Differential Equations on Complex Manifolds

Leray’s quantization of projective duality

“…Corollary 4.5 shows that Φ A interchanges generalized holomorphic functions on P * and P. It would be interesting to study more precisely the transform of the sheaf of ramified holomorphic functions on a hypersurface of P * . Related results are obtained in [28].…”

“…The arguments that we shall use here are very classical, and go back to Leray [21] (see also [28]). We begin with a geometric lemma.…”

About recognition of mirror-like objects

“…¬ÑÑÓAEË-ÐÂÕÞ àÕËØ ËÊÑÃÓÂÉÇÐËÌ ÏÑÅÖÕ ÃÞÕß ÐÂÌAEÇÐÞ Ô ÒÑÏÑÜßá ÒÓËÐÙËÒ ÔËÏÏÇÕÓËË ²ËÏÂÐÂ ë ºÄÂÓÙ [4,5].…”

On analytic continuability of the missing Cauchy datum for Helmholtz boundary problems

“…In view of existence results such as the theorem of Cauchy-Kovalevsky [5, Theorem 9.4.5], or the propagation of singularities of solutions to the analytic Cauchy problem [24], this question is central in the à priori estimation of the domain of analytic continuability of solutions across the boundary. This, in turn, is applicable, e.g., in the stability and convergence analysis of 'interior source methods', which is a family of promising numerical methods for direct and inverse elliptic boundary problems [2,3,8,9,10,11].…”

“…This is done for general linear, elliptic, second-order, analytic, exterior boundary problems in two independent variables and with piecewise analytic boundary in [18], for such interior and exterior problems with analytic boundary in [21], as well as for exterior three-dimensional Helmholtz problems in a half-space [19] or with axisymmetric boundary [20]. Another global approach can be found in Section 4 of Sternin and Shatalov [25], for three-dimensional Helmholtz problems with Neumann datum given on an algebraic surface; see Sternin and Shatalov [24] for a more general treatment. More recently, Kangro, Kangro and Nicolaides [23] proposed a local approach to the à priori analytic continuation of solutions of two-dimensional Dirichlet boundary problems for the Helmholtz equation across analytic pieces of the boundary.…”