Algebraic and analytic multiplier ideals and their applications

We show how the theory of multiplier ideals can be developed and discuss several applications of this theory. In the second section the same theory in the analytic setting is developed and several applications are given. Let X be a smooth algebraic variety and D an effective Q-divisor. We associate to D (or to the pair (X, D)) an ideal sheaf I(D) which controls the behavior of the fractional part of D and determines how close it is to have an simple normal crossing support. Other applications can be treated such as singularities of projective hypersurfaces and characterization of divisors. In the former case a result of Esnault-Viehweg concerning the least degree of hypersurfaces with multiplicity greater than or equal to a given positive integer at each point of a finite set is explained and proved in two different ways. A slight generalization is also given. Several vanishing and non-vanishing results including a global generation theorem are treated which will be used to prove...Show moreWe show how the theory of multiplier ideals can be developed and discuss several applications of this theory. In the second section the same theory in the analytic setting is developed and several applications are given. Let X be a smooth algebraic variety and D an effective Q-divisor. We associate to D (or to the pair (X, D)) an ideal sheaf I(D) which controls the behavior of the fractional part of D and determines how close it is to have an simple normal crossing support. Other applications can be treated such as singularities of projective hypersurfaces and characterization of divisors. In the former case a result of Esnault-Viehweg concerning the least degree of hypersurfaces with multiplicity greater than or equal to a given positive integer at each point of a finite set is explained and proved in two different ways. A slight generalization is also given. Several vanishing and non-vanishing results including a global generation theorem are treated which will be used to prove the results about singularities. In the second section the analytic analogues of the materials in section one are given and the characterization of analytic nef and good divisors are explained.Show less

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Algebraic and analytic multiplier ideals and their applications