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The structure of a stationary Gaussian process near a local maximum with a prescribed height$u$has been explored in several papers by the present author, see [5]–[7], which include results for moderate$u$as well as for$u$→±∞. In this paper we generalize these results to a homogeneous Gaussian field {ξ$(t) t ∈ R$^{n}}, with mean zero and the covariance function$r$($t$). The local structure of a Gaussian field near a high maximum has also been studied by Nosko, [8], [9], who obtains results of a slightly different type.

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Let X and Y be two complex manifolds, let$D$⊂$X$and$G$⊂$Y$be two nonempty open sets, let$A$(resp.$B$) be an open subset of ∂$D$(resp. ∂$G$), and let$W$be the 2-fold cross (($D$∪$A$)×$B$)∪($A$×($B$∪$G$)). Under a geometric condition on the boundary sets$A$and$B$, we show that every function locally bounded, separately continuous on$W$, continuous on$A$×$B$, and separately holomorphic on ($A$×$G$)∪($D$×$B$) “extends” to a function continuous on a “domain of holomorphy” $\widehat{W}$ and holomorphic on the interior of $\widehat{W}$ .

We study the asymptotic of the Bergman kernel of the spinc Dirac operator on high tensor powers of a line bundle.