Hassan Jolany

Existence of canonical metric on a projective variety was a
long standing conjecture -- the major part of this conjecture is
about varieties which do not have definite first Chern class (most of the
manifolds do not have definite first Chern class). There is a program
which is known as the Song-Tian program, for finding a canonical metric 
on canonical models of a projective variety, by using the minimal model
program. In this talk, we extend the Song-Tian program and give a Log
version of it. We investigate conical Kähler-Ricci flow
on holomorphic fiber spaces $\pi : X \to B$ whose generic fibers are
Calabi-Yau pairs, $c_1(K_B+D)<0$, where $D$ is a smooth, ample, and simple
normal crossing divisor on $B$. We show that there is a unique conical
Kahler Einstein metric on pair $(B, D)$ which is twisted by logarithmic
Weil-Petersson metric plus a current of integration, by introducing 
Log Yau-Vafa semi ricci flat metrics. We introduce a new logarithmic
canonical measure and show that its inverse is an analytic Zariski
decomposition. We present a similar result when log fibers have negative
first Chern class. For the positive case we show some partial results
also. Moreover, we show that along the conical Kähler Ricci flow the
metric collapsing of $(X, \omega(t))$ converges exponentially fast in
$C^0$-topology, hence in Gromov-Hausdorff topology, to $\omega_B$. This
strengthens previous works of Song-Tian and Tosatti-Weinkove-Yang.