Conférence GAGC 2016 (en 2017)

Twisted Kodaira-Spencer classes and their use in the study of invariants of surfaces in P4

Non programmé

50m

CIRM, Luminy

Orateur

Daniel Naie

Description

(joint work Igor Reider) Let $X$ be a projective surface. A twisted Kodaira-Spencer class is an element of the cohomology group $H^1(T_X(-D))$, with $D$ ``sufficiently positive''. We study the connection between the existence of a non-trivial twisted class and the geometry of $X$. In particular, we show that, for a minimal general type surface satisfying $c_2/c_1^2<5/6$, the non-vanishing of $H^1(T_X(-K_X))$ imposes the existence of configurations of rational curves on the surface. The techniques used to obtain this result are based on the interpretation of a non-trivial twisted class as an extension ---a short exact sequence of locally free sheaves on $X$---, and on the detailed study of this sequence. The above point of view and techniques are applied to the study of surfaces in ${\mathbb{P}}^4$. Indeed, a surface of non-negative Kodaira dimension contained in a hypersurface of degree $\le 5$ displays a natural non-trivial twisted class, allowing us to address the Hartshorne-Lichtenbaum problem for, and to slightly control the irregularity of these surfaces.