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2nd Meeting

AMSS, CAS

Xin Lv (Mainz) on behalf of Kang Zuo

On finiteness of CM jacobians of smooth hyperelliptic curves and superelliptic curves

Coleman’s conjecture predicts that for g sufficiently large there exists at most finitely many smooth complex projective curves of genus g (up to isomorphism) whose jacobians are CM abelian varieties. Based on the recent work by Tsimerman on the solution of André-Oort conjecture and my recent joint works with Ke Chen and Xin Lv on Oort conjectue for hyperelliptic curves and superelliptic curves we show Coleman’s conjecture holds true for those two cases.

Kwokwai Chan (CUHK)

Scattering diagrams and deformation theory

Given a Calabi-Yau manifold equipped with a Lagrangian torus fibration, we introduce a DGLA via Witten deformation, which is mirror to the Kodaira-Spencer DGLA that governs deformation of complex structures. We show that semi-classical limits of the corresponding Maurer-Cartan solutions give rise to scattering diagrams which have played a key role in the Gross-Siebert program. This realizes part of Fukaya's program in understanding mirror symmetry via the SYZ approach. This talk is based on joint work with Conan Leung and Ziming Ma.

Zhiyuan Li (Stanford)

Supersingular varieties

Artin and Shioda have introduced the definitions of supersingular K3 surfaces over fields with positive characteristic p. Their definitions are now known to be equivalent due to Tate conjecture. Moreover, there is a geometric description of supersingular K3 surfaces. Liedtke has recently shown that a K3 surface is supersingular if and only if X unirational when p > 3. In this talk, I will introduce different notions of supersingular varieties motivated from the K3 case, especially for supersingular Calabi-Yau threefolds and supersingular hyperkahlers. In particular, I will discuss the unirationality of these varieties and give some results in this direction.

Gerard van der Geer (Amsterdam)

Cycle classes of divisorial Maroni loci

To a general curve of genus g with a linear system of degree d one can associate a (d - 1) tuple of integers that describe the type of scroll spanned by the fibres of the linear system on the canonically embedded curve. This gives rise to the so-called Maroni stratification of the Hurwitz space H(d, g). We determine cycle classes of Maroni divisors in the compactified Hurwitz spaces. This is joint work with Alexis Kouvidakis.

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