*1st Meeting, May 10, 2015, BICMR, PKU*

Hélène Esnault (FU Berlin)

*Entropy of automorphisms on surfaces in positive characteristic*

We shall review the notion of entropy in this context and show the existence of automorphisms of maximal entropy, the logarithm of a degree 22 Salem number, on a supersingular K3 surface, which thus does not lift to characteristic zero. (Based on joint work with V. Srinivas, K. Oguiso and K. Oguiso-X. Yu).

Lawrence Ein (UIC)

*Asymptotic syzygies*

We discuss joint work with Robert Lazarsfeld on asymptotic syzgies. We discuss our work on the gonality conjecture, which says that one can determine the gonality of a smooth curve X of genus g from the shape of the minimal resolution of the section ring of a line bundle L on X, when degree of L is sufficiently large. We also plan to discuss the shape of the minimal resolution of the section ring of (X, L), where X is a smooth projective variety of dimension n and L is a sufficiently positive line bundle on X.

*2nd Meeting, September 26, 2015, 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.

*3rd Meeting, November 21, 2015, BICMR, PKU*

Yoshinori Gongyo (Tokyo)

*Rational points on log Fano threefolds over a finite field*

We prove the $W\mathcal{O}$-rationality of klt threefolds and the rational chain connectedness of klt Fano threefolds over a perfect field of characteristic p > 5. As a consequence, a klt Fano threefold over a finite field has a rational point. This is a joint work with Yusuke Nakamura and Hiromu Tanaka.

Jean-Louis Colliot-Thélène (Paris-Sud)

*Chow group of cycles of codimension two and third unramified cohomology*

Algebraic K-theory provides relations between the third unramified cohomology group (with torsion coefficients) of a smooth projective variety and the Chow group of codimension 2 cycles. This is used to study the image of such cycles under various cycle class maps into integral cohomogy. It is also used to investigate rationality questions for Fano hypersurfaces and for homogeneous spaces of connected linear algebraic groups. There are many open questions.

Shenghao Sun (YMSC, Tsinghua)

*Decomposition theorem and Independence of $\ell$*

The BBDG Decomposition theorem says that, over any algebraically closed base field, the $\ell$-adic intersection complex on an algebraic variety is taken to a direct sum of semisimple perverse sheaves, appropriately shifted in the derived category, under proper pushforwards. Each simple summand in the decomposition has a support by definition, and it is natural to expect that these supports remain unchanged as we vary the auxiliary choice of the prime $\ell$. We sketch the proof in the case when the base field is the algebraic closure of a finite field.

Kento Fujita (Kyoto)

*Optimal bounds for the volumes of Kahler-Einstein Fano manifolds*

We show that any n-dimensional Kahler-Einstein Fano manifold X satisfies that the anti-canonical volume is less than or equal to the value (n + 1)^n. Moreover, the equality holds if and only if X is isomorphic to the projective space.

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