Algebraic Geometry Seminar

Seminar information archive ~03/28Next seminarFuture seminars 03/29~

Date, time & place Friday 13:30 - 15:00 ハイブリッド開催/117Room #ハイブリッド開催/117 (Graduate School of Math. Sci. Bldg.)
Organizer(s) GONGYO Yoshinori, NAKAMURA Yusuke, TANAKA Hiromu

2015/04/13

15:30-17:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Frédéric Campana (Université de Lorraine)
An orbifold version of Miyaoka's semi-positivity theorem and applications (English)
[ Abstract ]
This `orbifold' version of Miyaoka's theorem says that if (X,D)
is a projective log-canonical pair with K_X+D pseudo-effective,
then its 'cotangent' sheaf $¥Omega^1(X,D)$ is generically semi-positive.
The definitions will be given. The original proof of Miyaoka, which
mixes
char 0 and char p>0 arguments could not be adapted. Our proof is in char
0 only.

A first consequence is when (X,D) is log-smooth with reduced boudary D,
in which case the cotangent sheaf is the classical Log-cotangent sheaf:
if some tensor power of $¥omega^1_X(log(D))$ contains a 'big' line
bundle, then K_X+D is 'big' too. This implies, together with work of
Viehweg-Zuo,
the `hyperbolicity conjecture' of Shafarevich-Viehweg.

The preceding is joint work with Mihai Paun.

A second application (joint work with E. Amerik) shows that if D is a
non-uniruled smooth divisor in aprojective hyperkaehler manifold with
symplectic form s,
then its characteristic foliation is algebraic only if X is a K3 surface.
This was shown previously bt Hwang-Viehweg assuming D to be of general
type. This result has some further consequences.