Ueda theory for compact curves with nodes

Abstract

Abstract. Let C be a compact complex curve included in a non-singular complex surface such that the normal bundle is topologically trivial. Ueda studied complex analytic properties of a neighborhood of C when C is non-singular or is a rational curve with a node. We propose an analogue of Ueda's theory for the case where C admits nodes. As an application, we study singular Hermitian metrics with semi-positive curvature on the anti-canonical bundle of the blow-up of the projective plane at nine points in arbitrary position.

Hermitian metrics on the anti-canonical bundle of the blow-up of the projective plane at nine points

“…In this case, K −1 X Z is semi-positive by a standard argument (see §2.1). In [K4], we showed that the conditions (i), (iii) and (iv) in Theorem 1.2 hold if α(N Z ) = e 2π √ −1θ holds for some Diophantine irrational number θ [K4,Theorem 1.4,Corollary 7.5]. Here we say that an irrational number θ is Diophantine if there exist positive numbers A and α such that min n∈Z |n − mθ| ≥ A • m −α .…”

“…As nothing is unclear on (singular) Hermitian metrics on K −1 X Z if it is not nef (see [K4, §7]), we assume that K −1 X Z is nef. Then, according to [K4,Proposition 7.10], there exists a reduced divisor Y Z ∈ |K −1 X Z | such that the restriction K −1 X Z | Y Z admits a flat connection (i.e. all the transition functions are C * -constant for a suitable choice of local trivializations) and that Y Z is the strict transform of either a smooth elliptic curve, a cycle of rational curves, a curve with a cusp, or three lines intersecting at a point of P 2 .…”

“…normal crossing ) singularities whose normalization consists of finite numbers of P 1 's and whose dual graph is a cycle graph (Note that a rational curve with a node is also a cycle of rational curves in our definition). We are mainly interested in the case where Y Z is a smooth elliptic curve or a cycle of rational curves, since the other cases have already been investigated in the proof of [K4,Proposition 7.10 (ii)].…”

“…Note that, when K −1 X Z is nef and Y Z has a singular point which is not a node (i.e. Y Z is a curve with a cusp or three lines intersecting at a point), it follows from the argument in the proof of [K4,Proposition 7.10] that the assertions (i), (iii), and (iv) in Theorem 1.2 are equivalent. Note also that, under the assumption in the theorem above, it is known that Y Z admits a strongly pseudoconcave neighborhoods system when N Z is not unitary flat (i.e.…”

“…Note also that, under the assumption in the theorem above, it is known that Y Z admits a strongly pseudoconcave neighborhoods system when N Z is not unitary flat (i.e. |α(N Z )| = 1, [U2] for a rational curve with a node, [K4,Theorem 1.6] in general). It is shown by Brunella that the assertions (i) and (iii) in Theorem 1.2 are equivalent to each other when Y Z is non-singular and X Z \ Y Z does not contain any compact complex curve [B,Theorem 1 (i)].…”