On convergence sets of formal power series

Abstract

BackgroundA formal power series f (x 0 , x 1 , . . . , x n ) with coefficients in C is said to be convergent if it converges absolutely in a neighborhood of the origin in C n+1 . A classical result of Hartogs (see [5]) states that a series f converges if and only if f z (t) := f (z 0 t, z 1 t, . . . , z n t) converges, as a series in t, for all z ∈ C n+1 . This can be interpreted as a formal analog of Hartogs' theorem on separate analyticity. Because a divergent power series still may converge in certain directions, it is natural and desirable to consider the set of all z ∈ C n+1 for which f z converges. Since f z (t) converges if and only if f w (t) converges for all w ∈ C n+1 on the affine line through z, ignoring the trivial case z = 0, the set of directions along which f converges can be identified with a subset of P n . The convergence set Conv(f ) of a divergent power series f is defined to be the set of all directions ξ ∈ P n such that f z (t) is convergent for some z ∈ π −1 (ξ ), where π : C n+1 \{0} → P n is the natural projection. For the case n = 1, Lelong [9] proved that the convergence set of a divergent series f (x 1 , x 2 ) is an F σ polar set (i.e. a countable union of closed sets of vanishing logarithmic capacity) in P 1 , and moreover, every F σ polar subset of P 1 is contained in the convergence set of a divergent series f (x 1 , x 2 ). The optimal result was later obtained by Sathaye (see [16]) who showed that the class of convergence sets of divergent power series f (x 1 , x 2 ) is precisely the class of F σ polar sets in P 1 . To study the collection Conv(P n ) of convergence sets of divergent series in higher dimensions we consider the class PSH ω (P n ) of ω-plurisubharmonic functions on P n with respect to the form ω := dd c log |Z| on P n . We show that Conv(P n ) contains projective hulls of AbstractThe convergence set of a divergent formal power series f (x 0 , . . . , x n ) is the set of all "directions" ξ ∈ P n along which f is absolutely convergent. We prove that every countable union of closed complete pluripolar sets in P n is the convergence set of some divergent series f. The (affine) convergence sets of formal power series with polynomial coefficients are also studied. The higher-dimensional analogs of the results of Sathaye

Pluripolar Hulls and Convergence Sets

“…In the present paper, we consider the class PSH ω (P n ) where ω is the form mentioned above, introduce relative ω-plurisubharmonic extremal functions and Property J in P n , and establish some results on pluripolar hulls and convergence sets. The main result of this paper generalizes the main result of [14].…”

“…Note that K * j is the pluripolar hull of the compact pluripolar set K j . Of course this is more general than the main result in [14], because K * j in general is neither compact nor a complete pluripolar set. In order to prove the main theorem, we introduce the notion of relative ω-plurisubharmonic extremal functions.…”

“…The above definition does not depend on ω; replacing ω by cω with c > 0 leads to an equivalent definition. In fact, a subset E of P n is a (complete) pluripolar set in P n if and only if the set U ∩ E is a (complete) pluripolar set in U for each affine open set U(see, e.g., [14]).…”

“…Property J defined in [14] for a set in C n is the very property one uses to prove that the set is a convergence set. We now introduce Property J in P n , which will be used to establish a connection between the union of the pluripolar hulls of a countable collections of compact pluripolar sets and a convergence set in P n (see Theorem 4.5).…”

“…Unfortunately, in general E k is not a complete pluripolar set, and E k does not have Property J (i.e., E k does not have Property J with respect to every point in P n \ E k ), though E k does have Property J with respect to each X ∈ P n \ ∪ ∞ j=1 K * j . Hence with some additional efforts we are able to modify parts of the reasoning in [14] to finish the proof here.…”