The Cohomological Genus and Symplectic Genus for 4-Manifolds of Rational or Ruled Types

Abstract

An important problem in low dimensional topology is to understand the properties of embedded or immersed surfaces in 4-dimensional manifolds. In this article, we estimate the lower genus bound of closed, connected, smoothly embedded, oriented surfaces in a smooth, closed, connected, oriented 4-manifold with the cohomology algebra of a rational or ruled surface. Our genus bound depends only on the cohomology algebra rather than on the geometric structure of the 4-manifold. It provides evidence for the genus minimizing property of rational and ruled surfaces.

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 $g\left(C\right)$ 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

$$g\left(C\right)=\frac{\left[C\right]·\left[C\right]+K\left(\right[C\left]\right)+2}{2}.$$

Here, $K\in H^2(X,\mathbb{Z})$ is the canonical class for the complex structure, $\left[C\right]\in H_2(X;\mathbb{Z})$ is the homology class of C and $\left[C\right]·\left[C\right]$ denotes the value of symmetric bilinear form

$$Q_X:H_2(X;\mathbb{Z})/Tor\times H_2(X;\mathbb{Z})/Tor\to \mathbb{Z}.$$

If oriented, smoothly embedded surfaces $C_1,C_2$ intersect transversely, then $\left[C_1\right]·\left[C_2\right]=Q_X(\left[C_1\right],\left[C_2\right])$ is the signed count of $C_1\cap C_2$. Thus $Q_X$ is called the intersection form of X. By Poincaré duality, we can identify homology group $H_2(X;\mathbb{Z})$ with cohomology group $H^2(X;\mathbb{Z})$ and $Q_X$ is equivalent to the cup product in cohomology ring

$$Q_X:H^2(X;\mathbb{Z})/Tor\times H^2(X;\mathbb{Z})/Tor\to \mathbb{Z},$$

which is a unimodular symmetric bilinear form, that is, $det\left(Q_X\right)=\pm 1$. Let $(b^{+}\left(X\right),b^{-}\left(X\right))$ be the signature of $Q_X$.

The adjunction formula shows that the genus of a complex curve C is completely determined by its homology class $\left[C\right]$, the symmetric bilinear form $Q_X$ 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 $A\in H^2(X;\mathbb{Z})$, we want to know the minimal genus of a connected surface representing A, or the minimal genus function

$$m{g}_X\left(A\right)=min\left\{g\left(C\right)\right|C:\mathrm{connected},\mathrm{smoothly}\mathrm{embedded}\mathrm{surface}\mathrm{with}\left[C\right]=PD\left(A\right)\}.$$

For rich history on this function, see the excellent surveys of Lawson [2,3].

Suppose X is the complex projective plane $\mathbb{C}{\mathbb{P}}^2$. If $C,C^{\prime }$ are smooth surfaces in X such that $\left[C\right]=\left[C^{\prime }\right]$ and C are complex curves, the Thom conjecture, which was proved in [4], says that

$$g\left(C^{\prime }\right)\ge g\left(C\right)=\frac{\left[C\right]·\left[C\right]+K\left(\right[C\left]\right)+2}{2}.$$

It shows that complex curves in $\mathbb{C}{\mathbb{P}}^2$ have a minimal genus within their homology classes.

If X has a symplectic structure $\omega$, we can also use a compatible, almost complex, structure to define the symplectic canonical class $K_{\omega }$. 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 $(X,\omega )$, each surface C satisfies the adjunction inequality:

$$g\left(C\right)\ge \frac{\left[C\right]·\left[C\right]+|K_{\omega }\left(\left[C\right]\right)|+2}{2}.$$

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 $X=S^2\times T^2$ has homology group $H_2(X;\mathbb{Z})\cong \mathbb{Z}\oplus \mathbb{Z}$. $H_2(X;\mathbb{Z})$ has a basis $B=[pt\times T^2],F=[S^2\times pt]$ with $B·B=F·F=0,$$B·F=1$ and $K=-2B$. Assume C is a complex torus on X with $\left[C\right]=aB+bF$, then it satisfies

$$g\left(C\right)=\frac{(aB+bF)·(aB+bF)+(-2B)·(aB+bF)+2}{2}=ab-b+1=1.$$

So $a=1$ or $b=0$ and $\left[C\right]=B+bF$ or $aB$. It is easy to construct an embedded torus in the class $B+bF$.

Later, an adjunction type lower bound h for $m{g}_X$ 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 $b^{+}\left(X\right)=1$. More precisely, let ${\Lambda }_X=H^{\ast }(X;\mathbb{Z})/Tor$ be the cohomology algebra of X modulo torsion. Then h is completely determined by ${\Lambda }_X$ when $b^{+}\left(X\right)=1$, and is called the cohomological genus function of X. This allows us to transform the study of $m{g}_X$, which depends on the geometric structure of X, to the computation of h, which is based on the algebraic structure of ${\Lambda }_X$ and is much easier to deal with than $m{g}_X$.

It is shown in [8] that the cohomological genus function h provides a sharp bound for $m{g}_X$ when $2\tilde{\chi }\left(X\right)+3\sigma \left(X\right)\ge 0$. Here, $\tilde{\chi }\left(X\right)$ is the modified Euler number of X and $\sigma \left(X\right)=b^{+}\left(X\right)-b^{-}\left(X\right)$ is the signature of X (see Section 2.1 for a precise definition). The sharp bound is realized when $X=Y\#l(S^1\times S^3)$ 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 $2\tilde{\chi }+3\sigma <0$ 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 $2\tilde{\chi }+3\sigma <0$. Then there exists a cohomology algebra isomorphism $\rho :{\Lambda }_X\to {\Lambda }_Z$ such that, for any class $A\in {\Lambda }_X^2$ with $A·A\ge 0$, the minimal genus of A is bounded below by the minimal genus of $\rho \left(A\right)$, $m{g}_X\left(A\right)\ge m{g}_Z\left(\rho \left(A\right)\right)$.

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 $2\tilde{\chi }+3\sigma <0$ and $\rho :{\Lambda }_X\to {\Lambda }_Z$ is any cohomology algebra isomorphism. Let $c_0={\rho }^{-1}(-K_Z)\in {\Lambda }_X^2$, where $K_Z$ is the canonical class of Z regarded as a complex surface. For all classes $A\in {\Lambda }_X^2$ with $A·A>0$ and $c_0·A\ge 0$, we have $m{g}_X\left(A\right)\ge m{g}_Z\left(\rho \left(A\right)\right)$. Specifically,

$$m{g}_X\left(A\right)\ge \frac{A·A-c_0·A+2}{2}.$$

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 $h\left(A\right)$ when $c_0·A<0$. 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 $2\tilde{\chi }+3\sigma <0$ when A satisfies $A·A>0$ and $c_0·A\ge 0$. In Section 4, we estimate h in the remaining case.

2. Cohomology Algebra of ${\mathit{b}}^{+}=\mathbf{1}$ Type and Cohomological Genus

2.1. Cohomology Algebra and Quadratic Form

Let $\Lambda ={\oplus }_{i=0}^4{\Lambda }^i$ be a finitely generated graded commutative algebra over $\mathbb{Z}$ with each summand ${\Lambda }^i$ a free abelian group, and a group isomorphism $p:{\Lambda }^4\cong \mathbb{Z}$. $(\Lambda ,p)$, or simply $\Lambda$, is called a cohomology algebra if, with respect to p, the products ${\Lambda }^i\times {\Lambda }^{4-i}\to {\Lambda }^4\cong \mathbb{Z}$ are duality pairings, in the sense that

$${\Lambda }^i\to {\mathrm{H}\mathrm{o}\mathrm{m}}_{\mathbb{Z}}({\Lambda }^{4-i},\mathbb{Z})$$

are isomorphisms of groups. Let $T:{\Lambda }^1\times {\Lambda }^1\to {\Lambda }^2$ be the skew-symmetric pairing and $\mathrm{\Gamma }:{\Lambda }^2\times {\Lambda }^2\to {\Lambda }^4\cong \mathbb{Z}$ the symmetric form.

Let $b_i(\Lambda )$ be the rank of ${\Lambda }^i$, ${\tilde{b}}_1(\Lambda )$ the rank of T, $\chi (\Lambda )=\sum {(-1)}^i{b}_i(\Lambda )$ the Euler number, and $\tilde{\chi }(\Lambda )=2+b_2(\Lambda )-2{\tilde{b}}_1(\Lambda )$ the modified Euler number. Denote the signature type of $\mathrm{\Gamma }$ by $(b^{+}(\Lambda ),b^{-}(\Lambda ))$, and let $\sigma (\Lambda )=b^{+}(\Lambda )-b^{-}(\Lambda )$ denote the signature. $\Lambda$ is called an algebra of $b^{+}=k$ type if $b^{+}(\Lambda )=k$. A class $\xi \in {\Lambda }^2$ is called characteristic if $\mathrm{\Gamma }(\xi ,\eta )\equiv \mathrm{\Gamma }(\eta ,\eta )mod2$ for any $\eta \in {\Lambda }^2$.

For X a smooth, closed, connected, oriented 4-manifold, let ${\Lambda }_X=H^{\ast }(X;\mathbb{Z})$/Tor. Then ${\Lambda }_X$ is a cohomology algebra. We define the cohomological invariants of X as $\ast \left(X\right)=\ast \left({\Lambda }_X\right),\ast =b_i,{\tilde{b}}_1,\chi ,\tilde{\chi },\sigma$.

Example 1.

The following algebra of $b^{+}=0$ type ${\Lambda }_S$ is modeled on $S^1\times S^3$: ${\Lambda }_S^2=0$, ${\Lambda }_S^1={\Lambda }_S^3=\mathbb{Z}$ with trivial T pairing.

Given two cohomology algebras ${\Lambda }_a$ and ${\Lambda }_b$, their direct sum ${\Lambda }_a\oplus {\Lambda }_b$ is defined as the cohomology algebra $\Lambda$ with ${\Lambda }^i={\Lambda }_a^i\oplus {\Lambda }_b^i$ for $1\le i\le 3$, and the bilinear pairings being the direct sum. It is clear that $b^{+}({\Lambda }_a\oplus {\Lambda }_b)=b^{+}\left){\Lambda }_a\right)+b^{+}\left({\Lambda }_b\right)$.

$\Lambda$ is called Lefschetz if T is non-degenerate, or equivalently, ${\tilde{b}}_1(\Lambda )=b_1(\Lambda )$. ${\Lambda }^{\prime }$ is called the Lefschetz reduction of $\Lambda$ if ${\Lambda }^{\prime }$ is Lefschetz and $\Lambda \cong {\Lambda }^{\prime }\oplus l{\Lambda }_S$ for some l. It is easy to see that ${\Lambda }^{\prime }$ is well defined up to isomorphism, so we will denote any Lefschetz reduction of $\Lambda$ by ${\Lambda }_{red}$. A simple but useful fact is that ${\tilde{b}}_1(\Lambda )={\tilde{b}}_1\left({\Lambda }_{red}\right)$ and $\tilde{\chi }(\Lambda )=\tilde{\chi }\left({\Lambda }_{red}\right)$.

Suppose now $\Lambda$ is a cohomology algebra of $b^{+}=1$ 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, ${\tilde{b}}_1(\Lambda )$ is always an even number.

Notice that the symmetric bilinear pairing $\mathrm{\Gamma }$ is unimodular. We will often abbreviate $\mathrm{\Gamma }(x,y)$ as $x·y$. It induces a unimodular quadratic form $Q:{\Lambda }^2\to \mathbb{Z}$ as $Q\left(x\right)=\mathrm{\Gamma }(x,x)$. $Q\left(x\right)$ is called the square of x. $\mathrm{\Gamma }$ is of even type if $Q\left(x\right)$ is even for any vector $x\in {\Lambda }^2$. Otherwise, $\mathrm{\Gamma }$ is called the odd type. $\mathrm{\Gamma }$ is called definite, or indefinite if min$\{b^{+},b^{-}\}=0$ or ≥1, respectively.

It is well known that indefinite unimodular symmetric forms are classified by their rank, signature and type (e.g., [11]). When $b^{+}=1$, the list of unimodular symmetric forms is

$$\langle 1\rangle \oplus n\langle -1\rangle ,U\oplus q(-E),\mathrm{where}n,q\in \mathbb{N}\cup \left\{0\right\}.$$

Here, $U=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ and E are the hyperbolic lattice and the (positive definite) $E_8$ lattice, respectively.

Remark 1.

It is conjectured in [12] that $q\le 1$ when $b^{+}=1$. If this is true, then for any Lefschetz type algebra Λ with $b^{+}=1$ and $2\tilde{\chi }+3\sigma <0$, there is a rational or ruled surface Z with ${\Lambda }_Z=\Lambda$.

2.2. Cohomological Genus Function h

A class $c\in {\Lambda }^2$ is called an adjunction class if it is characteristic and either of the following conditions is satisfied,

(I)

$c·c>\sigma (\Lambda )$,

(II)

$c·c\ge 2\tilde{\chi }(\Lambda )+3\sigma (\Lambda )$ and c pairs non-trivially with $\mathrm{Im}T$ when T is non-trivial.

Definition 1.

Let $A\in {\Lambda }^2$. For any class c of adjunction type, introduce the $c-$genus of A,

$$h_c\left(A\right)=\left\{\begin{array}{cc}\frac{A·A-|c·A|+2}{2}\hfill & ifA\ne 0\hfill \\ 0\hfill & ifA=0.\hfill \end{array}\right.$$

(1)

Let $h_{\Lambda }\left(A\right)=max{h}_c\left(A\right)$, where the maximum is taken among all adjunction classes of Λ. $h_{\Lambda }$ is called the cohomological genus of Λ. When there is no confusion, we just use $h\left(A\right)$ for $h_{\Lambda }\left(A\right)$.

For X, a smooth, closed, connected, oriented 4-manifold with $b^{+}\left(X\right)=1$ and any cohomology class $A\in {\Lambda }_X^2$ with $A·A\ge 0$, there is the inequality (Theorem 1.4 in [8])

$$m{g}_X\left(A\right)\ge h\left(A\right).$$

This inequality was proved in [7] when $b^{+}=1$ and $b_1=0$, via $L^2$ moduli spaces of manifolds with cylindrical ends. The proof in [8] uses the wall crossing formula for Seiberg–Witten invariants.

2.3. $Aut(\Lambda )$ and Adjunction Classes

Let $\Lambda$ be a cohomology algebra of $b^{+}=1$ type and $Aut(\Lambda )$ the automorphism group of $\Lambda$. Let ${\mathcal{C}}_{\Lambda }$ be the set of adjunction classes of $\Lambda$. We discuss the properties of this set and their consequences for the function h.

To compute or estimate $h\left(A\right)$, we want to minimize $|c·A|$ among all $c\in {\mathcal{C}}_{\Lambda }$.

If T is trivial, then there are only type I adjunction classes. A well-known result says that characteristic classes satisfy $c·c\equiv \sigma (\Lambda )\left(\mathrm{mod}8\right)$. So $c·c>\sigma (\Lambda )$ is equivalent to $c·c\ge \sigma (\Lambda )+8$. By direct computation, we also have $2\tilde{\chi }(\Lambda )+3\sigma (\Lambda )=\sigma (\Lambda )+8-4{\tilde{b}}_1(\Lambda )$.

Lemma 1.

When T is non-trivial, that is, ${\tilde{b}}_1(\Lambda )\ge 2$, 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, $h_c\left(A\right)$ is always an integer. It also satisfies $h_c\left(A\right)=h_c(-A)$, hence $h\left(A\right)=h(-A)$. We remark that the minimal genus function $m{g}_X$ also has this symmetry.

Lemma 2

([8]). ${\mathcal{C}}_{\Lambda }$ is preserved by $Aut(\Lambda )$. Consequently, h is invariant under $Aut(\Lambda )$. Under the natural identification between ${\Lambda }^2$ and ${\Lambda }_{red}^2$, ${\mathcal{C}}_{\Lambda }={\mathcal{C}}_{{\Lambda }_{red}}$. Consequently, $h_{\Lambda }=h_{{\Lambda }_{red}}$.

By this invariance property, to determine the function h, it suffices to pick an element, called reduced element, in each orbit of $Aut(\Lambda )$ and to calculate the value of h. When $2\tilde{\chi }+3\sigma \ge 0$, this was carried out in [7,8] completely.

3. Calculation of h in Terms of the Symplectic Genus

3.1. Cohomology Algebras of Rational or Ruled Type with $2\tilde{\chi }+3\sigma <0$

We say that an algebra $\Lambda$ is of rational or ruled type if $\Lambda \cong {\Lambda }_Z$ for some rational or ruled manifold Z. Notice that such $\Lambda$ is Lefschetz and of $b^{+}=1$ type, thus $2\tilde{\chi }(\Lambda )+3\sigma (\Lambda )=\sigma (\Lambda )+8-4{\tilde{b}}_1(\Lambda )$. Cohomology algebras of rational or ruled type with $2\tilde{\chi }+3\sigma <0$ can be listed as

1.

T trivial and $\mathrm{\Gamma }=\langle 1\rangle \oplus n\langle -1\rangle ,n\ge 10$,

2.

${\tilde{b}}_1(\Lambda )=2g\ge 4$ and $\mathrm{\Gamma }=U$,

3.

${\tilde{b}}_1(\Lambda )=2g\ge 4$ and $\mathrm{\Gamma }=\langle 1\rangle \oplus \langle -1\rangle ,$

4.

${\tilde{b}}_1(\Lambda )=2g\ge 2$ and $\mathrm{\Gamma }=U\oplus n\langle -1\rangle ,n\ge 1$.

3.1.1. Standard basis and reduced classes

We now introduce the notion of standard bases of ${\Lambda }^2$ in the four cases. The standard bases are equivalent under $Aut(\Lambda )$. Notice that, whenever T is trivial, we have $Aut(\Lambda )=Aut\left(\mathrm{\Gamma }\right)$.

We also introduce reduced classes under a choice of standard basis. As mentioned in the paragraph after Lemma 2, we only have to consider the $c-$genus for reduced classes in the computation of h.

Case (1): 

Let $n=b^{-}\ge 10$ and $\mathcal{B}=\{H,E_1,\dots ,E_n\}$ be a pairwise orthogonal basis for ${\Lambda }^2$ such that $H·H=1=-E_i·E_i$.

Reduced classes ([13,14]): $aH-{\sum }_{i=1}^n{b}_i{E}_i$ with $b_1\ge b_2\ge \cdots \ge b_n\ge 0$ and $a\ge b_1+b_2+b_3$.

For the remaining cases, assume $\mathrm{Im}T$ is generated by F. In particular, $F·F=0$. Note that F is preserved by $Aut(\Lambda )$ up to sign.

Case (2): 

Choose a basis $\mathcal{B}=\{F,B\}$ of ${\Lambda }^2$ with $B·B=0$ and $F·B=1$.

The automorphism group is ${\mathbb{Z}}_2$ generated by $-Id$.

Reduced classes: $aB+bF$ with $a>0$, or $a=0,b\ge 0$.

Case (3): 

Choose a basis $\mathcal{B}=\{F,B\}$ of ${\Lambda }^2$ with $B·B=1,F·B=1$.

The automorphism group is ${\mathbb{Z}}_2$ generated by $-Id$.

Reduced classes: $aB+bF$ with $a>0$, or $a=0,b\ge 0$.

Case (4): 

Choose a basis $\mathcal{B}=\{F,B,E_i\}$ of ${\Lambda }^2$ with $B·B=0$, $F·B=1$, $E_i·E_i=-1$, $E_i·E_j=E_i·F=E_i·B=0$, $1\le i\ne j\le n$.

Reduced classes (Definition 3.3 in [10]): $aB+bF-{\sum }_{i=1}^n{e}_i{E}_i$ with $a>0,a\ge e_1\ge e_2\ge \cdots \ge e_n\ge 0$ or $bF,b\ge 0$.

3.1.2. Geometric Standard Basis for Rational or Ruled Surfaces

For a rational surface Z, a standard basis is said to be geometric if each class is represented by an embedded sphere. For a ruled surface Z over a genus g surface, a standard basis is said to be geometric if F and $E_i$ are represented by embedded spheres, B is represented by an embedded surface of genus g.

Geometric standard bases always exist and are equivalent under $D\left(Z\right)$. Here, $D\left(Z\right)$ is the geometric automorphism group of Z, which is the image of $Dif{f}^{+}\left(Z\right)\to Aut(\Lambda \left(Z\right))$.

Lemma 3.

Any class of non-negative square in the four cases is equivalent to a reduced class under $Aut(\Lambda )$.

Proof. 

Lemma 3.4 (1) in [10] says that, for a rational or ruled surface, any class of non-negative square in the four cases is equivalent to a reduced class under $D\left(Z\right)$. Since $D\left(Z\right)$ is a subgroup of $Aut(\Lambda )$, any class of non-negative square is equivalent to a reduced class under $Aut(\Lambda )$. □

3.1.3. The Adjunction Classes

We describe the adjunction classes in terms of a standard basis.

Lemma 4.

Let c be an adjunction class. Then, c is of type I for a rational surface type, and of type II for an irrational ruled surface type.

Proof. 

For a rational surface type, there are only type I adjunction classes since T is trivial.

By Lemma 1, we just need to check that c is non-trivial on F. Let $c=\alpha B+\beta F$ for Cases (2) and (3), and $c=\alpha B+\beta F+{\sum }_{i=1}^n{\gamma }_i{E}_i$ with ${\gamma }_i$ odd for Case (4).

If c is trivial on F, then $\alpha =0$. So c cannot be of type I since $c·c=0=\sigma$ for Cases (2) and (3), and $c·c=-{\sum }_{i=1}^n{\gamma }_i^2\le -n=\sigma$ for Case (4). □

Generally, it is not easy to evaluate $h\left(A\right)$ since there are always infinitely many adjunction classes. We will use the notion of symplectic genus in [10] to evaluate $h\left(A\right)$ when $c_0·A\ge 0$.

3.1.4. The Adjunction Class $c_0$

Proposition 1.

Suppose a basis $\mathcal{B}$ is given in each case. There is a unique $c_0\in {\Lambda }^2$ such that

(i) 

$c_0$ is characteristic and $c_0·E_i=1$.

(ii) 

$c_0·c_0=2\tilde{\chi }+3\sigma$,

(iii) 

$c_0$ is reduced,

(iv) 

$|c_0·F|=2$ when T is non-trivial.

Explicitly, with respect to the standard basis $\mathcal{B}$, $c_0$ is given in each case as follows.

• Case (1): $c_0=3H-{\sum }_{i=1}^n{E}_i$
• Case (2): $c_0=2B+(2-2g)F$,
• Case (3): $c_0=2B+(1-2g)F$,
• Case (4): $c_0=2B+(2-2g)F-{\sum }_{i=1}^n{E}_i$.

Since standard bases are equivalent under $Aut(\Lambda )$, the associated $c_0$ are also equivalent under $Aut(\Lambda )$.

Proof. 

The existence of $c_0$ is given by the explicit expression.

To establish the uniqueness, note that the coefficient of each $E_i$ is 1 in Cases (1) and (4) by (i), and the coefficient of B is 2 in Cases (2)–(4) by (iii) and (iv). Finally, by (ii) and (iii), the coefficient of H is 3 in Case (1), and by (ii), the coefficient of F is uniquely determined in Cases (2)–(4). □

We write down $h_{c_0}\left(A\right)$ explicitly, when $c_0·A\ge 0$.

• Case (1): $\left(\genfrac{}{}{0pt}{}{a-1}{2}\right)-{\sum }_{i=1}^n\left(\genfrac{}{}{0pt}{}{b_i}{2}\right)$
• Case (2): $(a-1)(b-1)+ga$
• Case (3): $(a-1)(\frac{a}{2}+b-1)+ga$
• Case (4): $(a-1)(b-1)+ga-\frac{1}{2}{\sum }_{i=1}^n{e}_i(e_i-1)$.

3.2. Symplectic Genus of a Rational or Ruled Surface

To define the symplectic genus, first recall the notion of the $K-$symplectic cone. A class $K\in H^2(X;\mathbb{Z})$ is called a symplectic canonical class if it is the canonical class of some orientation-compatible symplectic structure on X. Let $\mathcal{K}$ be the set of symplectic canonical classes. For any $K\in \mathcal{K}$, we introduce the $K-$symplectic cone:

${\mathcal{C}}_K=\left\{\left[\omega \right]\right|\omega \mathrm{is}\mathrm{an}\mathrm{orientation}-\mathrm{compatible}\mathrm{symplectic}\mathrm{form},K_{\omega }=K\}$.

For $e\in {\Lambda }_X^2$, consider the subset ${\mathcal{K}}_e$ of $\mathcal{K}$:

$${\mathcal{K}}_e=\{K\in \mathcal{K}|\mathrm{there}\mathrm{is}\mathrm{a}\mathrm{class}\tau \in {\mathcal{C}}_K\mathrm{such}\mathrm{that}\tau ·e>0\}.$$

For $K\in {\mathcal{K}}_e$, define the $K-$symplectic genus ${\eta }_K\left(e\right)$ to be $\frac{1}{2}(K·e+e·e)+1$. Finally, the symplectic genus of e is defined as

$$\eta \left(e\right)=\underset{K\in {\mathcal{K}}_e}{max}{\eta }_K\left(e\right).$$

By Lemma 3.2 in [10], $\eta \left(e\right)$ has the following basic properties.

• $\eta \left(e\right)$ is no bigger than the minimal genus of e, and they are both equal to the $K-$symplectic genus ${\eta }_K\left(e\right)$ if e is represented by a connected $\omega -$symplectic surface for some symplectic form $\omega$ with $K_{\omega }=K$. Here $K_{\omega }$ is the symplectic canonical class of $\omega$
• $\eta \left(e\right)=\eta (-e)$, and $D\left(X\right)-$equivalent classes have the same $\eta$.

For a rational or ruled surface, the symplectic genus $\eta$ has additional properties. Let Z be a non-minimal rational or irrational ruled surface with a geometric standard basis. Let $c_0$ be the associated adjunction class. In [10], $c_0$ is denoted as $-K_0$.

• (Lemma 3.4 (5) in [10]) If e is a reduced class with non-negative square, then $-c_0\in \mathcal{K}\left(e\right)$ and consequently $\eta \left(e\right)$ is given by ${\eta }_{-c_0}\left(e\right)$.
• (Lemma 3.6 in [10]) The symplectic genus of any class with positive square or a primitive class with square 0 is non-negative.
• (Proposition 3.5 in [10]) Suppose e is a reduced class. If $e·e\ge \eta \left(e\right)-1$, then $e·e\ge 0$ and e is represented by a symplectic surface. Moreover, if e is either a class of positive square or a primitive class with square 0, e is represented by a connected symplectic surface, and therefore its minimal genus is given by its symplectic genus.

We are now ready to prove Theorem 1.

Proposition 2.

Let X be a 4-manifold with the cohomology ring of a rational surface or ruled surface Z with $2\tilde{\chi }+3\sigma <0$. Given a standard basis of ${\Lambda }_X^2$ and a geometric standard basis of ${\Lambda }_Z^2$, we can identify ${\Lambda }_X^2$ with ${\Lambda }_Z^2$ and let $c_0$ be the associated adjunction class. For a reduced class A with $A·A>0$, or $A·A=0$ and A is primitive, and $c_0·A\ge 0$, we have

$$m{g}_X\left(A\right)\ge h\left(A\right)=h_{c_0}\left(A\right)=m{g}_Z\left(A\right).$$

Proof. 

If $c_0·A\ge 0$, then $|c_0·A|=c_0·A$. Hence $h_{c_0}\left(A\right)={\eta }_{-c_0}\left(A\right)$.

Since $\eta \left(A\right)={\eta }_{-c_0}\left(A\right)$ by Lemma 3.4 (5) in [10], $A·A\ge \eta \left(A\right)-1$ is the same as

$$A·A\ge {\eta }_{-c_0}\left(A\right)-1=\frac{1}{2}(A·A-c_0·A),$$

which is the same as $A·A\ge -c_0·A$. This is certainly satisfied if $A·A\ge 0$ and $c_0·A\ge 0$.

By Proposition 3.5 in [10], $\eta \left(A\right)=m{g}_Z\left(A\right)$. Therefore we have

$$m{g}_X\left(A\right)\ge h\left(A\right)=h_{c_0}\left(A\right)=m{g}_Z\left(A\right),$$

if A is a reduced class with $A·A>0$, or $A·A=0$ and primitive, and $c_0·A\ge 0$. □

Proof of Theorem 1

Since a reduced class A with sufficiently big H coefficient in Case (1), or positive B coefficient and sufficiently big F coefficient in Cases (2)–(4) satisfies $A·A>0$ and $c_0·A\ge 0$, it follows from Proposition 2 that there are infinitely many classes as claimed. □

Remark 2.

It follows from Theorem 1 and Lemma 2 and Lemma 4.1 [8] that if $\tilde{Z}$ is a rational surface or a ruled surface with $2\tilde{\chi }+3\sigma <0$ and $Z=\tilde{Z}\#l(S^1\times S^3)$, then for any 4-manifold X with the cohomology ring of Z and an identification $\rho :{\Lambda }_X\to {\Lambda }_Z$, there are infinitely many homology classes $A\in {\Lambda }_X^2$ with $A·A>0$ and $m{g}_X\left(A\right)\ge m{g}_Z\left(\rho \left(A\right)\right)$.

If the conjecture on the even intersection form in [12] mentioned in Remark 1 holds, then any 4-manifold X with $b^{+}=1$ and $2\tilde{\chi }+3\sigma <0$ has the cohomology ring of a rational or ruled surface.

Example 2.

Let $Z=\mathbb{C}{\mathbb{P}}^2\#n\overline{\mathbb{C}{\mathbb{P}}^2},n\ge 10$. For any $b>0,n\ge 10$, $A=bnH-{\sum }_{i=1}^{n}b{E}_i$ satisfies $A·A>0,c_0·A\ge 0$.

By standard topological surgery, we can construct an embedded smooth surface C in Z such that $\left[C\right]=PD\left(A\right)$ and

$$g\left(C\right)=\frac{b^{2}n(n-1)}{2}-bn+1=h_{c_0}\left(A\right)$$

is the minimal genus.

4. Estimates of h When ${\mathit{c}}_{\mathbf{0}}·\mathit{A}<\mathbf{0}$

In this section, let X be a 4-manifold with the cohomology ring of a rational surface or ruled surface Z with $2\chi +3\sigma <0$. Given a standard basis of ${\Lambda }_X^2$ let $c_0$ be the associated canonical class. Let A be a reduced class with $A·A\ge 0$, and $c_0·A<0$. We will estimate $h\left(A\right)$.

4.1. Rational Surface Type (Case (1))

Let $\mathcal{B}=\{H,E_1,\dots ,E_n\}$ be a standard basis of ${\Lambda }^2$ such that $H^2=1=-E_i^2$. Let $A=aH-{\sum }_{i=1}^n{b}_i{E}_i$ be a reduced class with $A·A\ge 0$. So $b_1\ge b_2\ge \cdots \ge b_n\ge 0$ and $a\ge b_1+b_2+b_3$. For convenience, we assume that $b_n>0$. Otherwise, it reduces to a smaller n case.

Recall that $c_0=3H-{\sum }_{i=1}^n{E}_i$, so $c_0·A<0$ means that $3a-{\sum }_{i=1}^n{b}_i<0$. It is also clear that $3a-{\sum }_{i=1}^9{b}_i\ge 0$.

Let $c=kH-{\sum }_{i=1}^n{e}_i{E}_i$ be an adjunction class. Then, k and $e_i$ are odd and $c·c=k^2-\sum e_i^2\ge \sigma +8=9-n$.

Lemma 5.

Suppose $n\ge 10$, $A=aH-{\sum }_{i=1}^n{b}_i{E}_i$ is a reduced class with $A·A\ge 0$, and $3a-{\sum }_{i=1}^n{b}_i<0$. We can find an adjunction class $c_1$ such that $|c_1·A|\le b_{10}$. Consequently, we have the uniform and simple bound,

$$h\left(A\right)\ge h_{c_1}\left(A\right)=\frac{1}{2}(A·A-|c_1·A|+2)\ge \frac{1}{2}A·A-\frac{b_{10}}{2}+1.$$

Proof. 

Let $c_1=3H+{\sum }_{i=1}^n{ϵ}_i{E}_i$ where ${ϵ}_i=\pm 1$ are defined inductively as follows. Let $S_0=3a$, $S_p=3a+{\sum }_{i=1}^p{ϵ}_i{b}_i$, $1\le p\le n$ be the partial sums of $c_1·A=3a+{\sum }_{i=1}^n{ϵ}_i{b}_i$. Then,

$${ϵ}_{p+1}=-\mathrm{sgn}\left(S_p\right)=\left\{\begin{array}{cc}\hfill -1& \mathrm{if}{\mathrm{S}}_{\mathrm{p}}\ge 0,\hfill \\ \hfill 1& \mathrm{if}{\mathrm{S}}_{\mathrm{p}}<0,\hfill \end{array}\right.0\le p\le n-1.$$

(2)

The sign rule is to make the partial sums $S_p$ oscillate around 0 with minimal amplitudes.

The sign rule Equation (2) determines a subsequence of indices

$$10\le l_1<l_2<\cdots <l_m\le n,$$

(3)

where the sequence of partial sums $\left\{S_p\right\}$ change monotonicity.

Let $l_0=0$. When $l_j+1\le p\le l_{j+1}$, $0\le j\le m-1$, and j is even, we have ${ϵ}_p=-1$, and the subsequence $S_p$ strictly decreases until it becomes negative. So $-b_{l_{j+1}}\le S_{l_{j+1}}<0$, when j is even.

When $l_j+1\le p\le l_{j+1}$, $0\le j\le m-1$ and j are odd, we have ${ϵ}_p=1$, and the subsequence $S_p$ strictly increases until it becomes non-negative. So $0\le S_{l_{j+1}}<b_{l_{j+1}}$ when j is odd.

If $l_m=n$, then we have $|S_n|\le b_n$. If $l_m<n$, then the remaining string has two cases according to the sign of $S_{l_m}$. In all cases, we can verify that $|c_1·A|=|S_n|\le b_{l_m}$.

Since $l_m\ge 10$, we have a uniform bound $|c_1·A|\le b_{10}$. The genus bound follows immediately. □

Example 3.

Let $a=16,b_1=6,b_2=\cdots =b_9=5>b_{10}=b_{11}=3$. Then $c_1=3H-{\sum }_{i=1}^{10}{E}_i+E_{11}$. We have $A·A=2\ge 0,c_0·A=-4<0$, and $|c_1·A|=2<|c_0·A|=4$.

Similarly, for the sequence of primitive classes $A=(3s+1)H-(s+1)b_1-{\sum }_{i=2}^{9}s{E}_i-3E_{10}-3E_{11},s\ge 5$, $c_1=3H-{\sum }_{i=1}^{10}{E}_i+E_{11}$. Since $A·A=4s-18\ge 0$ and $c_1·A=2$ we have $h\left(A\right)\ge h_{c_1}\left(A\right)=2s-9$. Note that $h\left(A\right)$ could be arbitrarily large for this sequence of primitive classes.

Remark 3.

For primitive classes A with $A·A>0$ and $A_d=dA$, $h\left(A_d\right)$ is estimated in Theorem 11.1 and Corollary 11.2 in [7]. In particular, if A is characteristic, $h\left(A_d\right)\ge \left(\genfrac{}{}{0pt}{}{d}{2}\right)A·A$. It is remarked that the leading term for $h\left(A_d\right)$ is, by a factor of 2, better than the bounds for the divisible classes $A_d$ 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 $\{B,F\}$ a standard basis of ${\Lambda }^2$. Let $A=aB+bF$ be a reduced class with $A·A\ge 0$ and $c_0·A<0$. There is a uniform estimate for $h\left(A\right)$:

• $h\left(A\right)\ge (a-1)b+1$ when $B·B=0$.
• $h\left(A\right)\ge (a-1)(\frac{a}{2}+b)+1$ when $B·B=1$.

Proof. 

In the case $B·B=0$, $A·A=2ab\ge 0$ and $c_0·A=(2-2g)a+2b<0$ imply $(g-1)a>b\ge 0$. An adjunction class has the form $c=2pB+2qF$, and satisfies

$$c·c=8pq\ge 2\tilde{\chi }+3\sigma =8-8gandc·F=2p\ne 0.$$

We take adjunction class $c_1=2B$. Then $c_1·A=2b\ge 0$, and

$$h\left(A\right)\ge h_{c_1}\left(A\right)=\frac{1}{2}(2ab-2b+2)=(a-1)b+1.$$

In the case $B·B=1$, $A·A=a(a+2b)\ge 0$ and $c_0·A=(3-2g)a+2b<0$ imply $a>0,a+2b\ge 0$. An adjunction class has the form $c=2pB+(2q-1)F$, and satisfies

$$c·c=4p(p+2q-1)\ge 2\tilde{\chi }+3\sigma =8-8gandc·F=2p\ne 0.$$

We take adjunction class $c_2=2B-F$, then $c_2·A=a+2b\ge 0$, and

$$h\left(A\right)\ge h_{c_2}\left(A\right)=\frac{1}{2}(a^2+2ab-(a+2b)+2)=(a-1)(\frac{a}{2}+b)+1.$$

□

Remark 4.

This bound is uniform and very simple but not optimal in general. Consider the even intersection form case $B·B=0$. Suppose $A=aB+bF$ with $A·A=2ab\le 2g-2$.

1. 

Then, $c=2aB-2bF$ is an adjunction class and $c·A=0$. Therefore, we have $h\left(A\right)=h_c\left(A\right)=A·A+1=ab+1$.

2. 

For the divisible class $A^{\prime }=tA$, we also have $h\left(A^{\prime }\right)=h_c\left(A^{\prime }\right)=A^{\prime }·A^{\prime }+1$.

Remark 5.

The set of adjunction classes ${\mathcal{C}}_{\Lambda }$ is bigger when $b_1(\Lambda )$ increases. This is true since $\chi (\Lambda )$ becomes smaller as $b_1(\Lambda )$ increases. Consequently, we sometimes obtain stronger estimates for the same class $A=aB+bF$ when $b_1(\Lambda )$ increases, although the uniform estimate in Proposition 3 is independent of $b_1(\Lambda )$. For example, if $B·B=0$ and $g=2$, for $A=B+2F$, we can choose $c=2B-2F$ and get $h_c\left(A\right)=2$. When $B·B=0$ and $g>2$, we can choose $c=2B-4F$ and get $h\left(A\right)=h_c\left(A\right)=3$.

4.3. Non-Minimal Irrational Ruled Surface Type (Case (4))

Choose a standard basis $\mathcal{B}=\{F,B,E_1,\dots ,E_n\}$ for ${\Lambda }^2$ with $B·B=0=F·F$, $F·B=1$, $E_i·E_i=-1$, $E_i·E_j=E_i·F=E_i·B=0$. Recall that $g\ge 1$.

Let $A=aB+bF-{\sum }_{i=1}^n{e}_i{E}_i$ be a reduced class with $A·A=2ab-{\sum }_{i=1}^n{e}_i^2\ge 0$. So $a\ge e_1\ge \cdots \ge e_n\ge 0$, $b\ge 0$. We assume $e_1\ge 1$, otherwise it reduces to the $n=0$ case.

Recall that $c_0=2B+(2-2g)F-{\sum }_{i=1}^n{E}_i$. So $c_0·A<0$ implies $2b+(2-2g)a-{\sum }_{i=1}^n{e}_i<0$.

Lemma 6.

Let $A=aB+bF-{\sum }_{i=1}^n{e}_i{E}_i$ be a reduced class with $A·A\ge 0$, $e_1\ge 1$, and satisfy $2b+(2-2g)a-{\sum }_{i=1}^n{e}_i<0$. Then we have

$$h\left(A\right)\ge ab+1-\frac{1}{2}(\sum _{i=1}^n{e}_i^2+max\{2b-\sum _{i=1}^n{e}_i,e_1\}).$$

Proof. 

An adjunction class has the form $c=2pB+2qF+{\sum }_{i=1}^n{r}_i{E}_i$, where $p,q\in \mathbb{Z}$, $r_i$ are odd integers. c is of type II, so

$$c·c=8pq-\sum _{i=1}^n{r}_i^2\ge 2\tilde{\chi }+3\sigma =8-8g-n,\mathrm{and}c·F=2p\ne 0.$$

We will choose an adjunction class $c_1=2B+{\sum }_{i=1}^n{r}_i{E}_i$ with $r_i=\pm 1$. So,

$$c_1·A=2b+\sum _{i=1}^n{r}_i{e}_i.$$

If $2b-{\sum }_{i=1}^n{e}_i\ge 0$, then we take all $r_i=-1$, and get $c_1·A=2b-{\sum }_{i=1}^n{e}_i$.

If $2b-{\sum }_{i=1}^n{e}_i<0$, let $S_0=2b$, $S_j=2b+{\sum }_{i=1}^j{r}_i{e}_i$, $1\le j\le n$, be the partial sums of $c_1·A$. Applying the oscillating sequence method and sign rule Equation (2) in Section 4.1, we can deduce that $|c_1·A|=|S_n|\le e_1$.

Thus, we obtain an estimate $|c_1·A|\le max\{2b-{\sum }_{i=1}^n{e}_i,e_1\}$, and have the desired genus bound. □