No abstract uploaded!

Let X → B be a proper flat morphism between smooth quasiprojective varieties of relative dimension n, and L → X a line bundle which is ample on the fibers. We establish formulas for the first two terms in the Knudsen-Mumford expansion for det(πLk) in terms of Deligne pairings of L and the relative canonical bundle K. This generalizes the theorem of Deligne [1], which holds for families of relative dimension one. As a corollary, we show that when X is smooth, the line bundle η associated to X → B, which was introduced in Phong-Sturm [12], coincides with the CM bundle defined by Paul-Tian [10, 11]. In a second and third corollaries, we establish asymptotics for the K-energy along Bergman rays generalizing the formulas obtained in [11].

We provide an exhaustive description of all simply connected positively curved cohomogeneity one manifolds. The description yields a complete classification in all dimensions but 7, where in addition to known examples, our list contains one exceptional space and two infinite families not yet known to carry metrics of positive curvature. The infinite families also carry a 3-Sasakian metric of cohomogeneity one, which is associated to a family of selfdual Einstein orbifold metrics on the 4-sphere constructed by Hitchin.

We determine a global structure of the moduli space of self-dual metrics on 3CP2 satisfying the following three properties: (i) the scalar curvature is of positive type, (ii) they admit a non-trivial Killing field, (iii) they are not conformal to the LeBrun’s self- dual metrics based on the ‘hyperbolic ansatz’. We prove that the moduli space of these metrics is isomorphic to an orbifold R3/G, where G is an involution of R3 having two-dimensional fixed locus. In particular, the moduli space is non-empty and connected. We also remark that Joyce’s self-dual metrics with torus symmetry appear as a limit of our self-dual metrics. Our proof of the result is based on the twistor theory. We first determine a defining equation of a projective model of the twistor space of the metric, and then prove that the projective model is always birational to a twistor space, by determining the family of twistor lines. In determining them, a key role is played by a classical result in algebraic geometry that a smooth plane quartic always possesses twenty-eight bitangents.

The holonomy of the ambient metrics of Nurowski’s conformal structures associated to generic real-analytic 2-plane fields on oriented 5-manifolds is investigated. It is shown that the holonomy is always contained in the split real form G2 of the exceptional Lie group, and is equal to G2 for an open dense set of 2-plane fields given by explicit conditions. In particular, this gives an infinitedimensional family of metrics of holonomy equal to split G2. These results generalize work of Leistner-Nurowski. The inclusion of the holonomy in G2 is established by proving an ambient extension theorem for parallel tractors in the context of conformal geometry in general signature and dimension, which is expected to be of independent interest.