Irrational connected sums and the topology of algebraic surfaces

Abstract

Suppose W is an irreducible nonsingular projective algebraic 3-fold and V a nonsingular hypersurface section of W. Denote by V m {V_m} a nonsingular element of | m V | \left | {mV} \right | . Let V 1 {V_1} , V m {V_m} , V m + 1 {V_{m\, + \,1}} be generic elements of | V | \left | V \right | , | m V | \left | {mV} \right | , | ( m + 1 ) V | \left | {(m\, + \,1)V} \right | respectively such that they have normal crossing in W. Let S 1 m = V 1 ∩ V m {S_{1m}}\, = \,{V_1}\, \cap \,{V_m} and C = V 1 ∩ V m ∩ V m + 1 C\, = \,{V_1}\, \cap \,{V_m}\, \cap \,{V_{m + 1}} . Then S 1 m {S_{1m}} is a nonsingular curve of genus g m {g_m} and C is a collection of N = m ( m + 1 ) V 1 3 N\, = \,m\left ( {m + 1} \right )V_1^3 points on S 1 m {S_{1m}} . By [MM2] we find that ( ∗ ) V m + 1 ( \ast )\,{V_{m\, + \,1}} is diffeomorphic to V m − T ( S 1 m ) ¯ ∪ η V 1 ′ − T ( S 1 m ′ ) ¯ \overline {{V_m}\, - \,T({S_{1m}})} \,{ \cup _\eta }\,\overline {{V_1}’\, - \,T({S_{1m}}’)} , where T ( S 1 m ) T\left ( {{S_{1m}}} \right ) is a tubular neighborhood of S 1 m {S_{1m}} in V m {V_m} , V 1 ′ {V_1}’ is V 1 {V_1} blown up along C, S 1 m ′ {S_{1m}}’ is the strict image of S 1 m {S_{1m}} in V 1 ′ {V_1}’ , T ( S 1 m ′ ) T({S_{1m}}’) is a tubular neighborhood of S 1 m ′ {S_{1m}}’ in V 1 ′ {V_1}’ and η : ∂ T ( S 1 m ) → ∂ T ( S m ′ ) \eta :\,\partial T\left ( {{S_{1m}}} \right ) \to \partial T({S_m}’) is a bundle diffeomorphism. Now V 1 ′ {V_1}’ is well known to be diffeomorphic to V 1 # N ( − C P 2 ) {V_1}\, \# \,N\left ( { - C{P^2}} \right ) (the connected sum of V 1 {V_1} and N copies of C P 2 C{P^2} with opposite orientation from the usual). Thus in order to be able to inductively reduce questions about the structure of V m {V_m} to ones about V 1 {V_1} we must simplify the “irrational sum” ( ∗ ) ( \ast ) above. The general question we can ask is then the following: Suppose M 1 {M_1} and M 2 {M_2} are compact smooth 4-manifolds and K is a connected q-complex embedded in M i {M_i} . Let T i {T_i} be a regular neighborhood of K in M i {M_i} and let η : ∂ T 1 → ∂ T 2 \eta :\,\partial {T_1}\, \to \,\partial {T_2} be a diffeomorphism: Set V = M 1 − T 1 ¯ ∪ M 2 − T 2 ¯ V\, = \,\overline {{M_1}\, - \,{T_1}} \, \cup \,\overline {{M_2}\, - \,{T_2}} . How can the topology of V be described more simply in terms of those of M 1 {M_1} and M 2 {M_2} . In this paper we show how surgery can be used to simplify the structure of V in the case q = 1 , 2 q\, = \,1,\,2 and indicate some applications to the topology of algebraic surfaces.