Let M = M1 ×M2 be a product of complex manifolds. We prove that M cannot admit a complete Kähler metric with sectional curvature K < c < 0 and Ricci curvature Ric > d, where c and d are arbitrary constants. In particular, a product domain in Cn cannot cover a compact Kähler manifold with negative sectional curvature. On the other hand, we observe that there are complete Kähler metrics with nega...

A consistent N = 1 supersymmetric σ-model can be constructed, given a Kähler manifold by adding chiral matter multiplets. Their scalar components are covariant tensors on the underlying Kähler manifold. The Kähler U(1)-charges can be adjusted such that the anomalies cancel, using the holomorphic functions in which the Kähler potential transforms. The arbitrariness of the U(1)-charges of matter ...

We study the asymptotic behavior of the Kähler-Ricci flow on Kähler manifolds of nonnegative holomorphic bisectional curvature. Using these results we prove that a complete noncompact Kähler manifold with nonnegative bounded holomorphic bisectional curvature and maximal volume growth is biholomorphic to complex Euclidean space C . We also show that the volume growth condition can be removed if ...

The base space of an algebraically completely integrable Hamiltonian system acquires a rather special differential-geometric structure which plays an important role in modern physical theories such as Seiberg-Witten theory. This structure was formalised by D. Freed [1] as a Special Kähler manifold. In that paper he conjectured that there are no compact special Kähler manifolds other than flat o...

Problem 1.1. Given a compact constant scalar curvature Kähler manifold (M,J, g, ω), of complex dimension m := dimC M , and having defined △ := {(p1, . . . , pn) ∈ M n : ∃ a 6= b pa = pb}, characterize the set PW = {(p1, . . . , pn, α1, . . . , αn)} ⊂ (M n \ △) × (0,+∞) for which M̃ = Blp1,...,pnM , the blow up of M at p1, . . . , pn has a constant scalar curvature Kähler metric (cscK from now on...

Let (M,J) be a compact complex 2-manifold which which admits a Kähler metric for which the integral of the scalar curvature is nonnegative. Also suppose that M does not admit a Ricci-flat Kähler metric. Then if M is blown up at sufficiently many points, the resulting complex surface (M̃ , J̃) admits Kähler metrics with scalar curvature identically equal to zero. This proves Conjecture 1 of [16]. ...

The set of all Abelian simply transitive subgroups of the affine group naturally corresponds to the set of real solutions of a system of algebraic equations. We classify all simply transitive subgroups of the symplectic affine group by constructing a model space for the corresponding variety of solutions. Similarly, we classify the complete global model spaces for flat special Kähler manifolds ...

Suppose that M is a complete Kähler manifold such its holomorphic sectional curvature bounded from below by constant and radial also below. N strongly pseudoconvex complex Finsler above negative constant. In this paper, we establish Schwarz lemma for mappings f into N. As applications, obtain Liouville type rigidity result N, as well theorem bimeromorphic compact manifold.

where Ric(ωKS) is the Ricci form of gKS and LXωKS denotes the Lie derivative of ωKS along a holomorphic vector field X on M . If X = 0, then gKS is a Kähler-Einstein metric with positive scalar curvature. We will show that the second variation of Perelman’s Wfunctional is non-positive in the space of Kähler metrics with 2πc1(M) as Kähler class. Furthermore, if (M, gKS) is a Kähler-Einstein mani...