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Fibred algebraic surfaces and commutators in the symplectic group. (English) Zbl 1470.14022

The main results of the article under review concern the minimal number of singular fibres of fibrations of complex algebraic surfaces as well as the induced factorisations in the mapping class groups and the symplectic groups.
In the first, geometric part of the article, the authors consider fibrations \(f\colon S\to B\) from a smooth complex algebraic surface \(S\) over a curve \(B\) of genus \(b\) with (connected) fibres of genus \(g\). The Segre number of a fibre \(F_t\) is defined as \(\mu_t = \deg(\mathcal F)+D_tK_S-D_t^2\) where \(K_S\) is the canonical divisor of \(S\), \(D_t\) is the divisorial part of the critical scheme and and \(\mathcal F\) is a certain sheaf concentrated in the singular points of the reduction of the fibre. The first major result is the following Zeuthen-Segre-like formula, expressing that the total Segre number \(\mu = \sum_t\mu_t\) is measuring the failure for \(f\) to be a fibre bundle by means the topological Euler characteristic \(e(S) = c_2(S)\): \[\mathop{e}(S) = 4(g-1)(b-1)+\mu.\] Theorem 1.2 classifies the cases occurring where \(b=1\), \(g\geq 2\) and \(\mu = 3\) or \(\mu=4\). One explicit example is asserted to be the so-called Cartwright-Steger surface, which, according to Rito, has three singular fibres with single nodes as singularities. Moreover, the article contains a construction of a fibration with \(e(S) = \mu = 4\) with four nodes in the singular fibres, where \(S\) is the product of two curves of genus \(2\), as well as the construction of fibrations with \(e(S) =\mu = 4\) and \(p_g=q=2\), where the fibre genus is 4 and 10, respectively. Finally, the geometric part of the article under review investigates the minimal number of singular fibres where \(b\leq 1\); it is shown that such a non-isotrivial fibration has to have at least one singular fibre if \(b=1\) and at least \(3\) singular fibres if \(b=0\).
The second part of the article concerns certain factorisations in the mapping class group \(\mathcal Map_g\) and the symplectic group \(\mathrm{Sp}(2g,\mathbb Z)\), the existence of which are necessary conditions for the existence of fibrations as discussed in the first part. They arise as follows: As above, let \(f\colon S\to B\) be a fibration onto a curve \(B\) of genus \(b\) and fibres of genus \(g\), denote the points with singular fibres by \(p_1,\ldots,p_s\) and let \(B^*\subset B\) be their complement, i.e., the set of smooth fibres. Then, choosing a geometric basis, the fundamental group of \(B^*\) has the following presentation \[\pi_1(B^*) = \langle\alpha_1,\beta_1,\ldots,\alpha_b,\beta_b,\gamma_1\ldots,\gamma_s\mid\Pi_{i=1}^s\gamma_i\Pi_{j=1}^b[\alpha_j,\beta_j]\rangle.\] Therefore, homomorphisms \(\pi_1(B^*)\to G\) correspond to factorisations \(\Pi_{i=1}^s\delta_i\Pi_{j=1}^b[\alpha'_j,\beta'_j] = 1_G\). In the particular case of the mapping class group \(\mathcal Map_g = \mathcal Diff^+(C_g)/\mathcal Diff^0(C_g)\) (where \(C_g\) is a smooth curve of genus \(g\geq 1\)), we get such a homomorphism, and in particular such a factorisation, from the monodromy representation. By the correspondence result due to Moishezon, Kas and Matsumoto, on the other hand, any such factorisation in the mapping class group is induced from a (not necessarily holomorphic) Lefschetz fibration onto \(B\).
The mapping class group admits a canonical symplectic representation, given by its action on \(H_1(C_g;\mathbb Z)\cong \mathbb Z^{2g}\) in conjunction with the intersection form. Considering those factorisations in the symplectic group instead of the mapping class group is of course a weakening, but it grants the authors enough algebraic calculus to prove some very explicit factorisation results, like the following:
Let \(T\in \mathrm{Sp}(4,\mathbb Z)\) be the transvection at an element of the symplectic basis \(e_1,\ldots,e_4\), e.g., \(T(v) = v+(e_1,v)e_1\). Then \(T^m\) is a commutator if and only if \(m\) is even, cf. Theorem 5.1; the commutator is given explicitly on p. 215. In \(\mathrm{Sp}(2g,\mathbb Z)\) where \(g\geq 3\), every power of the analogous transvection \(T\) is shown to be a commutator; again, the commutator is completely explicit, cf. Theorem 7.21 and its proof on p. 224.
Let \(T_1\in\mathrm{Sp}(2,\mathbb Z)\) be defined analogously as above and let \(T_g\in\mathrm{Sp}(2g,\mathbb Z)\) be the orthogonal sum of \(g\) copies of \(T_1\). The article concludes with the analysis of the symplectic maps \(T_g\) and their powers in \(\mathrm{Sp}(2g,\mathbb Z)\) as well as in \(\mathrm{Sp}(2g',\mathbb Z)\) as \(g'\geq g\) via the standard inclusions.

MSC:

14D05 Structure of families (Picard-Lefschetz, monodromy, etc.)
14J29 Surfaces of general type
14J80 Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants)
32S50 Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants
32S20 Global theory of complex singularities; cohomological properties
20H99 Other groups of matrices
53D99 Symplectic geometry, contact geometry
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References:

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