We consider the Kähler-Ricci flow on complete finite-volume metrics that live on the complement of a divisor in a compact Kähler manifold X. Assuming certain spatial asymptotics on the initial metric, we compute the singularity time in terms of cohomological data on X. We also give a sufficient condition for the singularity, if there is one, to be type II.

Jost and Zuo’s theorem (adapting an earlier idea of Gromov [7]) states that if the Kähler form ω on a complete Kähler manifold satisfies ω = dβ, where β is a one-form of linear growth, then the only L harmonic forms lie in the middle dimension. An application of the same argument shows further that if G is a connected Lie group of isometries on a complete Riemannian manifold generated by Killin...

Consider a holomorphic torus action on a possibly non-compact Kähler manifold. We show that the higher cohomology groups appearing in the geometric quantization of the symplectic quotient are isomorphic to the invariant parts of the corresponding cohomology groups of the original manifold. For non-Abelian group actions on compact Kähler manifolds, this result was proved recently by Teleman and ...

We show that noncompact simply connected harmonic manifolds with volume density Θ p (r) = sinh n−1 r is isometric to the real hyperbolic space and noncompact simply connected Kähler harmonic manifold with volume density Θ p (r) = sinh 2n−1 r cosh r is isometric to the complex hyperbolic space. A similar result is also proved for Quaternionic Kähler manifolds. Using our methods we get an alterna...

We introduce a cohomological obstruction to solving the constant scalar curvature Kähler (cscK) equation twisted by a semipositive form, appearing in works of Fine and Song-Tian. Geometrically this gives an obstruction for a manifold to be the base of a holomorphic submersion carrying a cscK metric in certain “adiabatic” classes. In turn this produces many new examples of general type threefold...

We compute the moduli Kähler potential for M-theory on a compact manifold of G2 holonomy in a large radius approximation. Our method relies on an explicit G2 structure with small torsion, its periods and the calculation of the approximate volume of the manifold. As a verification of our result, some of the components of the Kähler metric are computed directly by integration over harmonic forms....

Let (M, I,ω,Ω) be a nearly Kähler 6-manifold, that is, an SU(3)-manifold with the (3,0)-form Ω and the Hermitian form ω which satisfies dω = 3λReΩ, d ImΩ = −2λω, for a non-zero real constant λ. We develop an analogue of Kähler relations on M , proving several useful identities for various intrinsic Laplacians onM . WhenM is compact, these identities bring powerful results about cohomology of M ...

For any Lagrangean Kähler submanifold M ⊂ T Cn, there exists a canonical hyper Kähler metric on T ∗M . A Kähler potential for this metric is given by the generalized Calabi Ansatz of the theoretical physicists Cecotti, Ferrara and Girardello. This correspondence provides a method for the construction of (pseudo) hyper Kähler manifolds with large automorphism group. Using it, a class of pseudo h...

The global holomorphic invariant αG(M) introduced by Tian[14], Tian and Yau[13] is closely related to the existence of Kähler-Einstein metrics. In his solution to the Calabi conjecture, Yau[19] proved the existence of a Kähler-Einstein metric on compact Kähler manifolds with nonpositive first Chern class. Kähler-Einstein metrics do not always exist in the case when the first Chern class is posi...