A consequence of a lower bound of the K-energy
We prove that over a Fano manifold having the K-energy of a the canonical class bounded from below, the Chen-Tian energy functional E_1 of the class is allso bounded from below.
math.AP↗
arXiv subjects
Nefton Pali
Publications and source records attributed to Nefton Pali.
Plurisubharmonic functions and positive currents of type (1,1) over almost complex manifolds
If $(X,J)$ is an almost complex manifold, then a function $u$ is said to be plurisubharmonic on $X$ if it is upper semi-continuous and its restriction to every local pseudo-holomorphic curve is subharmonic. As in the complex case, it is conjectured that plurisubharmonicity is equivalent to the fact that the $(1,1)$-current $i\partial_{_J}\bar{\partial}_{_J}u$ is positive, (the $(1,1)$-current $i\partial_{_J}\bar{\partial}_{_J}u$ need not be closed here). The conjecture is trivial if $u$ is of class ${\cal C}^2$. The result is elementary in the complex integrable case because the operator $i\partial_{_J}\bar{\partial}_{_J}$ can be written as an operator with constant coefficients in complex coordinates. Hence the positivity of the current is preserved by regularising with usual convolution kernels. This is not possible in the almost complex non integrable case and the proof of the result requires a much more intrinsic study. In this chapter we prove the necessity of the positivity of the $(1,1)$-current $i\partial_{_J}\bar{\partial}_{_J}u$. We prove also the sufficiency of the positivity in the particular case of an upper semi-continuous function $f$ which is continuous in the complement of the singular locus $f^{-1}(-\infty)$. For the proof of the sufficiency of the positivity in the general case of a real distribution $u$, we suggest a method depending on a rather delicate regularisation argument introduced by Demailly. This method consists of regularing the function $u$ by means of the flow induced by a Chern connection on the tangent bundle of the almost complex manifold.
math.DG↗
Chern connections and Chern curvature of the tangent bundle of almost complex manifolds
The $\bar{\partial}_{_{J}}$ operator over an almost complex manifold induces canonical connections of type $(0,1)$ over the bundles of $(p,0)$-forms. If the almost complex structure is integrable then the previous connections induce the canonical holomorphic structures of the bundles of $(p,0)$-forms. For $p=1$ we can extend the corresponding connection to all Schur powers of the bundle of $(1,0)$-forms. Moreover using the canonical $\C$-linear isomorphism betwen the bundle of $(1,0)$-forms and the complex cotangent bundle $T^*_{X,J}$ we deduce canonical connections of type $(0,1)$ over the Schur powers of the complex cotangent bundle $T^*_{X,J}$. If the almost complex structure is integrable then the previous $(0,1)$-connections induces the canonical holomorphic structures of those bundles. In the non integrable case those $(0,1)$-connections induces just the holomorphic canonical structures of the restrictions of the corresponding bundles to the images of smooth $J$-holomorphic curves. We introduce the notion of Chern curvature for those bundles. The geometrical meaning of this notion is a natural generalisation of the classical notion of Chern curvature for the holomophic vector bundles over a complex manifold. We have a particular interest for the case of the tangent bundle in view of applications concerning the regularisation of $J$-plurisubharmonic fonctions by means of the geodesic flow induced by a Chern connection on the tangent bundle. This method has been used by Demailly in the complex integrable case. Our specific study in the case of the tangent bundle gives an asymptotic expanson of the Chern flow which relates in a optimal way the geometric obstructions caused by the torsion of the almost complex structure, and the non symplectic nature of the metric.
math.DG↗
Une caractérisation différentielle des faisceaux analytiques cohérents sur une variété complexe
We give a generalization, in the context of sheaves, of a classical result of Grothendieck concerning the integrability of connections of type $(0,1)$ over a ${\cal C}^{\infty}$ vector bundle over a complex manifold. We introduce the notion of $\bar{\partial}$-coherent sheaf, which is a ${\cal C}^{\infty}$ notion, and we prove the existence of an (exact) equivalence between the category of coherent analytic sheaves and the category of $\bar{\partial}$-coherent sheaves. The principal difficulty of the proof is the solution of a quasi-linear differential equation with standard $\bar{\partial}$ as its principal term. We are able to find a solution of this differential equation, using a rapidly convergent iteration scheme of Nash-Moser type.
math.AG↗