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