Complex Ball Quotients and New Symplectic 4-manifolds with Nonnegative Signatures
CCOMPLEX BALL QUOTIENTS AND NEW SYMPLECTIC -MANIFOLDS WITH NONNEGATIVE SIGNATURES ANAR AKHMEDOV, S ¨UMEYRA SAKALLI, AND SAI-KEE YEUNG
Abstract.
We present the various constructions of new symplectic 4-manifoldswith non-negative signatures using the complex surfaces on the BMY line c = 9 χ h , the Cartwright-Steger surfaces, the quotients of Hirzebruch’s cer-tain line-arrangement surfaces, along with the exotic symplectic 4-manifoldsconstructed in [5, 13]. In particular, our constructions yield to (i) an irre-ducible symplectic and infinitely many non-symplectic 4-manifolds that arehomeomorphic but not diffeomorphic to (2 n − CP n − CP for eachinteger n ≥
9, (ii) the families of simply connected irreducible nonspin sym-plectic 4-manifolds that have the smallest Euler characteristics among the allknown simply connected 4-manifolds with positive signatures and with morethan one smooth structure. We also construct a complex surface with posi-tive signature from the Hirzebruch’s line-arrangement surfaces, which is a ballquotient. Introduction
This article is a continuation of the previous work, carried out in ([2], [1], [4], [3],[5], [6], [7], [9], [8], [10], [11], [13], [12]), on the geography of symplectic 4-manifolds.For some background and concise history on symplectic geography problem, werefer the reader to the introduction found in [10], [7], and [11].Our work here is greatly motivated and influenced by the recent work of DonaldCartwright, Vincent Koziarz, and third author in [18] and the earlier work of GopalPrasad and the third author in [31, 32]. The main purpose of our article is to con-struct new minimal symplectic 4-manifolds that are interesting with respect to thesymplectic geography problem. Starting from Cartwright-Steger surfaces, and theirnormal covers on Bogomolov-Miyaoka-Yau line c = 9 χ h , the Hirzebruch’s line-arrangement surfaces and their quotients, by forming their symplectic connectedsum with the exotic symplectic 4-manifolds constructed in [5, 13], or the product4-manifolds Σ g × Σ h , and applying the sequence of Luttinger surgeries along thelagrangian tori, we obtain a family of new symplectic 4-manifolds with non-negativesignatures. As a consequence of our work, we produce (i) an irreducible symplecticand infinitely many non-symplectic 4-manifolds that are homeomorphic but notdiffeomorphic to (2 n − CP n − CP for each integer n ≥
9, (ii) the familiesof simply connected irreducible nonspin symplectic 4-manifolds that have the small-est Euler characteristics among the all known simply connected 4-manifolds withpositive signature and with more than one smooth structure. We also constructa complex surface on Bogomolov-Miyaoka-Yau line c = 9 χ h using Hirzebruch’scertain line-arrangement surface. Date : January 15, 2019.2020
Mathematics Subject Classification.
Primary 57R55; Secondary 57R17, 32Q55. a r X i v : . [ m a t h . S G ] F e b ANAR AKHMEDOV, S¨UMEYRA SAKALLI, AND SAI-KEE YEUNG
Before stating our main results, let us fix some notations that will be usedthroughout this paper. Given two 4-manifolds, X and Y , we will denote theirconnected sum by X Y . For a positive integer k ≥
2, the connected sum of k copies of X will be denoted by kX . Let CP denote the complex projectiveplane, with its standard orientation, and let CP denote the underlying smooth4-manifold CP equipped with the opposite orientation. Our main results are thefollowing theorems. Theorem 1.
Let M be (2 n − CP n − CP for any integer n ≥ . Then thereexist an infinite family of irreducible symplectic and an infinite family of irreduciblenon-symplectic -manifolds that all are homeomorphic but not diffeomorphic to M . The theorem above improves one of the main results of [6, 13] where exoticirreducible smooth structures on (2 n − CP n − CP for n ≥
25 and for n ≥
12 were constructed, respectively. The next theorem improves the main resultsof [6, 10, 13] for the positive signature cases.
Theorem 2.
Let M be one of the following -manifolds. (i) (2 n − CP n − CP for any integer n ≥ . (ii) (2 n − CP n − CP for any integer n ≥ .Then there exist an infinite family of irreducible symplectic -manifolds and aninfinite family of irreducible non-symplectic -manifolds that are homeomorphicbut not diffeomorphic to M . The second theorem above, which deals with the cases of signature equal 1 and2, can be extended to the signature grater than equal 3 cases as well.Let us recall that exotic irreducible smooth structures on (2 n − CP n − CP for n ≥
12, on (2 n − CP n − CP for n ≥
14, on (2 n − CP n − CP for n ≥
13, and on (2 n − CP n − CP for n ≥
15 were constructedin [13] (see also earlier work in [6] and [10]).Our paper is organized as follows. In Sections 2 and 3, we discuss some back-ground material and collect some building blocks that are needed in our construc-tions of symplectic 4-manifolds. In Sections 4, 5, 6, we present the proofs of ourmain results. A preliminary report on this work has been presented by the firstauthor at Purdue University and by the second author at MPIM and in variousresearch seminars and workshops since November 2018.2.
Complex surfaces on Bogomolov-Miyaoka-Yau line
Fake projective planes.
A fake projective plane is a smooth complex surfacewhich is not the complex projective plane, but has the same Betti numbers as thecomplex projective plane. The first fake projective plane was constructed by DavidMumford in 1979 using p-adic uniformization [29]. He also showed that therecould only be a finite number of such surfaces. Two more examples were foundby Ishida and Kato [25] in 1998, and another by Keum [26] in 2006. In 2007 [31](see also Addendum [32]), the third author and Gopal Prasad almost completelyclassified fake projective planes by proving that they fall into “28 classes”. Using thearithmeticity of the fundamental group of fake projective planes, and the formulafor the covolume of principal arithmetic subgroups, they found twenty eight distinctclasses of fake projective planes. For a very small number of classes, they left openthe question of existence of fake projective planes in that class, but conjectured that
OMPLEX BALL QUOTIENTS AND NEW SYMPLECTIC 4-MANIFOLDS 3 there are none. Finally, Donald Cartwright and Tim Steger verified their conjectureand found all the fake projective planes, up to isomorphism, in each of the 28 classes[17].
Example 1.
In this example, we recall some properties of fake projective plane M . We refer the reader to [37] and also [36] , where a complete classification of allsmooth surfaces of general type with Euler number is given. There are pairs offake projectives planes as classified in [31, 32, 18] and one Cartwright-Steger surfaceto be explained in .For fake projective planes, the Euler characteristic and the Betti numbers of M are e ( M ) = 3 , b ( M ) = 0 and b ( M ) = 1 . M is a minimal complex surfaceof general type with σ ( M ) = 1 , c ( M ) = 3 e ( M ) = 9 and χ h ( M ) = 1 . Theintersection form of M is odd, and has rank . The fundamental group Π of M isa torsion-free cocompact arithmetic subgroup of P U (2 , , thus M is a ball quotient B C / Π . For pairs of fake projecitve planes, the canonical line bundle K M isdivisible by , i.e., there is a line bundle L such that K M = 3 L . For the remainingfour pairs of fake projective planes, we know that K = 3 H + τ for some torsion linebundle τ . It was mentioned that the class of H can be represented by a symplecticsurface H of self-intersection (see discussion in [27] , pages 212-213), but noticethat Taubes result concerning existence of pseudoholomorphic curves does not applyto the classes H or H in this case ( [34] ). By considering the classes pH for anypositive integer p ≥ , we can produce symplectic surface H ( p ) of self-intersection p and the genus g ( H ( p )) = 1 + 1 / pH · pH + 3 H · pH ) = 1 + p ( p + 3) / . Thesesymplectic surfaces in M are quite useful. Using the symplectic connected sumoperation [21] , the pair ( M, H ( p )) (or the covers of M on BMY line) and the knotsurged homotopy elliptic surfaces E ( n ) K along with the symplectic submanifold S K can be used to construct exotic symplectic -manifolds with positive signature andnear the Bogomolov-Miyaoka-Yau line c = 9 χ h . The symplectic surface S K abovein E ( n ) K is a higher genus section of self-intersection − n resulting from − n spheresection of E ( n ) under the the knot surgery along a fibered knot K of genus g . Tomake this construction work, one needs to set n = p and g = 1 + p ( p + 3) / .Since π ( E ( n ) K \ S K ) is trivial and π ( H ( p )) surjects into π ( M ) , the resultingsymplectic -manifold is simply connected. Complex surfaces of Cartwright and Steger.
The study of enumeratingthe set of all fake projective planes in the so-called class C in the notation of [31]led Donald Cartwright and Tim Steger to discover a complex surface with irregu-larity q = 1 and Euler characteristic e = 3, named as Cartwright-Steger surface.Cartwright and Steger showed that a certain maximal arithmetic subgroup ¯Γ of P U (2; 1) contains a torsion-free subgroup Π of index 864 which has abelianization Z . Such subgroup Π is unique up to conjugation. Furthermore, for each ineger n ≥
1, Π has a normal subgroup Π n of index n . Let M n = B ( C ) / Π n denote thequotient of a complex hyperbolic space by a torsion free lattice Π n of P U (2; 1).The Euler characteristic of M n is e ( M n ) = ne ( M ) = 3 n . M n is a minimal complexsurface of general type with σ ( M n ) = n , c ( M n ) = 3 e ( M n ) = 9 n and χ h ( M n ) = n .The intersection form of M is odd, indefinite and modulo torsion is isomorphic to3 (cid:104) (cid:105) ⊕ (cid:104)− (cid:105) . The Betti numbers of M are: 1 , , , ,
1. It is known that the Al-banese map of M gives rise to an Albanese fibration with generic fiber of genus 19[18]. ANAR AKHMEDOV, S¨UMEYRA SAKALLI, AND SAI-KEE YEUNG
The covers of Cartwright-Steger surface.
Let us recall the following from[18]. In one of our constructions we will be using the curves b ( M c ) or b − ( M c ) inProposition 2.4 of [18]. For simplicity, let us consider D = b ( M c ).Recall that in the notation of [17] and [31], the maximal arithmetic lattice con-sidered in this case is denoted by Γ summarized in Theorem 1 of [18]. The latticeof the Cartwright-Steger surface is denoted by Π with generators given by a , a , a explained in Theorem 2 of [18].The map π : M = B C / Π → B C / Γ is a covering map of order 864. The quotient B C / Γ is represented by the right hand side of Figure 1 of [18]. D is a component of π − ( D A ) in the picture and π − ( D A ) is an immersed totally geodesic curve. Thesingularities of D could only be found in π − ( P ) and π − ( P ). D is a componentof genus 4 in π − ( D A ). According to Proposition 2.4 of [18], the only singularpoints of the curve D is given by a point of normal crossing given by n − ( D ) = 2.By Proposition 2.4(d) and its proof in [18], Π M \ M has genus 4 by the Riemann-Hurwitz formula, and we can find explicit generators u i , v i of Π M such that[ u , v ][ u , v ][ u , v ][ u , v ] = 1. When M = b ( M c ), the following eight elementsgenerate Π M : p = a a − a − j a − a − j ,p = a a a a a j a − j a − a − a − ,p = j a − a − a j a − a − a − ,p = j a a a − a − j a a a − , p = a a a j a − j a a a − ,p = a a a a a a − ,p = a a j a a − a − a j ,p = j a − j a a a a a − , and satisfy the single relation p − p − p p p p − p p − p − p − p p p − p p − p = 1 . Here the group elements such as j is given by of [18]. The above gives a set ofgenerators for π ( ˆ D ), where ˆ D is the normalization of D .To compute i ∗ ( π ( D )) ⊂ π ( M ), where i : D → M is the inclusion, note that i ∗ ( π ( D )) is generated by i ∗ p j , j = 1 , . . . ,
8, together with loops around the nodalpoint. The following two elements π and π of Π satisfy π ( b − ( O )) ∈ b ( M c ), aretaken from the third table on page 41 of the arXiv version of the paper [18]: π := a a a − , π := a a − a − a a − a − a − Consider the subgroup G of G given by G = < G | p , p , p , p , p , p , p , p , π π − >. By using the Magma program, one can verify that G is a normal subgroup ofΠ of order 4. Moreover, it can be verified that the quotient group is Z × Z . Nolarger subgroup of G containing G could be found and hence G is our candidate i ∗ ( π ( D )). The authors are grateful to Donald Cartwright for his help with Magmacomputations.In conclusion, we have the following. Proposition 1. D is an immersed totally geodesic curve satisfying the followingproperties. (1) The normalization ˆ D of D is a Riemann surface of genus . (2) D · D = − . OMPLEX BALL QUOTIENTS AND NEW SYMPLECTIC 4-MANIFOLDS 5 (3) i ∗ π ( D ) is a normal subgroup of π ( M ) of index , and π ( M ) /i ∗ π ( D ) = Z × Z . Denote by H the covering group π ( M ) /i ∗ π ( D ) = Z × Z in Proposition 1.We have 1 → i ∗ π ( D ) → π ( M ) → H. Consider now a normal unramified covering ˜ M of M with covering group given by H . Let p : ˜ M → M be the covering map. From construction, p − ( D ) consists offour connected components. Let E be one such connected component. Then fromconstruction, inclusion i ∗ π ( E ) → π ( ˜ M ) is an isomorphism. Hence we have Lemma 1. E is a curve of self-intersection − on ˜ M . The normalization of E isa Riemann surface of genus . Moreover, i ∗ π ( E ) → π ( ˜ M ) is an isomorphism. This follows from construction. Note that a neighborhood of D in M is isomor-phic to a neighborhood of E in ˜ M , as the covering is a normal covering with π ( ˜ M )a normal subgroup of Π. Lemma 2.
The Chern numbers of ˜ M are given by c ( ˜ M ) = 36 , c ( ˜ M ) = 12 . This follows from the fact that the Chern numbers involved are multiplicative.
Lemma 3. ˜ M CP contains a symplectic genus curve Σ of self intersection − that carries the fundamental group of ˜ M CP Proof.
It was shown in Lemma 1 that ˜ M contains a curve E of self intersection − E is 4 as well. We symplectically blow up E at its self inter-section, so that it becomes square − e intersectsit twice. We symplectically smooth two intersections points of the proper transformof E with e , which gives us genus 5 symplectic curve Σ of self intersection − M CP . Since i ∗ π ( E ) → π ( ˜ M ) is an isomorphism, we see that Σ carriesthe fundamental group of ˜ M CP . (cid:3) Luttinger surgery and symplectic cohomology (2 n − S × S )We briefly review the Luttinger surgery, and collect some symplectic buildingblocks that will be used later in our constructions. For the details on Luttingersurgery, the reader is referred to the papers [28] and [14]. Definition 1.
Let X be a symplectic -manifold with a symplectic form ω , and thetorus Λ be a Lagrangian submanifold of X . Given a simple loop λ on Λ , let λ (cid:48) be asimple loop on ∂ ( ν Λ) that is parallel to λ under the Lagrangian framing. For anyinteger n , the (Λ , λ, /n ) Luttinger surgery on X defined to be the X Λ ,λ (1 /n ) =( X − ν (Λ)) ∪ φ ( S × S × D ) , the /n surgery on Λ with respect to λ under theLagrangian framing. Here φ : S × S × ∂ D → ∂ ( X − ν (Λ)) denotes a gluing mapsatisfying φ ([ ∂ D ]) = n [ λ (cid:48) ] + [ µ Λ ] in H ( ∂ ( X − ν (Λ)) , where µ Λ is a meridian of Λ . It is shown in [14] that X Λ ,λ (1 /n ) possesses a symplectic form that restricts tothe original symplectic form ω on X \ ν Λ. The proof of the following lemma is easyto verify and is left to the reader as an exercise.
Lemma 4. (1) π ( X Λ ,λ (1 /n )) = π ( X − Λ) /N ( µ Λ λ (cid:48) n ) , where N ( µ Λ λ (cid:48) n ) denote the smallestnormal subgroup of π ( X − Λ) that contains µ Λ λ (cid:48) n (2) σ ( X ) = σ ( X Λ ,λ (1 /n )) and e ( X ) = e ( X Λ ,λ (1 /n )) . ANAR AKHMEDOV, S¨UMEYRA SAKALLI, AND SAI-KEE YEUNG
Luttinger surgeries on product manifolds Σ n × Σ and Σ n × T . Recallfrom [19, 4] that for each integer n ≥
2, there is a family of irreducible pairwisenon-diffeomorphic 4-manifolds { Y n ( m ) | m = 1 , , , . . . } that have the same integercohomology ring as (2 n − S × S ). Y n ( m ) are obtained by performing 2 n + 3Luttinger surgeries (cf. [14, 28]) and a single m torus surgery on Σ × Σ n . These2 n + 4 torus surgeries are performed as follows( a (cid:48) × c (cid:48) , a (cid:48) , − , ( b (cid:48) × c (cid:48)(cid:48) , b (cid:48) , − , ( a (cid:48) × c (cid:48) , a (cid:48) , − , ( b (cid:48) × c (cid:48)(cid:48) , b (cid:48) , − , (1) ( a (cid:48) × c (cid:48) , c (cid:48) , +1) , ( a (cid:48)(cid:48) × d (cid:48) , d (cid:48) , +1) , ( a (cid:48) × c (cid:48) , c (cid:48) , +1) , ( a (cid:48)(cid:48) × d (cid:48) , d (cid:48) , + m ) , together with the following 2( n −
2) additional Luttinger surgeries( b (cid:48) × c (cid:48) , c (cid:48) , − , ( b (cid:48) × d (cid:48) , d (cid:48) , − , . . . , ( b (cid:48) × c (cid:48) n , c (cid:48) n , − , ( b (cid:48) × d (cid:48) n , d (cid:48) n , − . Here, a i , b i ( i = 1 ,
2) and c j , d j ( j = 1 , . . . , n ) denote the standard loops thatgenerate π (Σ ) and π (Σ n ), respectively. See Figure 1 for a typical Lagrangiantori along which the surgeries are performed. xx yxx y y yx ya i a i a i b i d j a i d j b i c j c j c j d j Figure 1.
Lagrangian tori a (cid:48) i × c (cid:48) j and a (cid:48)(cid:48) i × d (cid:48) j Since m -torus surgery is non-symplectic for m ≥
2, the manifold Y n ( m ) is sym-plectic only when m = 1. Using the Lemma 4, we see that the Euler characteristicof Y n ( m ) is 4 n − π ( Y n ( m )) is generated by a i , b i , c j , d j ( i = 1 , j = 1 , . . . , n ) and the following relations hold in π ( Y n ( m )):[ b − , d − ] = a , [ a − , d ] = b , [ b − , d − ] = a , [ a − , d ] = b , (2) [ d − , b − ] = c , [ c − , b ] = d , [ d − , b − ] = c , [ c − , b ] m = d , [ a , c ] = 1 , [ a , c ] = 1 , [ a , d ] = 1 , [ b , c ] = 1 , [ a , c ] = 1 , [ a , c ] = 1 , [ a , d ] = 1 , [ b , c ] = 1 , [ a , b ][ a , b ] = 1 , n (cid:89) j =1 [ c j , d j ] = 1 , [ a − , d − ] = c , [ a − , c − ] = d , . . . , [ a − , d − n ] = c n , [ a − , c − n ] = d n , [ b , c ] = 1 , [ b , d ] = 1 , . . . , [ b , c n ] = 1 , [ b , d n ] = 1 . OMPLEX BALL QUOTIENTS AND NEW SYMPLECTIC 4-MANIFOLDS 7
The surfaces Σ × { pt } and { pt } × Σ n in Σ × Σ n are not affected by the aboveLuttinger surgeries, and descend to surfaces in Y n ( m ). They are symplectic sub-manifolds in Y n (1). Let us denote these symplectic submanifolds n Y n (1) by Σ andΣ n . Note that [Σ ] = [Σ n ] = 0 and [Σ ] · [Σ n ] = 1. Let µ (Σ ) and µ (Σ n ) denotethe meridians of these surfaces in Y n ( m ). The above construction easily generalizesto Σ × Σ n . We will denote the resulting smooth manifold in this case as Z n ( m ).Next, we consider a slightly different construction. Let us fix integers n ≥ m ≥
1. Let Y n (1 , m ) denote smooth 4-manifold obtained by performing thefollowing 2 n torus surgeries on Σ n × T :( a (cid:48) × c (cid:48) , a (cid:48) , − , ( b (cid:48) × c (cid:48)(cid:48) , b (cid:48) , − , (3) ( a (cid:48) × c (cid:48) , a (cid:48) , − , ( b (cid:48) × c (cid:48)(cid:48) , b (cid:48) , − , · · · , · · · ( a (cid:48) n − × c (cid:48) , a (cid:48) n − , − , ( b (cid:48) n − × c (cid:48)(cid:48) , b (cid:48) n − , − , ( a (cid:48) n × c (cid:48) , c (cid:48) , +1) , ( a (cid:48)(cid:48) n × d (cid:48) , d (cid:48) , + m ) . Let a i , b i ( i = 1 , , · · · , n ) and c, d denote the standard generators of π (Σ n )and π ( T ), respectively. Since all the torus surgeries listed above are Luttingersurgeries when m = 1 and the Luttinger surgery preserves minimality, Y n (1 /p, /q )is a minimal symplectic 4-manifold. The fundamental group of Y n (1 /p, m/q ) isgenerated by a i , b i ( i = 1 , , · · · , n ) and c, d , and the Lemma 4 implies that thefollowing relations hold in π ( Y n (1 , m )):[ b − , d − ] = a , [ a − , d ] = b , [ b − , d − ] = a , [ a − , d ] = b , (4) · · · , · · · , [ b − n − , d − ] = a n − , [ a − n − , d ] = b n − , [ d − , b − n ] = c, [ c − , b n ] − m = d, [ a , c ] = 1 , [ b , c ] = 1 , [ a , c ] = 1 , [ b , c ] = 1 , [ a , c ] = 1 , [ b , c ] = 1 , · · · , · · · , [ a n − , c ] = 1 , [ b n − , c ] = 1 , [ a n , c ] = 1 , [ a n , d ] = 1 , [ a , b ][ a , b ] · · · [ a n , b n ] = 1 , [ c, d ] = 1 . Let us denote by Σ (cid:48) n ⊂ Y n (1 , m ) a genus n surface that desend from the surfaceΣ n × { pt } in Σ n × T .4. Construction of a smooth complex algebraic surface with K = 144 and χ h = 16In this section, we construct a smooth complex algebraic surface with invariants K = 144 and χ h = 16. This complex surface of general type is on the BMYline c = 9 χ h , and thus is a ball quotient. It is obtained as an abelian coveringof the complex projective plane branched over an arrangement of 12 lines shownas in Figure 2, known in the literature as the Hesse configuration. Such complexsurfaces with bigger invariants, K and χ h , was initially studied by by FriedrichHirzebruch (for example, see [24], page 134). Our construction is motivated and ANAR AKHMEDOV, S¨UMEYRA SAKALLI, AND SAI-KEE YEUNG similar in spirit to that of Bauer-Catanese in [16], where the complex ball quotientsobtained from a complete quadrangle arrangement in CP .In CP , let us consider the Hesse arrangement H , which is a configuration of9 points p i (1 ≤ i ≤
9) and 12 lines l j (1 ≤ j ≤ p i and each point lies at the intersection of 4 of the lines l j (see Figure 2). We blow up CP at the points p , · · · , p , and denote the blow upmap by π : T := (cid:100) CP → CP . Let E i be the exceptional divisor corresponding tothe blow up at the point p i for i = 1 , · · · ,
9. In the sequel, we will slightly abuseour notation and denote the proper transform of a line l j using the same symbol,or ˜ l j when distinction needed.Let us now take the formal sum of the proper transforms l j of the 12 lines of thearrangement and the 9 exceptional divisors E i ’s, and denote it by D . The divisor D in T has only simple normal crossings. The homology classes of simple closedloops around the l j ’s and the E i ’s generate H ( T − D, Z ). Let us denote a loopencircling a line E i or l j by using the same letter. Then for each i = 1 , · · · ,
9, theclass of E i can be written as a sum of the homology classes of 4 loops around the 4lines intersecting E i . To illustrate this, notice that we have E = l + l + l + l and similar relations hold for the other E i ’s. Moreover, the sum of the homologyclasses of 12 loops l j ’s are 0, which shows that H ( T − D, Z ) is a free group of rank11. l l l l l l l l l l l l p p p p p p p p p q q q q q q q q q q q q Figure 2.
Hesse arrangement in CP It is known that a surjective homomorphism ϕ : Z (cid:39) H ( T − D, Z ) → ( Z / Z ) determines an abelian ( Z / Z ) -cover p : W → T = (cid:100) CP . We need that p isbranched exactly in D . Notice that there are various epimorphisms satisfying this. OMPLEX BALL QUOTIENTS AND NEW SYMPLECTIC 4-MANIFOLDS 9
To illustrate one, let us define ϕ as follows: ϕ ( l ) = ϕ ( l ) = ϕ ( l ) = (1 , ,ϕ ( l ) = (2 , , ϕ ( l ) = (0 , , ϕ ( l ) = (2 , ,ϕ ( l ) = (2 , , ϕ ( l ) = (1 , , ϕ ( l ) = (2 , ,ϕ ( l ) = ϕ ( l ) = (1 , , ϕ ( l ) = (0 , . Then ϕ ( E ) = ϕ ( l + l + l + l ) = (0 , , ϕ ( E ) = ϕ ( l + l + l + l ) = (0 , ,ϕ ( E ) = ϕ ( l + l + l + l ) = (1 , , ϕ ( E ) = ϕ ( l + l + l + l ) = (2 , ,ϕ ( E ) = ϕ ( l + l + l + l ) = (1 , , ϕ ( E ) = ϕ ( l + l + l + l ) = (1 , ,ϕ ( E ) = ϕ ( l + l + l + l ) = (0 , , ϕ ( E ) = ϕ ( l + l + l + l ) = (1 , ,ϕ ( E ) = ϕ ( l + l + l + l ) = (0 , ϕ ( l + l + l + l + l + l ) (cid:54) = (0 , ϕ gives a ( Z / Z ) Galois cover branched exactly in D (see Lemma 2.3, part 1in [16], also [38]).We also note that the following are linearly independent: ϕ ( E ) and ϕ ( l i ), i = 1 , , , ϕ ( E ) and ϕ ( l i ), i = 2 , , , ϕ ( E ) and ϕ ( l i ), i = 3 , , , ϕ ( E ) and ϕ ( l i ), i = 1 , , , ϕ ( E ) and ϕ ( l i ), i = 2 , , , ϕ ( E ) and ϕ ( l i ), i = 3 , , , ϕ ( E ) and ϕ ( l i ), i = 1 , , , ϕ ( E ) and ϕ ( l i ), i = 2 , , , ϕ ( E ) and ϕ ( l i ), i = 3 , , , D has simple normal crossings, we deduce that the total space W issmooth (see Lemma 1.4 in [38]).Let us compute some invariants of the surface W , and verify that c ( W ) = K W =144 and χ h ( W ) = 16. Let H be the divisor class corresponding to the invertiblesheaf O (1) on CP . The canonical sheaf w CP of CP is O ( − −
1) = O ( −
3) whichcorresponds to the canonical divisor − H . Then, the canonical divisor K Y of Y is − H + (cid:80) i =1 E i where we denoted the pullback of H by itself. By using the canonicaldivisor formula for abelian covers (Proposition 4.2 in [30]), we compute K W = π ∗ (cid:16) ( − H + (cid:88) i =1 E i ) + 23 (cid:88) i =1 E i + 23 (12 H − (cid:88) i =1 ( E i ) (cid:17) = π ∗ (cid:16) H − (cid:88) i =1 E i (cid:17) . Since H · E i = 0 , H = 1 and E i = −
1, the above equality gives K W = 9(25 −
9) =144.The Euler number e ( W ) of W can be found as follows. e ( W ) = 9 e ( (cid:100) CP = CP CP ) − · e ( CP ) + 4 ·
48 = 48 . Thus c ( W ) = 3 c ( W ), and W is a ball quotient. Since 12 χ h ( W ) − c ( W ) = e ( W ), we have χ h ( W ) = 16. In summary, we proved the following theorem. Theorem 3.
There exists a smooth complex algebraic surface W with invariants c ( W ) = 144 and χ h ( W ) = 16 constructed as ( Z / Z ) -cover of CP branched overthe Hesse configuration. Now we consider the map p ◦ π : W → CP , where π is the blow up map, p isthe abelian cover. Let us take p , one of the blown up points in CP which is theintersection point of l , l , l , l (see Figure 2). The pencil of lines in CP passingthrough p lifts to a fibration on W . To determine the genus of the generic fiber ofthis fibration, we take a line K passing through p such that its only intersectionwith the lines l , l , l , l is p . In addition, K intersects the remaining 8 linesof the arrangement. These 8 intersection points and the point p are 9 branchpoints on K . The preimage of K − E in W , which is the generic fiber of the givenfibration, is a degree 3 cover of K − E (cf. [15], p.241), branched at 9 points.For the determination of the genus g of the surface above K − E , we apply theRiemann-Hurwitz ramification formula(5) 2 g − −
2) + 9 · ⇒ g = 7 . Therefore, generic fibers are of genus 7 surfaces. Moreover, there are at least 9distinct fibrations in W coming from the points p i ’s.Let us consider the 12 lines l j of the Hesse arrangement and determine theirinverse images in W under p ◦ π . We observe that on each l j , j = 1 , · · · ,
12, thereare 5 branch points. By the Riemann-Hurwitz formula, we have(6) 2 g − −
2) + 5 · ⇒ g = 3 . Therefore, they lift to genus 3 curves. To find their self-intersections, we apply theadjunction formula. Firstly, we note that each l j is blown up at three points, say p k , p l , p m . For its proper transform ˜ l j in (cid:100) CP , we have(7) [˜ l j ] = H − E k − E l − E m . Thus, K W · [Σ ] = π ∗ (cid:16) (5 H − (cid:88) i =1 E i ) · ( H − E k − E l − E m ) (cid:17) = 3(5 − − −
1) = 6 . Using the adjunction formula 2 g − K W · [Σ ] + [Σ ] , we have [Σ ] = − E i , there are 4 branch points. Thus,their preimages are genus 2 curves in W :(8) 2 g − −
2) + 4 · ⇒ g = 2 . Similarly as above, K W · [Σ ] = π ∗ (cid:16) (5 H − (cid:88) i =1 E i ) · ( E i ) (cid:17) = 3and by the adjunction formula we have 2 g − K W · [Σ ] + [Σ ] ; which showsthat [Σ ] = − CP passing through p and take theline l . The preimage of its proper transform ˜ l is a genus three surface Σ withself-intersection − W . The exceptional divisors E , E and E intersecting˜ l lift to genus 2 curves with self-intersections −
1, each of which intersects Σ transversally once. Notice that the lift of E gives rise to a section, and the unionof lifts of the exceptional divisors E , E , and the proper transform of intersecting ˜ l corresponds to a singular fiber of the given fibration. We symplectically resolve theirthree transversal intersection points and obtain genus 9 symplectic submanifoldof W with self intersection +1. As in Section 2.3 of [13], we have the followingproposition. Proposition 2. W CP contains an embedded symplectic genus curve Σ withself intersection . Furthermore, there is a surjection f ∗ : π (Σ ) → π ( W CP ) . Constructions of Symplectic 4-Manifolds with PositiveSignatures from Hirzebruch’s line-arrangement surface
In what follows, we will construct families of simply connected, minimal, sym-plectic and smooth 4-manifolds with positive signatures, by making use of the com-plex surface W CP , which we constructed above using the Hesse configuration.By Proposition 2, we know that W CP contains an embedded symplectic genus9 curve of self-intersection zero. We endow W CP with the symplectic structureinduced from the K¨ahler structure. It will be the first building block in our con-struction, which has the following invariants: e ( W CP ) = 49, σ ( W CP ) = 15, c ( W CP ) = 143 and χ h ( W CP ) = 16. Our second piece will be a minimal,simply connected and symplectic 4-manifold X g,g +2 ([13], Theorem 3.12 and Section5): Theorem 4.
For any integer g ≥ , there exist a minimal symplectic 4-manifold X g,g +2 obtained via Luttinger surgery such that (i) X g,g +2 is simply connected (ii) e ( X g,g +2 ) = 4 g +2 , σ ( X g,g +2 ) = − , c ( X g,g +2 ) = 8 g − , and χ ( X g,g +2 ) = g . (iii) X g,g +2 contains the symplectic surface Σ of genus with self-intersection and two genus g surfaces with self-intersection − intersecting Σ positivelyand transversally. Symplectic and smooth 4-manifolds with signatures equal to 12.
Wenow present our first construction of exotic symplectic 4-manifolds with σ = 12.Our first building block is the complex surface W CP containing genus 9 sym-plectic surface with self-intersection 0. The second building block is obtained fromthe symplectic 4-manifold X , , in the notation of Theorem 4. We will use the factthat X , contains a symplectic genus two surface Σ with self-intersection 0 andtwo genus 7 symplectic surfaces with self intersections − positivelyand transversally.Let us review the construction of X , (see [13] for the details). We take acopy of T × { pt } and { pt } × T in T × T equipped with the product symplecticform, and symplectically resolve the intersection point of these dual symplectictori. The resolution produces symplectic genus two surface of self intersection+2 in T × T . By symplectically blowing up this surface twice, in T CP , weobtain a symplectic genus 2 surface Σ with self-intersection 0, with two − T CP withΣ × Σ along the genus two surfaces Σ and Σ ×{ pt } . By performing the sequence of appropriate ± T CP ) Σ =Σ ×{ pt } (Σ × Σ ), weobtain the symplectic 4-manifold X , ([13]). It can be seen from the constructionthat, X , contains a symplectic surface Σ with self intersection 0 and two genus7 surfaces S and S with self intersections − . Notice that the surfaces S and S result from the internalsum of the punctured exceptional spheres in T CP \ ν (Σ ) and the puncturedgenus 7 surfaces in Σ × Σ \ ν (Σ ×{ pt } ). Moreover, X , contains a pair of disjointLagrangian tori T and T of self-intersections 0 such that π ( X \ ( T ∪ T )) = 1.Note that these Lagrangian tori descend from Σ × Σ , and survive in X , aftersymplectic connected sum and the Luttinger surgeries. This is because there areat least two pairs of Lagrangian tori in Σ × Σ that were away from the standardsymplectic surfaces Σ ×{ pt } and { pt }× Σ , and the Lagrangian tori that were usedfor Luttinger surgeries. Also, the fact that π ( X , \ ( T ∪ T )) = 1 is explained indetails in [6] (see proof of Theorem 8, page 272). Next, we symplectically resolve theintersection of Σ and one of the genus 7 surfaces, say S , in X , . This producesthe genus 9 surface Σ of square +1 intersecting the other genus 7 surface S withself-intersection −
1. We blow up Σ at one point (away from its intersection pointwith S ). Thus we obtained a genus 9 surface Σ (cid:48) of square 0 inside X , CP .Since the two symplectic building blocks W CP and X , CP contain sym-plectic genus 9 surfaces of self intersections zero, we can form their symplecticconnected sum along these surfaces. Let Y = ( W CP ) Σ =Σ (cid:48) ( X , CP ) . Lemma 5.
The symplectic manifold Y has e ( Y ) = 112 , σ ( Y ) = 12 .Proof. We have stated the topological invariants of W CP above and by Theo-rem 4, we have e ( X , ) = 30, σ ( X , ) = −
2. Applying the symplectic connectedsum formula, we compute the topological invariants of Y as above. (cid:3) Next, we proceed by following the same lines in [13], Section 5 and the referencestherein. We show that Y is symplectic and simply connected, using Gompf’s Sym-plectic Connected Sum Theorem and Van Kampen’s Theorem, respectively. Us-ing Freedman’s classification theorem for simply-connected 4-manifolds, the lemmaabove and the fact that W CP contains genus two surface of self-intersection − , we conclude that Y is homeomorphic to 61 CP CP . Since Y is symplectic, by Taubes’s theorem, Y has non-trivial Seiberg-Witten invariant.Next, using the connected sum theorem, we deduce that the Seiberg-Witten invari-ant of 61 CP CP is trivial. Therefore, Y is not diffeomorphic to 61 CP CP .Furthermore, Y is a minimal symplectic 4-manifold by Usher’s Minimality Theorem[35]. Since symplectic minimality implies smooth minimality, Y is also smoothlyminimal, and thus is smoothly irreducible.Moreover, as explained above, Y contains a pair of disjoint Lagrangian tori T and T of self-intersection 0 such that π ( Y \ ( T ∪ T )) = 1. We can perturbthe symplectic form on Y in such a way that one of the tori, say T , becomessymplectically embedded. We perform a knot surgery, (using a knot K with non-trivial Alexander polynomial) on Y along T to obtain irreducible 4-manifold ( Y ) K that is homeomorphic but not diffeomorphic to Y . By varying our choice of theknot K , we can realize infinitely many pairwise non-diffeomorphic, irreducible 4-manifolds, either symplectic or nonsymplectic. (see Theorem 3.7 in [13]) OMPLEX BALL QUOTIENTS AND NEW SYMPLECTIC 4-MANIFOLDS 13
Symplectic and smooth 4-manifolds with signatures equal to 11.
Inthis section, we will construct simply connected, minimal, symplectic and smooth 4-manifolds with signature is equal to 11 in two different ways. The first constructiongives the exotic 59 CP CP and the second gives the exotic 63 CP CP .In first construction, one of our building block is again W CP , containing asymplectic genus 9 surface Σ of square 0. To obtain the second symplectic buildingblock, we form the symplectic connected sum of T CP with Σ × Σ along thegenus two surfaces Σ and Σ × { pt } . Let X , = ( T CP ) Σ =Σ ×{ pt } (Σ × Σ ) . Similar to the discussion above, we see that X , contains a symplectic surfaceΣ with self intersection 0 and two genus 6 surfaces with self intersections − . Furthermore, X , contains asymplectic genus 7 surface Σ of square 0 resulting from the internal sum of apunctured genus one surface in T CP \ ν (Σ ) and a punctured genus 6 surfaceΣ in Σ × Σ \ ν (Σ ×{ pt } ). In addition, Σ intersects Σ positively and transverselyonce. We symplectically resolve this intersection and get symplectic genus 9 surfaceof self intersection +2. We blow it up at two points and hence we obtain symplecticgenus 9 surface Σ (cid:48)(cid:48) of square 0 inside X , CP .Next, we form the symplectic connected sum of W CP and X , CP alongthe symplectic genus 9 surfaces Σ and Σ (cid:48)(cid:48) of squares zero. Let V = ( W CP ) Σ =Σ (cid:48)(cid:48) ( X , CP ) . The invariants of X , CP are as follows. e ( X , CP ) = 28, σ ( X , CP ) = −
4, thus we have
Lemma 6.
The symplectic manifold V has e ( V ) = 109 , σ ( V ) = 11 . We conclude as above that V is symplectic, simply connected and an exoticcopy of 59 CP CP which is also smoothly minimal. As in the previous case,by performing knot surgery, we obtain infinitely many pairwise non-diffeomorphic,irreducible, symplectic and non-symplectic 4-manifolds.Let us now build another simply connected, minimal, symplectic and smooth4-manifolds with signature 11, but with different χ . Our first building block is Y = ( W CP ) Σ =Σ (cid:48) ( X , CP ) constructed above. We note that Y contains agenus 2 surface of self intersection − W , which is not affected by thesymplectic connected sum operation in the construction of Y . For instance, one canconsider E in (cid:100) CP , which is the exceptional sphere coming from the blow-up of thepoint p in the Hesse arrangement. We have shown that the exceptional sphereslift to genus 2 surfaces of self intersections − W . Take one of the preimages of E inside W . It is a symplectic genus 2 surface of square − W , anddescends to Y after the symplectic connected sum. Let us denote it by Σ (cid:48) . On theother hand, we take copies of T × { pt } and { pt } × T in T × T equipped withthe product symplectic form, and symplectically resolve the intersection point ofthese dual symplectic tori. The resolution produces symplectic genus two surfaceof self intersection +2 in T × T . By symplectically blowing up this surface, weobtain a symplectic genus 2 surface Σ ⊂ T CP of self-intersection +1, with the − Y with T CP along the genus two surfaces Σ (cid:48) and Σ . Let L = Y Σ (cid:48) =Σ (cid:48) ( T CP ) . The invariants of T CP are as follows. e ( T CP ) = 1, σ ( T CP ) = −
1, thuswe have
Lemma 7. e ( L ) = 117 , σ ( L ) = 11 . We can conclude that L is symplectic, simply connected, an exotic copy of63 CP CP , which is also smoothly minimal. As in the previous case, by per-forming knot surgery we realize infinitely many pairwise non-diffeomorphic, irre-ducible, symplectic and nonsymplectic 4-manifolds.6. Constructions of exotic -manifolds with nonnegative signaturesfrom Cartwright-Steger surfaces In this section, we will construct families of simply connected non-spin symplecticand smooth 4-manifolds with nonnegative signatures and small χ . We consider thecomplex surface ˜ M that we constructed from Cartwright-Steger surfaces in Section2.3), with c ( ˜ M ) = 36 and e ( ˜ M ) = 12. Using the formulas σ = ( c − e ) / χ =( e + σ ) /
4, we have σ ( ˜ M ) = χ ( ˜ M ) = 4. Recall that by Lemma 3, ˜ M CP containsa genus 5 symplectic curve Σ of self intersection − i ∗ π (Σ ) → π ( ˜ M CP )being a surjective homomorphism. In our construction of symplectic 4-manifoldswith nonnegative signatures, ˜ M CP along with Σ will serve as our first buildingblock. For our second building block, we will use the minimal, simply connected andsymplectic 4-manifolds X g,g +2 and X g,g +1 [5] (see also [13]) for which the followingtheorems hold: Theorem 5.
For any integer g ≥ , there exist a minimal symplectic 4-manifold X g,g +2 obtained via Luttinger surgery such that (i) X g,g +2 is simply connected (ii) e ( X g,g +2 ) = 4 g +2 , σ ( X g,g +2 ) = − , c ( X g,g +2 ) = 8 g − , and χ ( X g,g +2 ) = g . (iii) X g,g +2 contains the symplectic surface Σ of genus with self-intersection and genus g surfaces with self-intersection − intersecting Σ positivelyand transversally. Theorem 6.
There exist a minimal symplectic 4-manifold X g,g +1 obtained viaLuttinger surgery such that (i) X g,g +1 is simply connected (ii) e ( X g,g +1 ) = 4 g +1 , σ ( X g,g +1 ) = − , c ( X g,g +2 ) = 8 g − , and χ ( X g,g +1 ) = g . (iii) X g,g +1 contains the symplectic surface Σ of genus with self-intersection ,genus Σ g +1 symplectic surface with self-intersection intersecting Σ posi-tively and transversally.Proof. For the details of the constructions of X g,g +2 and X g,g +1 , we refer the readerto [5] and Section 5 of [13]. (cid:3) OMPLEX BALL QUOTIENTS AND NEW SYMPLECTIC 4-MANIFOLDS 15
Symplectic and smooth manifolds with ( σ, χ ) = (1 , . To constructsimply connected, symplectic and smooth 4-manifolds with ( σ, χ ) = (1 , M CP containing genus 5 curve Σ of self intersection − X , in thenotation of Theorem 5.For the convenience of the reader, we briefly review the construction of X , .Take a copy of T × { pt } and { pt } × T in T × T equipped with the productsymplectic form, and symplectically resolve the intersection point of these dualsymplectic tori. The resolution produces symplectic genus two surface of self in-tersection +2 in T × T . By symplectically blowing up this surface twice, in T CP , we obtain a symplectic genus 2 surface Σ with self-intersection 0, withtwo − has a dual symplectictorus T of self intersection zero intersecting Σ positively and transversally at onepoint. Next, we form the symplectic connected sum of T CP with Σ × Σ along the genus two surfaces Σ and Σ × { pt } . By performing the sequence of 8appropriate ± T CP ) Σ =Σ ×{ pt } (Σ × Σ ), we obtainthe symplectic 4-manifold X , .It can be seen from the construction that there are genus 3 curves of self inter-sections 0 inside X , . Each of them comes from the internal sum of the one of thepunctured tori in T CP \ ν (Σ ) and one of the punctured genus two surfacesin Σ × Σ \ ν (Σ × { pt } ). Such a genus 3 curve of square zero intersects Σ pos-itively and transversally at one point. We symplectically resolve this intersectionand obtain a genus 5 surface Σ (cid:48) of square +2 in X , .Since the two symplectic building blocks ˜ M CP and X , contain symplecticgenus 5 surfaces of self intersections − and Σ (cid:48) . Let M , = ( ˜ M CP ) Σ =Σ (cid:48) X , . Lemma 8. σ ( M , ) = 1 , χ h ( M , ) = 10 , e ( M , ) = 39 and c ( M , ) = 81 .Proof. We have σ ( M , ) = σ ( ˜ M CP ) + σ ( X , ) = 3 + ( −
2) = 1 and χ h ( M , ) = χ ( ˜ M CP ) + χ ( X , ) + (5 −
1) = 4 + 2 + 4 = 10. Using the formulas c = 3 σ + 2 e and e = 4 χ − σ , we compute e ( M , ) and c ( M , ) as given. (cid:3) Let us now show that M , is an exotic copy of 19 CP CP . Notice that M , is symplectic and simply connected, which follows from Gompf’s Symplec-tic Connected Sum Theorem [21] and Seifert-Van Kampen’s Theorem respectively.Using Freedman’s classification theorem for simply-connected 4-manifolds and thelemma above, M , is homeomorphic to 19 CP CP . Since M , is symplec-tic, by Taubes’s theorem it has a non-trivial Seiberg-Witten invariant. Next, byappealing to the connected sum theorem for the Seiberg-Witten invariants, we de-duce that the Seiberg-Witten invariant of 19 CP CP is trivial. Thus, M , is not diffeomorphic to 19 CP CP . Furthermore, M , is a minimal symplec-tic 4-manifold by Usher’s Minimality Theorem [35]. Since symplectic minimalityimplies smooth minimality, M , is also smoothly minimal, and thus is smoothlyirreducible [23]. As in Section 5, by performing appropriate generalized torus surg-eries, we realize infinitely many pairwise non-diffeomorphic, irreducible, symplecticand nonsymplectic 4-manifolds (see [5] for the further details of such construction). Symplectic and smooth manifolds with ( σ, χ ) = (0 , . In this construc-tion, our first building block is again ˜ M CP containing genus 5 symplectic curveΣ of self intersection −
2. For our second building block, we use X , in the notationof Theorem 5.Let us recall the construction of X , . In constructing X , , we first obtain a sym-plectic genus 2 surface Σ with self-intersection 0, with two − T CP . In addition, there are symplectic tori T of self intersections zero each of which intersects Σ positively and transversallyonce. Next, we form the symplectic connected sum of T CP with Σ × Σ along the genus two surfaces Σ and Σ × { pt } . By performing the sequence of 6appropriate ± T CP ) Σ =Σ ×{ pt } (Σ × Σ ), we obtainthe symplectic 4-manifold X , . Therefore, we see that X , contains a symplecticsurface Σ with self intersection 0 and two tori T and T with self intersections − . Note that T and T re-sult from the internal sum of the punctured exceptional spheres in T CP \ ν (Σ )and the punctured tori in Σ × Σ \ ν (Σ ×{ pt } ). Moreover, there are genus 2 curvesof self intersections 0 inside X , . Each of them comes from the internal sum ofthe one of the punctured tori in T CP \ ν (Σ ) and one of the punctured tori inΣ × Σ \ ν (Σ × { pt } ). Such a genus 2 curve Σ (cid:48) of square zero intersects Σ pos-itively and transversally at one point. We symplectically resolve the intersectionsof Σ with T and Σ with Σ (cid:48) . Thus we obtain a genus 5 surface Σ of square +3in X , . By blowing up Σ at one point, we obtain a genus 5 surface Σ (cid:48) of square+2 in X , CP .Since the two symplectic building blocks ˜ M CP and X , CP contain sym-plectic genus 5 surfaces of self intersections − and Σ (cid:48) . Let M , = ( ˜ M CP ) Σ =Σ (cid:48) ( X , CP ) . Lemma 9. σ ( M , ) = 0 , χ ( M , ) = 9 , e ( M , ) = 36 and c ( M , ) = 72 .Proof. We have σ ( M , ) = σ ( ˜ M CP ) + σ ( X , CP ) = 3 + ( −
3) = 0 and χ ( M , ) = χ ( ˜ M CP ) + χ ( X , CP ) + (5 −
1) = 4 + 1 + 4 = 9. Consequently, wecompute e ( M , ) and c ( M , ) as given in the statement. (cid:3) As above, we show that M , is an exotic copy of 17 CP CP and M , is alsosmoothly irreducible. As in Section 5, by performing generalized torus surgeries,we realize infinitely many pairwise non-diffeomorphic, irreducible, symplectic andnonsymplectic 4-manifolds (see [5] for the further details of such construction).6.3. Symplectic and smooth manifolds with ( σ, χ ) = (2 , . In this case,the first symplectic building blocks is ˜ M CP along the genus 5 curve Σ of selfintersection −
2. Our the second symplectic building block is X , in the notationof Theorem 6, which was constructed in [5].Let us recall the construction of X , . We take a copy of T ×{ pt } and the braidedtorus T β representing the homology class 2[ { pt } × T ] in T × T (see [5], page 4for the construction of T β ). The tori T × { pt } and T β intersect at two points.We symplectically blow up one of these intersection points, and symplecticallyresolve the other intersection point to obtain the symplectic genus two surface of self OMPLEX BALL QUOTIENTS AND NEW SYMPLECTIC 4-MANIFOLDS 17 intersection 0 in T CP (see [5], pages 3-4). The symplectic genus 2 surface Σ hasa dual symplectic torus T of self intersections zero intersecting Σ positively andtransversally at one point. We form the symplectic connected sum of T CP withΣ × Σ along the genus two surfaces Σ and Σ ×{ pt } . By performing the sequenceof 4 appropriate ± T CP ) Σ =Σ ×{ pt } (Σ × Σ ), weobtain the symplectic 4-manifold X , constructed in [5]. It can be seen from theconstruction that, X , contains a symplectic surface Σ with self intersection 0,resulting from the internal sum of the punctured torus in T CP \ ν (Σ ) andone of the punctured genus two surfaces in Σ × Σ \ ν (Σ × { pt } ). Σ intersectsΣ positively and transversally at one point. (The reader may see Section 5.3and Figure 7 in [13] showing the construction steps for a similar case.) We nowsymplectically resolve their intersection which gives genus five surface Σ (cid:48) of selfintersection +2 in X , .Let M , = ( ˜ M CP ) Σ =Σ (cid:48) ( X , ) . Lemma 10. σ ( M , ) = 2 , χ h ( M , ) = 10 , e ( M , ) = 38 and c ( M , ) = 82 .Proof. We have σ ( M , ) = σ ( ˜ M CP ) + σ ( X , ) = 3 + ( −
1) = 2 and χ h ( M , ) = χ ( ˜ M CP )+ χ ( X , )+(5 −
1) = 4+2+4 = 10. Consequently, we compute e ( M , )and c ( M , ). (cid:3) Similarly, using the Lemma 10 and the above mentioned theorems, we see that M , is an exotic copy of 19 CP CP , which is smoothly irreducible.6.4. Symplectic and smooth manifolds with ( σ, χ ) = (1 , . Similar to theprevious cases, we use ˜ M CP containing genus 5 curve Σ of self intersection − X , CP in the notation of Theorem 6, constructed in [5].To construct X , , we first obtain a symplectic genus two surface of self intersec-tion 0 in T CP as follows. Let us take a copy of T × { pt } and the braided torus T β representing the homology class 2[ { pt } × T ] in T × T . The tori T × { pt } and T β intersect at two points. We symplectically blow up one of these two inter-section points, and symplectically resolve the other intersection point to obtain thesymplectic genus two surface Σ of self intersection 0 in T CP . Note that the ex-ceptional sphere S intersects Σ positively and transversally twice. Next, we formthe symplectic connected sum of T CP with Σ × Σ along the genus two sur-faces Σ and Σ × { pt } . By performing the sequence of 6 appropriate ± T CP ) Σ =Σ ×{ pt } (Σ × Σ ), we obtain the symplectic 4-manifold X , . It was shown in [5], X , is an exotic copy of CP CP . Observe that as aresult of the internal sum of the twice punctured sphere S in T CP \ ν (Σ ) andthe twice punctured tori in Σ × Σ \ ν (Σ × { pt } ), we acquire a symplectic genus2 surface of self intersection − X , intersecting Σ positively and transversallytwice. We symplectically resolve the two intersections and get symplectic genus5 surface of square +3 in X , . We blow up this surface at one point and obtainsymplectic genus 5 surface Σ (cid:48) of self intersection +2 in X , CP .Let us define M , = ( ˜ M CP ) Σ =Σ (cid:48) ( X , CP ) . Lemma 11. σ ( M , ) = 1 , χ h ( M , ) = 9 , e ( M , ) = 35 and c ( M , ) = 73 . Proof.
We have σ ( M , ) = σ ( ˜ M CP ) + σ ( X , CP ) = 3 + ( −
2) = 1 and χ ( M , ) = χ ( ˜ M CP ) + χ ( X , CP ) + (5 −
1) = 4 + 1 + 4 = 9. Consequently, wecompute e ( M , ) and c ( M , ) as given. (cid:3) Similarly, using the Lemma 11 and the above mentioned theorems, we show thatthe minimal symplectic 4-manifold M , is an exotic copy of 17 CP CP . Remark 1.
In this remark, we discuss how to obtain a minimal symplectic -manifold with the fundamental group Z and the invariants ( σ, χ ) = (0 , . Since e = 4 χ − σ = 32 , such a symplectic -manifold yields to a homology CP CP with π ∼ = Z . Since the covering group of the complex surface M (see Proposition1) is Z × Z , it has a degree two unramified covering. Let us consider the normalunramified covering M of M with covering group given by index two subgroup H (cid:48) of π ( M ) . Let p : M → M be the covering map. Notice that in this casethe pull-back of D under this Z covering is not isomorphic to the fundamentalgroup of the ambient manifold, but rather a normal subgroup of index . Using thesymplectic pair ( M CP , Σ ) instead of ( ˜ M CP , Σ ) , and ( X , , Σ (cid:48) ) in our aboveconstructions (see 6.3) leads to a minimal symplectic -manifold with ( σ, χ ) = (0 , and π ∼ = Z . Previously no such examples were known. Acknowledgments
The first author was partially supported by a Simons Research Fellowship andCollaboration Grant for Mathematicians from the Simons Foundation. He wouldlike to thank the Departments of Mathematics at Purdue and at Harvard Univer-sities for their hospitality, where part of this work was completed. The secondauthor would like to thank Max Planck Institute for Mathematics in Bonn for itssupport and hospitality. The third author is partially supported by NSF grantDMS-1501282. All authors would like to thank Donald Cartwright for his helprelated to Magma computations.
References [1] A. Akhmedov,
Small exotic -manifolds , Algebr. Geom. Topol. (2008), 1781–1794.[2] A. Akhmedov, Construction of symplectic cohomology S × S , G¨okova Geometry and Topol-ogy Proceedings, (2007), 36–48.[3] A. Akhmedov, S. Baldridge, R. ˙I. Baykur, P. Kirk and B. D. Park, Simply connected minimalsymplectic -manifolds with signature less than −
1, J. Eur. Math. Soc. (2010), 133–161.[4] A. Akhmedov and B. D. Park, Exotic smooth structures on small -manifolds , Invent. Math. (2008), 209–223.[5] A. Akhmedov and B. D. Park, Exotic smooth structures on small -manifolds with oddsignatures , Invent. Math. (2010), 577–603.[6] A. Akhmedov and B. D. Park, New Symplectic -manifolds with positive signature , Journalof Gokova Geometry and Topology, (2008), 1–13.[7] A. Akhmedov and B. D. Park, Geography of Simply Connected Spin Symplectic 4-Manifolds ,Math. Res. Letters, (2010), no. 3, 483–492.[8] A. Akhmedov and B. D. Park, Geography of simply-connected nonspin symplectic -manifolds with positive signature. II , Canadian Mathematical Bulletin, 2020: DOI:https://doi.org/10.4153/S0008439520000533.[9] A. Akhmedov and B. D. Park, Geography of Simply Connected Spin Symplectic 4-Manifolds,II , C. R. Acad. Sci. Par. Ser. I., (2019), 296–298.[10] A. Akhmedov, M. Hughes, and B. D. Park:
Geography of simply-connected nonspin -manifolds with positive signature , Pacific J. Math., (2), 2013, 257–282. OMPLEX BALL QUOTIENTS AND NEW SYMPLECTIC 4-MANIFOLDS 19 [11] A. Akhmedov, B. D. Park, and G. Urzua,
Spin symplectic 4-manifolds near Bogomolov-Miyaoka-Yau line , Journal of Gokova Geometry and Topology, (2010), 55–66.[12] A. Akhmedov, B. D. Park, and S. Sakallı, Exotic smooth structures on connected sums of S × S , preprint (2019).[13] A. Akhmedov and S. Sakallı, On the geography of nonspin symplectic 4-manifolds with non-negative signature , Topology and its Applications, 206 (2016), 24-45.[14] D. Auroux, S. K. Donaldson and L. Katzarkov,
Luttinger surgery along Lagrangian tori andnon-isotopy for singular symplectic plane curves , Math. Ann. (2003), 185–203.[15] W. P. Barth, K. Hulek, C. A. M. Peters, A. Van de Ven.:
Compact complex surfaces , Springer-Verlag, Berlin Heidelberg, Second Enlarged Edition, 2004.[16] I. C. Bauer and F. Catanese,
A Volume Maximizing Canonical Surface In 3-Space , Comment.Math. Helv., , 2008, 387–406.[17] D. Cartwright and T. Steger, Enumeration of the 50 fake projective planes , C. R. Acad. Sci.Paris, Ser. (2010), 11–13.[18] D. Cartwright, V. Koziarz, and S-K.,
On the Cartwright-Steger surface , J. Algebraic Geom.26 (2017), 655-689; long arXiv version, arXiv:1412.4137.[19] R. Fintushel, B. D. Park and R. J. Stern,
Reverse engineering small -manifolds , Algebr.Geom. Topol. (2007), 2103–2116.[20] M. H. Freedman, The topology of four-dimensional manifolds , J. Differential Geom. (1982), 357–453.[21] R. E. Gompf, A new construction of symplectic manifolds , Ann. of Math. (1995), 527–595.[22] I. Hambleton and M. Kreck,
On the classification of topological -manifolds with finite fun-damental group , Math. Ann. (1988), 85–104.[23] M. J. D. Hamilton and D. Kotschick, Minimality and irreducibility of symplectic four-manifolds , Int. Math. Res. Not. , Art. ID 35032, 13 pp.[24] F. Hirzebruch,
Arrangements of Lines and Algebraic Surfaces , Arithmetic and geometry :papers dedicated to I.R. Shafarevich on the occasion of his sixtieth birthday / TN: 982123,volume 2, 1983, 113–140.[25] M.-N. Ishida and F. Kato,
The strong rigidity theorem for non-Archimedean uniformization ,Tohoku Math. J. (1998), 537–555.[26] J. Keum, A fake projective plane with an order automorphism , Topology, (2006), 919–927.[27] D. Kotschick, The Seiberg-Witten invariants of symplectic four-manifolds , SeminaireN.Bourbaki, (1995-96), 195–220.[28] K. M. Luttinger,
Lagrangian tori in R , J. Differential Geom. (1995), 220–228.[29] D. Mumford, An algebraic surface with K ample, K = 9 , p g = q = 0, Amer. J. Math. (1979), 223–244.[30] R. Pardini, Abelian covers of algebraic varieties , J. Reine Angew. Math. 417, 1991, 191–214.[31] G. Prasad and S-K. Yeung,
Fake projective planes , Invent. Math. (2007), 321–370.[32] G. Prasad and S-K. Yeung,
Addendum to “Fake projective planes” Invent. Math. 168, 321-370 (2007) , Invent. Math. (2010), 213–227.[33] C. H. Taubes,
The Seiberg-Witten invariants and symplectic forms , Math. Res. Lett. (1994),809–822.[34] T-J Li, C. H. Taubes (Personal Communication).[35] M. Usher, Minimality and symplectic sums , Int. Math. Res. Not. , Art. ID 49857, 17pp.[36] S-K. Yeung,
Classification of surfaces of general type with Euler number
3, J. Reine Angew.Math. 679 (2013), 1-22.[37] S.-K. Yeung,
Foliations associated to harmonic maps on some complex two ball quotients ,Sci. China Math. 60 (2017), 1137-1148; Erradum, ibid 63(2020), 1645.[38] V. Kulikov,
Old and new examples of surfaces of general type with p g = 0, Izvestiya: Math-ematics, 2004, , pp.965–1008. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA
Email address : [email protected] Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR,72701, USA
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