Slopes of Lefschetz fibrations and separatin vanishin cycles Yusuf Z. Gürtaş Abstract. In this article we find upper and lower bounds for the slope of enus hyperelliptic Lefschetz fibrations. We demonstrate the connection between the slope of enus hyperelliptic Lefschetz fibrations and the number of separatin vanishin cycles: we show that λ > 4 4/ if and only if the fibration contains separatin vanishin cycles. We also improve the existin bound on s/n, the ratio of number of separatin vanishin cycles to the number of non-separatin vanishin cycles, for hyperelliptic Lefschetz fibrations of enus 2. In particular we show that s < n when 6. M.S.C. 2010: 57M07, 57R17, 20F38. Key words: low dimensional topoloy; symplectic topoloy; mappin class roup: Lefschetz fibration; vanishin cycle; slope. 1 Introduction A Lefschetz fibration over S 2 is a smooth map f : X S 2 from a compact, connected, oriented, smooth 4- manifold X with the followin properties : 1. f has finitely many critical values q 1, q 2,..., q k in S 2, 2. each of the preimaes f 1 q 1 ), f 1 q 2 ),..., f 1 q k ) consists of exactly one critical point, say p 1, p 2,..., p k in X, 3. around each of the points p 1, p 2,..., p k and q 1, q 2,..., q k there are local charts, areein with the orientations of X and S 2, on which f is locally iven as z 1, z 2 ) z 2 1 + z 2 2 in complex coordinates. The fibers over B = {q 1, q 2,..., q k } are called sinular fibers. The points in S 2 \ B are called reular values and the fibers over them are called reular fibers. It s a consequence of this definition that the restriction of f to f 1 S 2 \ B ) is a fiber bundle over S 2 \ B with fibers diffeomorphic to Σ, a compact, connected, oriented surface Balkan Journal of Geometry and Its Applications, Vol.19, No.1, 2014, pp. 42-53. c Balkan Society of Geometers, Geometry Balkan Press 2014.
Slopes of Lefschetz fibrations and separatin vanishin cycles 43 of enus. We also refer to as the enus of the fibration f : X S 2, [8]. The monodromy around each of the sinular fibers is iven by a positive Dehn twist about a simple closed curve in Σ, which is called a vanishin cycle. A vanishin cycle is a simple closed curve on reular fibers that collapses to a point on a sinular fiber as one ets near a critical point. Choosin a reference point q S 2 \ B one can characterize the fibration f by its monodromy homomorphism 1.1) ψ : π 1 S 2 \ B ) M, where M = π 0 Diff + Σ ) ) is the mappin class roup of Σ, [1, 5]. We can assume that each vanishin cycle is homotopically nontrivial because we can eliminate those that are trivial by blowin down the fibration to obtain another one that is relatively minimal,[4, 5, 8]. We call a vanishin cycle γ nonseparatin if Σ \ γ is connected. Otherwise we call it separatin. γ γ nonseparatin Σ 2 separatin Σ 2 Fiure 1: A separatin and a nonseparatin vanishin cycle on Σ 2 f q k q 2 q 1 D Fiure 2: Fibration restricted to D S 2 It s possible to arrane the critical points of f : X S 2 in such a way that they are distinct and each sinular fiber contains only one of them as we assumed in the definition of Lefschetz fibration. Let s now take a disc D within S 2 includin the set
44 Yusuf Z. Gürtaş of critical values B and restrict the fibration to f 1 D), [4]. The crucial information about the fibration lies over this part of S 2, Fiure 2). Let s define now the monodromy homomorphism in 1.1) explicitly. Let q be a reular value in D and choose α 1, α 2,..., α k in D as the enerators of π 1 S 2 \ B ) as shown in Fiure 3. As we o around q 1 alon α 1 a smooth fiber bundle over α 1 with fiber Σ is formed and the way we identify the fibers over the inital and final points of α 1, both of which are q, with a model surface Σ ives us a diffeomorphism of Σ. The isotopy class of this map is an element of M by definition. It turns out that this mappin class is realized by a positive Dehn twist about a homotopically) nontrivial simple closed curve in Σ, call it γ 1. Let s denote the Dehn twist about γ 1 by D γ1. We do the same for α 2 and obtain another element of M, call it D γ2. It s not difficult to see that the map ψ in 1.1) respects composition both in the domain and in the rane and we have 1.2) ψ [α 1 α 2 ]) = ψ[α 1 ] [α 2 ]) = ψ[α 1 ])ψ[α 2 ]) = D γ2 D γ1. D α k q k q 2 α 2 q 1 α 1 q Fiure 3: Generators of π 1 S 2 \ B ) Strictly speakin, the last equality in 1.2) must have D γ1 D γ2 on the riht hand side but it s customary to compose elements of the mappin class roup from riht to left. Therefore it shouldn t cause a problem as lon as we keep that little detail in mind. Continuin the same way until we o around the last critical value q k and composin alon we obtain 1.3) ψ [α 1 α 2 α k ]) = D γk D γ2 D γ1. It s clear from Fiure 3 that [α 1 α 2 α k ] = [ D] in π 1 S 2 \ B ) and [ D] = Id in π 1 S 2 \ B ) ; therefore we have D γk D γ2 D γ1 = Id in M. This shows that a enus Lefschetz fibration over S 2 ives us a word that is equal to identity in the mappin class roup of the fiber Σ. The converse is also true: Every such word that is equal
Slopes of Lefschetz fibrations and separatin vanishin cycles 45 to identity in the mappin class roup of Σ defines a Σ fibration over S 2. It s important to note two thins here: First one is, this correspondence is not one-to-one. In order to have a one-to-one correspondence we have to consider equivalence classes of Lefschetz fibrations and words in the mappin class roup. On one side we have isomorphism classes of enus Lefschetz fibrations and on the other side we have equivalence classes of words with positive exponents that are equal to identity in M. Two such words in M are equivalent if it s possible to obtain one from the other throuh cyclic permutation of the twists D γi, conjuatin the word by an element of M, or throuh ) elementary transformations such as replacin D γi+1 D γi by D γi+1 D γ 1 i+1 D γi D γi+1, [4, 5, 8]. The correspondence between isomorphism classes of Lefschetz fibrations and equivalence classes of words in the mappin class roup as defined above would be one-to-one. Secondly the words in the mappin class roup must carry only positive exponents. It s due to orientation preservin condition for the charts in the definition of Lefschetz fibrations that we allow only positive Dehn twists. If we relax the definition to allow orientation reversin ones then the 4-manifold X can no loner be shown to be symplectic, [4]. One of the oals that we seek in studyin Lefschetz fibrations is to obtain information about symplectic 4-manifolds because Lefschetz fibrations are rouhly topoloical descriptions of symplectic 4- manifolds due to the companion theorems of Donaldson and Gompf, [4, 5]. The reader is referred to [5] for a thorouh review of Lefschetz fibrations and symplectic 4-manifolds. All Lefschetz fibrations throuhout this article are assumed to have enus 2 fiberin over S 2. It s known that the 4- manifold X in the definition of Lefschetz fibration carries an almost complex structure, [5]; therefore it makes sense to define its holomorphic Euler characteristic χ h and first Chern class c 1. Let 1.4) χ h := 1 4 σ + e) and c2 1 := 2e + 3σ, where e is the Euler characteristic and σ is the sinature of the 4- manifold X. The slope λ f of a fibration f : X S 2 is defined as λ f := K 2 f /χ f, where 1.5) K 2 f := c 2 1 + 8 1) and χ f := χ h + 1. It is known that λ f 4 4 for a relatively minimal holomorphic enus Lefschetz fibration, and this bound is sharp since all of the known hyperelliptic Lefschetz fibrations over S 2 with no separatin vanishin cycle satisfy λ f = 4 4/ [10]. For example consider the words c1 c 2 c 2 2+1 c 2 c 1 ) 2 = 1 1.6) c 2 c 2 c 1 ) 4+2 = 1 c 2+1 c 2 c 1 ) 2+2 = 1 in the hyperelliptic mappin class roup H.
46 Yusuf Z. Gürtaş c 1 c 2 c 4 c 6 c 2 c 3 c 5 c 2+1 Fiure 4: Generators of hyperelliptic mappin class roup Let X 1, X 2, X 3 be the Lefschetz fibrations defined by those words, respectively. The Euler characteristic of each of these 4-manifolds can be computed usin the formula 1.7) ex) = 22 2) + µ, where µ is the number of sinular fibers, which is equal to the number of twists, [4]. Here we don t make a distinction between the simple closed curves c i and the Dehn twists about them in order to keep the notation simple ) Therefore ex 1 ) = 22 2) + 42 + 1) = 4 + 8 ex 2 ) = 22 2) + 24 + 2) = 8 2 + 4 ex 3 ) = 22 2) + 2 + 1)2 + 2) = 4 2 + 2 + 6. For hyperelliptic Lefschetz fibrations, local sinature formulas have been computed by Endo, [3]. The local contribution of a nonseparatin vanishin cycle to the sinature is + 1 2 + 1. Therefore we can compute the sinatures of X 1, X 2, X 3 as Usin 1.4) we obtain σx 1 ) = + 1 42 + 1) = 4 + 1) 2 + 1 σx 2 ) = + 1 24 + 2) = 4 + 1) 2 + 1 σx 3 ) = + 1 2 + 1 2 + 1)2 + 2) = 2 + 1)2. χ h X 1 ), c 2 1X 1 )) = 1, 4 4) χ h X 2 ), c 2 1X 2 )) = 2 + 1, 4 2 12 + 8) χ h X 2 ), c 2 1X 2 )) = 1 2 2 1 2 + 1, 22 8 + 6). When we compute the slopes of each of these three fibrations usin 1.5) we see that they are all equal to 4 4/. Monden ave examples of nonholomorphic Lefschetz fibrations violatin the bound λ f 4 4/ by usin inverse lantern substitution to lower the slope, [7]. The reader
Slopes of Lefschetz fibrations and separatin vanishin cycles 47 is also referred to [7] for a short list of articles where more examples of Lefschetz fibrations proven to be nonholomorphic usin various techniques can be found. We will write λ instead of λ f throuhout this article for simplicity. The connection between λ and the number of separatin vanishin cycles in an hyperelliptic Lefschetz fibration seems to be unaccounted for in the literature. In the next section we will prove Theorem 2.2 that reveals this connection. Let s be the number of separatin vanishin cycles and n be the number of those that are non-separatin. Recall that a Lefschetz fibration over S 2 can not contain only separatin vanishin cycles, Corollary 8, [8]). Therefore Theorem 2.2 should be understood as a fibration containin a mixture of separatin and non-separatin vanishin cycles. An interestin quantity that is worth calculatin at this point is the proportion of the number of separatin cycles within a fibration, in particular its ratio to the s number of non-separatin vanishin cycles,. We do not find any estimates in the n literature on this ratio except for 1.8) s n 5 due to A.Stipsicz, [9], and 1.9) s n 5 6 n for Lefschetz pencils due to V. Braunardt and D. Kotschick, [2]. We ll assume that n > 0 throuhout this article. Therefore s/n is always defined. Let r := s/n for a Lefschetz fibration of enus. There isn t enouh evidence to justify that the bounds 1.8) and 1.9) could actually be sharp. On the contrary, all of the known examples suest that r should not be too hih. In this article we will improve the bound on r for hyperelliptic Lefschetz fibrations and prove Theorem 1.1. For an hyperelliptic Lefschetz fibration of enus 2 we have We will also prove r 3 + 2 4 1) 2 + 1 n 1). Theorem 1.2. For a relatively minimal holomorphic Lefschetz fibration of enus 2 we have r 3 + 2 4 n 2 n. 2 Main Section The sinature of a enus hyperelliptic Lefschetz fibration X S 2 is iven by σ = + 1 [/2] 2 + 1 n + 4h h) s h s, 2 + 1 h=1
48 Yusuf Z. Gürtaş where s h is the number of separatin vanishin cycles which separate the surface into [ ] [/2] two components one with enus h and s = s h, [3]. Let 2 x = [/2] h=1 h=1 h h) s h. The other invariants of X that will be used throuhout this article are: Euler characteristic e = n + s 4 1), usin 1.7), holomorphic Euler characteristic χ h = 1 4 e + σ) = 1 n + s 4 1) + 1 4 2 + 1 n + 4x ) 2 + 1 s 2.1) n + 4x = 1), 4 2 + 1) and square of the first Chern class c 2 1 c 2 1 = 2e + 3σ = 2 n + s 4 1)) + 3 + 1 2 + 1 n + 4x ) 2 + 1 s = 2n s 8 1) 3 + 1 2 + 1 n + 12x 2 + 1, where s is the number of separatin vanishin cycles and n is the number of nonseparatin vanishin cycles. Lemma 2.1. s 2x for 2. Proof. It s not difficult to see that s 1) x by definition of x and s. Therefore s x 1 The proof follows from the fact that and s x 1. 2 for 2. 1 We will use this lemma to prove the theorem that shows the connection between λ and s : Theorem 2.2. A enus hyperelliptic Lefschetz fibration X S 2 satisfies λ > 4 4 if and only if s 0, i.e., it contains separatin vanishin cycles. Proof. The slope λ of the fibration is iven as 2.2) +1 λ = c2 1 + 8 1) χ h + 1) = 2n s 3 2+1 n + 12x = 4 n 1) s 2 + 1) + 12x n + 4x n+4x 42+1). 2+1
Slopes of Lefschetz fibrations and separatin vanishin cycles 49 Assume s 0. Then x 0 and we have n 1) s 2 + 1) + 12x λ 4 4/) = 4 4 + 4/ n + 4x = 4 2s2 s + 8x + 4x n + 4x) 2 + 1) 4x s) = 4 > 0, n + 4x) because 4x > s by Lemma 2.1. and all other factors are positive. Therefore λ 4 4/) = 0 if and only if 4x = s; i.e., if and only if s = 0. Note that if every fiber is smooth then λ = 12, otherwise λ < 12, [10]. Proposition 2.3. For a enus Lefschetz fibration the slope is iven by 2.3) λ = 12 n + s χ h + 1. Proof. By definition λ = c2 1 + 8 1) χ h + 1 = 12χ h e + 8 1) χ h + 1 = 12χ h + 12 12 e 4 1) χ h + 1 n + s 4 1)) 4 1) = 12 + χ h + 1 = 12 n + s χ h + 1 Remark 2.1. Usin χ h = 1 4 e + σ) = 1 n + s 4 1) + σ), we can substitute 4 2.4) σ + n + s = 4 χ h + 1) in 2.3), in order to obtain 2.5) n + s λ = 12 4 σ + n + s = 12 4 1 + σ n+s. Solvin the first equality for σ ives 2.6) σ = λ 8 12 λ n + s), i.e., σ n + s = λ 8 12 λ, which relates the sinature of a Lefschetz fibration to the total number of vanishin cycles throuh scalar multiplication and the averae sinature σ n+s per vanishin cycle to the slope. In particular σ > 0 corresponds to λ > 8 and σ < 0 corresponds to λ < 8 just as c 2 1 > 8χ h and c 2 1 < 8χ h correspond to σ > 0 and σ < 0, respectively.
50 Yusuf Z. Gürtaş σ Remark 2.2. When λ = 10 the averae sinature must be 1. This can never n + s happen because the sinature contribution of each vanishin cycle is either -1, or 0, or +1 and accordin to the handlebody decomposition of Lefschetz fibrations the first handle attached alon the first vanishin cycle, which can be arraned to be a non-separatin one by cyclically permutin, will always result in a 4- manifold with σ 0 sinature, [8]. Therefore n+s < 1. This proves Proposition 2.4. The slope of a Lefschetz fibration satisfies λ < 10. Corollary 2.5. A enus Lefschetz fibration satisfies the bound c 2 1 < 10χ h + 2 2. More is true if the Lefschetz fibration is hyperelliptic: Proposition 2.6. For a enus hyperelliptic Lefschetz fibration we have 2.7) λ 10 2 + s χ h + 1. Proof. First we estimate χ h as + 1 2 + 1 n + χ h = 1 4 σ + e) = 1 4 1 n 4 2 + 1 + 4 2 2 + 1 2.8) 2 ) s [/2] h=1 4 1) = 1 n 4 2 + 1 + 1 s 2 1) := M, 4 2 + 1 4h h) s h s + n + s 4 1) 2 + 1 ) usin the fact that h h) 2 2 ) and [/2] h=1 s h = s. Now, use this to write Euler characteristic as 2.9) e = n + s 4 1) 4 2 + 1) 1 n = 4 2 + 1 + 1 s 2 4 2 + 1 = The estimate 4 2 + 1) M + 1 ) s + 4 4. ) 1) + 1 ) s + 4 4 σ n s 4 = n + s 4 1) 2s + 4 2) = e 2s + 4 2), Corollary 9, [8]), can be used to write 2.10) χ h = 1 4 σ + e) 1 4 e 2s + 4 2) + e) = 1 2 e 1 2 s + 2, and usin 2.9) we obtain χ h 1 4 2 + 1) M + 1 ) s + 4 4 ) 1 2 2 s + 2 = 2 2 + 1 M 1 2 s + 3 2 2.
Slopes of Lefschetz fibrations and separatin vanishin cycles 51 We will solve this for s s 4 2 + 1 M 2χ h + 6 4 4 and use it in estimatin 4 2 + 1) c 2 1 = 12χ h e = 12χ h M + 1 ) s + 4 4 ) = 12χ h 4 2 + 1 M + 1) s 4 + 4 12χ h 4 2 + 1 M + 4 2 + 1 M 2χ h + 6 4 4 s 4 + 4 = 10χ h + 2 4 s. Now, λ = c2 1 + 8 1) χ h + 1 10χ h + 2 4 s + 8 1) χ h + 1 and we have λ 10 2 + s χ h + 1. = 10χ h + 10 10 2 s, χ h + 1 For hyperelliptic Lefschetz fibrations we can do even better: Proposition 2.7. The slope of an hyperelliptic enus Lefschetz fibration satisfies 2.11) 4 1 Proof. The sinature satisfies the bound + 4s 2 + 1) 3 4) n + 4s 1)) λ 10 2 2 + s n 2. σ = + 1 2 + 1 n + 4x 2 + 1 s + 1 4s 1) n + 2 + 1 2 + 1 s = + 1 2 + 1 n + 2 5 2 + 1 s, because s 1) x by definition of x and s. Now, usin 2.6) we can write + 1 2 + 1 n + 2 5 2 + 1 s 8 λ n + s), 12 λ and solvin this for λ ives the first inequality. To prove the second inequality we bein with the fact that χ h + 1 > 0, as we can see it from 2.4) because σ < n+s see Remark 2.2 above or Corollary 9 in [8] ). Also usin 2.10) we can write χ h 1 2 e 1 2 s + 2 = 1 2 n + s 4 1)) 1 2 s + 2 = 1 2 n, which can be rewritten as 1 χ h + 1 2 n 2. Now, addin 10 to both sides after multiplyin by 2 + s proves the second inequality thanks to Proposition 2.6.
52 Yusuf Z. Gürtaş Remark 2.3. We wrote 2.11) in that particular form instead of simplifyin it in order to emphasize the fact that it is another proof for Theorem 2.2 and that 4 4 λ < 10 for hyperelliptic Lefschetz fibrations. Proof. of Theorem 1.1) Usin the bound σ n s 4 Corollary 9, [8]) we et 1 n s σ) 1. Then 4 1 4 n s σ) = 1 4 ives 1 1 4 n s + 1 2 + 1 n + 4x )) 2 + 1 s = 1 3 + 2) n 4x 4 2 + 1 3 + 2) n 4x, i.e., x 1 n 3 + 2) 2 + 1). 2 + 1 4 Usin the estimate 1)s x one more time, we have 1) s 1 n 3 + 2) 2 + 1). 4 Dividin throuh by n 1) ives r = s n 3 + 2 4 1) 2 + 1 n 1). Corollary 2.8. For an hyperelliptic Lefschetz fibration of enus 6 we have s < n. Remark 2.4. One can prove Theorem 1.1 by solvin 4 1 + 4s 2 + 1) 3 4) n + 4s 1)) 10 2 2 + s n 2 for s n as well, 2.11). Also, solvin n + s λ = 12 4 n + s + σ 10 2 2 + s n 2 for σ results in σ n s 4, which is another proof for Proposition 2.6 thanks to Corollary 9, [8]). Finally, solvin 2.12) 4 1 λ = 12 4 1 + σ n+s for σ n + s + 1 2 + 1, σ n + s ives which shows that the averae sinature per vanishin cycle is at least + 1 2 + 1 for relatively minimal holomorphic Lefschetz fibrations. For hyperelliptic Lefschetz fibrations equality holds when s = 0, [3], and it s strict inequality when s > 0 by virtue of Theorem 2.2.
Slopes of Lefschetz fibrations and separatin vanishin cycles 53 Note that violatin the bound λ 4 4/ is equivalent to violatin the averae sinature bound in 2.12). In other words the averae sinature bound 2.12) can be used as a simple tool to prove that a Lefschetz fibration is nonholomorphic. Xiao conjectured that λ > 4 4/ for non-hyperelliptic holomorphic Lefschetz fibrations which was proved for some low enus by Konno, [6]. Now we will prove Theorem 1.2 usin 2.12). Proof. of Theorem 1.2) By Corollary 7 in [8] we have σ n s. Since we assume n > 0 we can conclude that σ n s 2, [9]. Combinin this with 2.12) we et + 1 2 + 1 Solvin for s/n yields r 3 + 2 4 n 2 n. σ n + s n s 2. n + s Acknowledements. Many thanks to Hurşit Önsiper for helpful and encourain conversations. The author is also rateful to Hisaaki Endo for his insihtful comments. References [1] D. Auroux, A stable classification of Lefschetz fibrations, Geometric Topoloy electronic) 9 2005), 203 217. [2] V. Braunardt and D. Kotschick, Clusterin of critical points in Lefschetz fibrations and the symplectic Szpiro inequality, Trans. Amer. Math. Soc. 355 2003), 3217 3226. [3] H. Endo, Meyer s sinature cocyle and hyperelliptic fibrations, Math. Ann. 316 2000), 237 257. [4] T. Fuller, Lefschetz fibrations of 4-dimensional manifolds, Cubo Mat. Educ. 5 3) 2003), 275 294. [5] R. Gompf and A. Stipsicz, An Introduction to 4-manifolds and Kirby Calculus, AMS Graduate Studies in Mathematics 20, 1999. [6] K. Konno, Non-hypereilliptic fibrations of small enus and certain irreular canonical surfaces, Ann. Sc. Norm. Sup. Pisa. Cl. Sci. 20 4) 1993), 575 595. [7] N. Monden, Lefschetz fibrations with small slope, Pacific Journal of Mathematics 267 1) 2014), 243 256. [8] B. Ozbaci, Sinatures of Lefschetz fibrations, Pacific Journal of Mathematics 202 1) 2002), 99 118. [9] A. Stipsicz, On the number of vanishin cycles in Lefschetz fibrations, Math. Res. Lett. 6 1999), 449 456. [10] G. Xiao, Fibered alebraic surfaces with low slope, Math. Ann. 276 1987), 449 466. Author s address: Yusuf Z. Gürtaş Department of Mathematics and Computer Sciences, Queensborouh Community Collee, CUNY 222-05 56th Avenue Bayside, NY 11364, USA. E-mail: yurtas@qcc.cuny.edu