This paper is a sequel of [8]. In algebraic geometry, when discussing the compactification of the moduli space of complex manifold X with ample canonical bundle KX , it is necessary to consider holomorphic degeneration family π : X → B, where Xt = π(t) are smooth for t 6= 0, X and X0 are QGorenstein, such that the canonical bundle of Xt for t 6= 0 and the dualizing sheaf of X0 are ample. We wil...

For every positively curved Kähler-Einstein manifold in four dimensions we construct an infinite family of supersymmetric solutions of type IIB supergravity. The solutions are warped products of AdS3 with a compact seven-dimensional manifold and have non-vanishing five-form flux. Via the AdS/CFT correspondence, the solutions are dual to two-dimensional conformal field theories with (0, 2) super...

the only justification for the einstein velocity addition law appeared to be its empirical adequacy, so that the intrinsic beauty and harmony in einstein addition remained for a long time a mystery to be conquered. accordingly, the aim of this expository article is to present (i) the einstein relativistic vector addition, (ii) the resulting einstein scalar multiplication, (iii) the einstein rel...

This paper relates the boundary term in the Chern-Gauss-Bonnet formula on 4-manifolds M with the renormalized volume V , as defined in the AdS/CFT correspondence, for asymptotically hyperbolic Einstein metrics on M . In addition we compute and discuss the differential or variation dV of V , or equivalently the variation of the L norm of the Weyl curvature, on the space of such Einstein metrics....

We present a general numerical method for investigating prescribed Ricci curvature problems on toric Kähler manifolds. This method is applied to two generalisations of Einstein metrics, namely Ricci solitons and quasi-Einstein metrics. We begin by recovering the Koiso–Cao soliton and the Lü–Page–Pope quasi-Einstein metrics on CP2]CP (in both cases the metrics are known explicitly). We also find...

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...

The goal of the paper is to deliberate conformal Ricci soliton and *-conformal within framework paracontact geometry. Here we prove that if an ?-Einstein para-Kenmotsu manifold admits soliton, then it Einstein. Further have shown 3-dimensional para-cosymplectic flat satisfies where vector field conformal. We also constructed some examples verify our results.

For the most general supersymmetric solutions of type IIB supergravity consisting of a warped product of AdS5 with a five-dimensional manifold M5, we construct an explicit consistent Kaluza-Klein reduction on M5 to minimal D = 5 gauged supergravity. Thus, any solution of the gauged supergravity can be uplifted on M5 to obtain an exact solution of type IIB supergravity. We also show that for gen...

In 1983, Futaki [2] introduced his invariants which generalize the obstruction of Kazdan-Warner to prescribe Gauss curvature on S. The Futaki invariants are defined for any compact Kähler manifold with positive first Chern class that has nontrivial holomorphic vector fields. Their vanishing are necessary conditions to the existence of Kähler-Einstein metric on the underlying manifold. Let M be ...

Let Z be a compact complex (2n+1)-manifold which carries a complex contact structure, meaning a codimension-1 holomorphic sub-bundle D ⊂ TZ which is maximally non-integrable. If Z admits a Kähler-Einstein metric of positive scalar curvature, we show that it is the Salamon twistor space of a quaternion-Kähler manifold (M, g). If Z also admits a second complex contact structure D̃ 6= D, then Z = C...