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Medientyp:
E-Artikel
Titel:
On Cohomology Vanishing with Polynomial Growth on Complex Manifolds with Pseudoconvex Boundary
Beteiligte:
Ohsawa, Takeo
Erschienen:
European Mathematical Society - EMS - Publishing House GmbH, 2023
Erschienen in:
Publications of the Research Institute for Mathematical Sciences, 59 (2023) 4, Seite 759-768
Sprache:
Nicht zu entscheiden
DOI:
10.4171/prims/59-4-3
ISSN:
0034-5318;
1663-4926
Entstehung:
Anmerkungen:
Beschreibung:
The \bar{\partial} cohomology groups with polynomial growth H^{r,s}_{\mathrm{p.g.}} will be studied. It will be shown that, given a complex manifold M , a locally pseudoconvex bounded domain \Omega\Subset M satisfying certain geometric boundary conditions and a holomorphic vector bundle E\to M , H^{r,s}_{\mathrm{p.g.}}(\Omega,E)=0 holds for all s\geq1 if E is Nakano positive and r=\dim{M} . It will also be shown that H^{r,s}_{\mathrm{p.g.}}(\Omega,E)=0 for all r and s with r+s>\dim{M} if, moreover, \operatorname{rank}E=1 . By the comparison theorem due to Deligne, Maltsiniotis (Astérisque 17 (1974), 141–160) and Sasakura (Inst. Math. Sci. 17 (1981), 371–552), it follows in particular that, for any smooth projective variety X , for any ample line bundle L\to X and for any effective divisor D on X such that [D]|_{| D|}\geq0 , the algebraic cohomology H^{s}_{\mathrm{alg}}(X\setminus | D | , \Omega_X^r(L)) vanishes if r+s>\dim{X} .