Irrational connected sums and the topology of algebraic surfaces
Authors: Mandelbaum Richard 
Published: July 2010

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove upper V Subscript m Baseline minus upper T left-parenthesis upper S Subscript 1 m Baseline right-parenthesis With bar union Subscript eta Baseline ModifyingAbove upper V prime 1 minus upper T left-parenthesis upper S prime Subscript 1 m right-parenthesis With bar">

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.