# Meeting in June 2015

• Robert Laterveer: Hard Lefschetz for Chow groups. Inspired by the conjectural Bloch-Beilinson filtration, we formulate a conjectural hard Lefschetz property for the Chow groups of a smooth projective algebraic variety over the complex numbers. This property can be verified in some special cases – roughly speaking, for varieties for which the self-product has vanishing middle-dimensional Griffiths group, and for varieties that have finite-dimensional motive (in the sense of Kimura). As we will explain, closely related (but slightly different) results can also be deduced from recent work of Charles Vial.
• Susanna Zimmermann: An Abelian quotient of the real Cremona group. The Cremona group of the complex plane contains many normal subgroups, all of which are of infinite index. There is no proper normal subgroup containing elements of degree 1 (not equal to the identity map), 2, 3 and 4. What about the Cremona group of the real plane? I will present an abelian quotient of it, which implies the existence of normal subgroups of index equal to any given power of 2, all of them containing every map of degree 1, 2, 3, and 4.
• Giulia Battiston: A Galois descent theory for inseparable field extension. Let $L/K$ be a Galois separable field extension, then classical Galois descent theory describes algebraic objects over $K$, such as for example $K$-varieties, as being equivalent to algebraic objects over $L$ endowed with a $Gal(L/K)$-action which is $\sigma$-linear. If $L/K$ is not separable, though, such a theory does not apply for the simple reason that the field of $Gal(L/K)$-invariants is strictly bigger than $K$. We will present how this inconvenient can be bypassed using the automorphism group of truncated polynomials over L and hence obtaining a Galois descent theory for inseparable extensions.
• Zsolt Patakfalvi: Projectivity of the moduli space of KSBA stable pairs and applications. KSBA (Kollár-Shepherd-Barron-Alexeev) stable pairs are higher dimensional generalizations of (weighted) stable pointed curves. I will present a joint work in progress with Sándor Kovács on proving the projectivity of this moduli space, by showing that certain Hodge-type line bundles are ample on it. I will also mention applications to the subadditivity of logarithmic Kodaira dimension, and to the ampleness of the CM (Chow-Mumford) line bundle.