Extension of closed currents to the projective space (Q282854)

Let \(X\) be a complex projective variety of dimension \(p\) in the projective space \({\mathbb P}_n\). If \(\theta\) is a smooth closed differential \((q,q)\)-form on \(X\), then it can be ``extended'' to a closed \((q,q)\)-current \(S\) on \({\mathbb P}_n\) such that \(S|_X=\theta\), the restriction being defined by means of the blow-up with center \(X\).NEWLINENEWLINEIn the note, such an ``extension'' is constructed by means of an integral kernel \(G\) (a current of bidegree \((p,p)\), closed in \({\mathbb P}_n\times X\)) as \(S=\pi_{1*}(G\wedge\pi_2^*\theta)\), where \(\pi_1\) and \(\pi_2\) are the projections of \({\mathbb P}_n\times X\) to \({\mathbb P}_n\) and \(X\), respectively. In addition, a current \(U\) is constructed in a way that \(S-dd^cU\) is smooth and any point \(x\in X\setminus E\) has a neighborhood \(\Omega\) such that \((dd^cU)|_\Omega= (dd^cU|_\Omega)\) for a negligible closed algebraic set \(E\); this gives an answer to a problem raised by Grothendick.NEWLINENEWLINEThe construction is also extended from smooth forms \(\theta\) to algebraic cycles on \(X\). In particular, it is shown that an algebraic subset \(Z\) is a complete intersection if and only if its ``extension'' \(S\) is a positive current.NEWLINENEWLINEUsing a similar idea, the author defines ``restrictions'' \(S|_X\) of positive closed currents \(S\) to \(X\), provided \(S|_{{\mathbb P}_n\setminus X}\) is closed, positive and of finite mass. A variant of the Lefschetz hyperplane section theorem is proved for transverse restrictions \(S|_Y\).