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# 3rd Meeting

## 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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