1. Introduction
Let X be a smooth, closed, connected, oriented 4-manifold and C a closed, connected, smoothly embedded, oriented surface in X. We are interested in the topology of C or, specifically, the genus of C.
If
X is a complex surface and
C is a complex curve, the well known adjunction formula in algebraic geometry (for example, see Sections 1.4 and 2.1 in [
1]) shows that
Here,
is the canonical class for the complex structure,
is the homology class of
C and
denotes the value of symmetric bilinear form
If oriented, smoothly embedded surfaces
intersect transversely, then
is the signed count of
. Thus
is called the intersection form of
X. By Poincaré duality, we can identify homology group
with cohomology group
and
is equivalent to the cup product in cohomology ring
which is a unimodular symmetric bilinear form, that is,
. Let
be the signature of
.
The adjunction formula shows that the genus of a complex curve
C is completely determined by its homology class
, the symmetric bilinear form
and the canonical class
K (or the complex structure). This observation suggests that the genus of embedded surfaces should be explored from the viewpoint of homology theory. To fix a homology class
, we want to know the minimal genus of a connected surface representing
A, or the minimal genus function
For rich history on this function, see the excellent surveys of Lawson [
2,
3].
Suppose
X is the complex projective plane
. If
are smooth surfaces in
X such that
and
C are complex curves, the Thom conjecture, which was proved in [
4], says that
It shows that complex curves in have a minimal genus within their homology classes.
If
X has a symplectic structure
, we can also use a compatible, almost complex, structure to define the symplectic canonical class
. The symplectic Thom conjecture, proved in [
5,
6], states that a symplectic surface has minimal genus within its homology class, and, under some conditions for
, each surface
C satisfies the adjunction inequality:
In both cases, the lower bound of genus depends on the intersection form and the complex or symplectic structure. Moreover, the computation is descended to linear spaces and can be achieved by elementary linear algebra.
For example, the product of Riemann surfaces
has homology group
.
has a basis
with
and
. Assume
C is a complex torus on
X with
, then it satisfies
So or and or . It is easy to construct an embedded torus in the class .
Later, an adjunction type lower bound
h for
was introduced in [
7] by Strle, and was extended to a more general situation in [
8]. See also the beautiful reformulation in [
9]. Unlike the well known adjunction bounds in [
4,
5,
6], this bound is purely in terms of the cohomology algebra of
X under the assumption
. More precisely, let
be the cohomology algebra of
X modulo torsion. Then
h is completely determined by
when
, and is called the cohomological genus function of
X. This allows us to transform the study of
, which depends on the geometric structure of
X, to the computation of
h, which is based on the algebraic structure of
and is much easier to deal with than
.
It is shown in [
8] that the cohomological genus function
h provides a sharp bound for
when
. Here,
is the modified Euler number of
X and
is the signature of
X (see
Section 2.1 for a precise definition). The sharp bound is realized when
with
Y being a rational or ruled surface.
In this note, we will deal with the complementary case. We speculate that rational surfaces or ruled surfaces with have the same property.
Conjecture 1. Let X be a smooth 4-manifold with the cohomology algebra of a rational surface or ruled surface Z with . Then there exists a cohomology algebra isomorphism such that, for any class with , the minimal genus of A is bounded below by the minimal genus of , .
We are able to partially verify the conjecture.
Theorem 1. Let X be a smooth 4-manifold with the cohomology algebra of a rational surface or ruled surface Z with and is any cohomology algebra isomorphism. Let , where is the canonical class of Z regarded as a complex surface. For all classes with and , we have . Specifically, Various terminologies will be reviewed in
Section 2. One important ingredient of the proof of Theorem 1 is the symplectic genus introduced in [
10].
In this note, we will also make some explicit estimates of when . In this case, our estimates are not optimal, but are easy to compute and effective.
The organization of this note is as follows: in
Section 2, we review the notions of cohomology algebra, cohomological genus and adjunction classes from [
8], and their general properties, which are needed throughout the article. In
Section 3, we prove Theorem 1 and explicitly compute
h for a rational or ruled type cohomology algebra with
when
A satisfies
and
. In
Section 4, we estimate
h in the remaining case.
2. Cohomology Algebra of Type and Cohomological Genus
2.1. Cohomology Algebra and Quadratic Form
Let
be a finitely generated graded commutative algebra over
with each summand
a free abelian group, and a group isomorphism
.
, or simply
, is called a cohomology algebra if, with respect to
p, the products
are duality pairings, in the sense that
are isomorphisms of groups. Let
be the skew-symmetric pairing and
the symmetric form.
Let be the rank of , the rank of T, the Euler number, and the modified Euler number. Denote the signature type of by , and let denote the signature. is called an algebra of type if . A class is called characteristic if for any .
For X a smooth, closed, connected, oriented 4-manifold, let /Tor. Then is a cohomology algebra. We define the cohomological invariants of X as .
Example 1. The following algebra of type is modeled on : , with trivial T pairing.
Given two cohomology algebras and , their direct sum is defined as the cohomology algebra with for , and the bilinear pairings being the direct sum. It is clear that .
is called Lefschetz if T is non-degenerate, or equivalently, . is called the Lefschetz reduction of if is Lefschetz and for some l. It is easy to see that is well defined up to isomorphism, so we will denote any Lefschetz reduction of by . A simple but useful fact is that and .
Suppose now
is a cohomology algebra of
type. A special feature is that the image of
T is either 0 or 1 dimensional, see [
8], Paragraph 2.1.1. Since
T is skew symmetric,
is always an even number.
Notice that the symmetric bilinear pairing is unimodular. We will often abbreviate as . It induces a unimodular quadratic form as . is called the square of x. is of even type if is even for any vector . Otherwise, is called the odd type. is called definite, or indefinite if min or ≥1, respectively.
It is well known that indefinite unimodular symmetric forms are classified by their rank, signature and type (e.g., [
11]). When
, the list of unimodular symmetric forms is
Here, and E are the hyperbolic lattice and the (positive definite) lattice, respectively.
Remark 1. It is conjectured in [12] that when . If this is true, then for any Lefschetz type algebra Λ
with and , there is a rational or ruled surface Z with . 2.2. Cohomological Genus Function h
A class is called an adjunction class if it is characteristic and either of the following conditions is satisfied,
- (I)
,
- (II)
and c pairs non-trivially with when T is non-trivial.
Definition 1. Let . For any class c of adjunction type, introduce the genus of A,Let , where the maximum is taken among all adjunction classes of Λ.
is called the cohomological genus of Λ.
When there is no confusion, we just use for . For
X, a smooth, closed, connected, oriented 4-manifold with
and any cohomology class
with
, there is the inequality (Theorem 1.4 in [
8])
This inequality was proved in [
7] when
and
, via
moduli spaces of manifolds with cylindrical ends. The proof in [
8] uses the wall crossing formula for Seiberg–Witten invariants.
2.3. and Adjunction Classes
Let be a cohomology algebra of type and the automorphism group of . Let be the set of adjunction classes of . We discuss the properties of this set and their consequences for the function h.
To compute or estimate , we want to minimize among all .
If T is trivial, then there are only type I adjunction classes. A well-known result says that characteristic classes satisfy . So is equivalent to . By direct computation, we also have .
Lemma 1. When T is non-trivial, that is, , a type I adjunction class is also of type II if and only if it pairs non-trivially with ImT.
Since an adjunction class c is characteristic, is always an integer. It also satisfies , hence . We remark that the minimal genus function also has this symmetry.
Lemma 2 ([
8]).
is preserved by . Consequently, h is invariant under . Under the natural identification between and , . Consequently, . By this invariance property, to determine the function
h, it suffices to pick an element, called reduced element, in each orbit of
and to calculate the value of
h. When
, this was carried out in [
7,
8] completely.
4. Estimates of h When
In this section, let X be a 4-manifold with the cohomology ring of a rational surface or ruled surface Z with . Given a standard basis of let be the associated canonical class. Let A be a reduced class with , and . We will estimate .
4.1. Rational Surface Type (Case (1))
Let be a standard basis of such that . Let be a reduced class with . So and . For convenience, we assume that . Otherwise, it reduces to a smaller n case.
Recall that , so means that . It is also clear that .
Let be an adjunction class. Then, k and are odd and .
Lemma 5. Suppose , is a reduced class with , and . We can find an adjunction class such that . Consequently, we have the uniform and simple bound, Proof. Let
where
are defined inductively as follows. Let
,
,
be the partial sums of
. Then,
The sign rule is to make the partial sums oscillate around 0 with minimal amplitudes.
The sign rule Equation (
2) determines a subsequence of indices
where the sequence of partial sums
change monotonicity.
Let . When , , and j is even, we have , and the subsequence strictly decreases until it becomes negative. So , when j is even.
When , and j are odd, we have , and the subsequence strictly increases until it becomes non-negative. So when j is odd.
If , then we have . If , then the remaining string has two cases according to the sign of . In all cases, we can verify that .
Since , we have a uniform bound . The genus bound follows immediately. □
Example 3. Let . Then . We have , and .
Similarly, for the sequence of primitive classes , . Since and we have . Note that could be arbitrarily large for this sequence of primitive classes.
Remark 3. For primitive classes A with and , is estimated in Theorem 11.1 and Corollary 11.2 in [7]. In particular, if A is characteristic, . It is remarked that the leading term for is, by a factor of 2, better than the bounds for the divisible classes in [15,16]. 4.2. Minimal Irrational Ruled Surface Type (Cases (2) and (3))
Let c be an adjunction class. By Lemma 1, c is of type II.
Proposition 3. Let Λ be a cohomology algebra of irrational ruled surface type and a standard basis of . Let be a reduced class with and . There is a uniform estimate for :
Proof. In the case
,
and
imply
. An adjunction class has the form
, and satisfies
We take adjunction class
. Then
, and
In the case
,
and
imply
. An adjunction class has the form
, and satisfies
We take adjunction class
, then
, and
□
Remark 4. This bound is uniform and very simple but not optimal in general. Consider the even intersection form case . Suppose with .
- 1.
Then, is an adjunction class and . Therefore, we have .
- 2.
For the divisible class , we also have .
Remark 5. The set of adjunction classes is bigger when increases. This is true since becomes smaller as increases. Consequently, we sometimes obtain stronger estimates for the same class when increases, although the uniform estimate in Proposition 3 is independent of . For example, if and , for , we can choose and get . When and , we can choose and get .
4.3. Non-Minimal Irrational Ruled Surface Type (Case (4))
Choose a standard basis for with , , , . Recall that .
Let be a reduced class with . So , . We assume , otherwise it reduces to the case.
Recall that . So implies .
Lemma 6. Let be a reduced class with , , and satisfy . Then we have Proof. An adjunction class has the form
, where
,
are odd integers.
c is of type II, so
We will choose an adjunction class
with
. So,
If , then we take all , and get .
If
, let
,
,
, be the partial sums of
. Applying the oscillating sequence method and sign rule Equation (
2) in
Section 4.1, we can deduce that
.
Thus, we obtain an estimate , and have the desired genus bound. □