SISSA Algebraic Geometry Seminars

[Ugo Bruzzo]

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SISSA ALGEBRAIC GEOMETRY SEMINARS
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On Tuesday, June 21st, we shall have two talks in Algebraic
Geometry. Both will take place in the Dubrovin Lecture Room (136).


Dimitri Markushevich (Lille). 2:30 pm
A case study in complex crystallographic groups: point group SL(2,7)

Abstract: A complex crystallographic (CC) group Γ is a discrete group of affine transformations of the complex space C^n acting with compact quotient. Any such group is an extension of a finite linear group G, called the point group, by a lattice L of maximal rank 2n. A CC group is of reflection type (a CCR group) if it is generated by affine reflections. A conjecture of Bernstein-Schwarzman suggests that the quotient C^n/Γ is a weighted projective space when Γ is irreducible; this is a natural generalization of Shephard-Todd-Chevalley theorem for finite linear groups generated by reflections. The conjecture is known in dimension 2 and for CCR groups of Coxeter type, that is those whose point group G is conjugate to a real Coxeter group. In the talk the case of a genuinely complex CCR group Γ in dimension 3 will be discussed, with quasi-simple point group G of order 336. In this case C^n/Γ can be interpreted as the quotient of the Jacobian of Klein's quartic curve by its full automorphism group {±1}×H, where H is Klein's simple group of order 168.

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Justin Sawon (North Carolina). 4:00 pm
Mirror symmetry for generalized Kummer varieties

Abstract: The generalized Kummer variety K_n of an abelian surface A is the fibre of the natural map Hilb^{n+1}A->Sym^{n+1}A->A. Debarre described a Lagrangian fibration on K_n whose fibres are the kernels of JacC->A, where C are curves in a fixed linear system in A.
In this talk we consider the dual of the Debarre system, constructed in a similar way to the duality between SL- and PGL-Hitchin systems described by Hausel and Thaddeus. We conjecture that these dual fibrations are mirror symmetric, in the sense that their (stringy) Hodge numbers are equal, and we verify this in a few cases. In fact, there is another isotrivial Lagrangian fibration on K_n. We can describe its dual fibration and verify the mirror symmetry relation in many more cases.


U. Bruzzo