M ay 2 00 8 A Bergman kernel proof of the Kawamata subadjunction theorem

The main purpose of the following article is to give a proof of Y. Kawa-mata's celebrated subadjunction theorem in the spirit of our previous work on Bergman kernels. We will use two main ingredients : an L 2 m –extension theorem of Ohsawa-Takegoshi type (which is also a new result) and a more complete version of our former results. §0 Introduction Let X be a projective manifold and let Θ be a closed positive current of (1,1)– type on X. A quantitative measure of the logarithmic singularities of Θ is its critical exponent introduced e.g. in [10] as follows: C(X, Θ) := sup{c ≥ 0 : exp(−cϕ Ω) ∈ L 1 (Ω)} for all coordinate set Ω ⊂ X. The function ϕ Ω above is a local potential of Θ on the coordinate set Ω (see the paragraph 3 for the expression of the normalization we use). We remark that the above notion makes sense, since two local potentials of the same current differ by a smooth function. One can see that the notion of critical exponent of a closed positive (1, 1)–current is the analytic counterpart of the log canonical threshold in algebraic geometry. Indeed, if we have Θ = [D] where D is some effective Q–divisor on X, then C(X, Θ) = 1 if and only if D is log canonical. Pushing the analogy with the algebraic geometry a little bit further, one can easily imagine what the notion of center of a current Θ with C(X, Θ) = 1 should be : we have to consider the components of the multiplier ideal sheaf associated to Θ. However, it is not yet known that the multiplier ideal sheaf of such a current is strictly contained in the structural sheaf of X ; therefore, in this article we will consider exclusively psh functions with analytic singularities (but nevertheless, it is possible to obtain partial results concerning the psh functions whose singularities admit accurate enough approximations with analytic singularities). Concerning the minimal center of a Q-effective and log canonical divisor D we have the deep subadjunction theorem of Y. Kawamata, stating that the restriction of the canonical bundle of the ambient manifold twisted with D to the center is linearly equivalent to the canonical class of the center plus a closed positive current. We establish now the general framework for our results : a) The current Θ has analytic singularities …