connections to cluster varieties.

5) Some applications to mathematical physics (e.g. the role of

Jeffrey-Kirwan residues in mirror symmetry).

References:

[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.

## Log Calabi-Yau geometry

Lecturer:

Course Type:

PhD Course

Academic Year:

2021-2022

Period:

March-June

Duration:

40 h

Description:

Research Group: