The paper concerns an enumerative result in local/global study of variation of Hodge structure (VHS) associated to projective fibrarions \(f:S \to B\) where \(X\) and \(B\) are in general quasiprojective varieties and \(\dim B=1\). In many cases the numerical identities (inequalities) on VHS maybe understood as a measure of the complexity of the twist of the vHS when they degenerate. These identities are also interesting of their own from algebraic geometry point of view. The concept sits as a bridge between algebraic geometry and Hodge theory. Various techniques from geometry may be applied to the VHS in different cases, see [V. González-Alonso et al., “On the rank of the flat unitary factor of the Hodge bundle”, Preprint, arXiv:1709.05670; C. Peters, Pure Appl. Math. Q. 12, No. 1, 75–104 (2016; Zbl 1378.14029); K. Zuo, Asian J. Math. 4, No. 1, 279–301 (2000; Zbl 0983.32020)] for instance.
Apart from the singular fibers the degree \(k\) cohomologies of the fibres \(H^k(f^{-1}(b), \mathbb{Q})\) define a local system \(H\) of \(\mathbb{Q}\)-vector spaces. Equivalently one has a Gauss-Manin connection \(\nabla:H \otimes \mathcal{O}_B \to H \otimes \Omega_B^1\) whose kernel is (exactly) \(H\) [\(\nabla\) is formal derivation with respect to the parameter in the base \(B\)]. It satisfies the Griffiths transversality \(\nabla F^p \subset F^{p-1} \otimes \Omega_B^1\) where \(F^{\bullet}\) is the (decreasing) Hodge filtration on \(\mathcal{H}=H \otimes_{\mathbb{Q}} \mathcal{O}_B=\bigoplus_{p+q=k} \mathcal{H}^{p,q}\) [each \(F^p\) is a smooth complex vector bundle]. The well defined maps \[ \theta_p=\mathrm{Gr}_F^p \nabla:F^p/F^{p+1} \to F^{p-1}/F^p \otimes \Omega_B^1 \] are called Kodaira-Spencer (KS) maps. The paper specifically considers the KS-map for a VHS of a fibration by curves of genus \(g\) [where also more interest is in \(g \geq 2\)]. In this case we just have one KS-map \[ \theta: \mathcal{H}^{1,0} \to \mathcal{H}^{0,1} \otimes \Omega_B^1 \] The authors employ a well known decomposition of Hodge bundle due to Fujita [second Fujita decomposition] as \[ \mathcal{H}^{1,0}=f_*\omega_{S/B}=\mathcal{A} \oplus \mathcal{U} \] where \(\mathcal{A}\) is an ample sheaf and \(\mathcal{U}\) is a unitary flat bundle. The flatness of \(\mathcal{U}\) means that it can be written as \(\mathcal{U}=\mathbb{U} \otimes \mathcal{O}_B\) where \(\mathbb{U}\) is a local system defined as the kernel of the restriction of the Gauss-Manin connection \(\nabla\) to \(\mathcal{H}^{1,0}\). One trivially has \(\mathcal{U} \subset \ker \theta=:K\), [F. Catanese and M. Dettweiler, Int. J. Math. 27, No. 7, Article ID 1640001, 25 p. (2016; Zbl 1368.14019); P. Frediani et al., J. Inst. Math. Jussieu 19, No. 4, 1389–1408 (2020; Zbl 1469.14020); T. Fujita, Proc. Japan Acad., Ser. A 54, 183–184 (1978; Zbl 0412.32029)].
The major effort of the article is to give a constructive evidence that the last inclusion can be strict and with large gap. To this purpose they give a detailed study of the deformation theory of the general fibre of \(f\) defined by zero divisors on the curve. Specifically for any smooth curve of genus \(g \geq 2\) and any \(r=0, \dots , g-1\) they construct a non-isotrivial deformation of the curve \(C\) over a quasiprojective \(B\) such that \(\text{rk}(K)=r\) and \(\text{rk}(\mathcal{U}) \leq (g+1)/2\) [Theorem 1.1]. As a corollary they show that when the jacobian \(J(C)\) is simple there is a deformation of \(C\) with \(\mathcal{U}=0\) [Corollary 1.2].
A tool of use is that of deformation of the curve \(C\) defined by supporting divisors [Definition 2.1]. We say an effective divisor \(D\) on \(C\) supports \(\xi \in H^1(C, T_C)\) if \[ \xi \in \ker (H^1(C, T_C) \to H^1(C,T_C(D)) \] We call \(D\) minimal if any \(D' <D\) does not support \(\xi\). A minimal supporting divisor of a smooth family of curves \(f: \mathcal{C} \to B\) is an effective divisor \(\mathcal{D} \subset \mathcal{C}\) such that on a general fiber \(\mathcal{C}_b\) the restriction \(\mathcal{D}_b=\mathcal{D}|_{\mathcal{C}_b}\) is the minimal supporting divisor of the infinitesimal deformation \(\xi_b\) of \(C_b\) induced by \(f\), see [E. Horikawa, J. Math. Soc. Japan 25, 372–396 (1973; Zbl 0254.32022)]. The author also employ a notion of deforming a general fibre along the points on an effective divisor.
Definition. One says the family \(f\) is obtained from \(\pi=\pi_o:C \to C'\) by moving some branch points \(q_1, \dots , q_k \in C'\) (while keeping the remaining points \(q_{k+1}, \dots , q_n\) fixed) if there exist some maps \(\tilde{q}_1, \dots , \tilde{q}_k:B \to C'\) which are injective around \(b=o\) and such that \(\pi_b:\mathcal{C}_b \to C'\) is ramified over \(\tilde{q}_1(b), \dots , \tilde{q}_k(b)\) with the same ramification type as \(b=o\).
A main result of Section 3 is Lemma 3.3: Lemma. If \(f\) is minimally supported on a divisor \(D\) with \(D.\ \mathcal{C}_b=d\) and \(h^0(\mathcal{C}_b, \mathcal{O}_{\mathcal{C}_b}(D|_{\mathcal{C}_b}))=1\) [in this case we call the divisor on the fiber to be rigid] for general \(b \in B\), then \(\text{rk}(K)=g-d\) and \(\text{rk}(\mathcal{U}) \leq (g+1)/2\) [Lemma 3.3]. In this case \(\mathcal{U}\) has finite monodromy [Lemma 3.4].
The proof uses the fact that \(H^0(C, \omega_C(-D)) \subset \ker (\cup \ \xi)\) and hence \[ \dim \ker (\cup \ \xi) \geq g-(\deg D -r(D)) \] mentioned in Lemma 2.2 in the text. Theorem 3.5 provides the construction of a nontrivial families of curves with \(K\) of any given rank between \(1, \dots , g-1\). A basic fact used in the proof of Theorem 3.5 is a geometric interpretation of supporting divisors. Let \[ \phi:C \to \mathbb{P}(H^0(C, \omega_C^{\otimes 2}(-D)^{\vee}) \] be the canonical embedding, define \(\langle D \rangle :=\bigcap_{D \leq \phi^*H} H\) be the intersection of all hyperplanes cutting out a divisor on \(C\) that contains \(D\). Assume \(\xi \in H^1(C,T_C)\) be a non-zero first order infinitesimal deformation which defines \([\xi] \in \mathbb{P}(H^1(C,T_C))=\mathbb{P}(H^0(C, (\omega_C^{\otimes 2})^{\vee})\). Then the divisor \(D\) supports \(\xi\) if and only if \([\xi] \in \langle D \rangle\). The construction uses deformations minimally supported in a given rigid effective divisor of degree \(d \in \{1, \dots , g \}\). Because the map \(\mathrm{Div}^d(C) \to \mathrm{Pic}^d(C)\) is generically 1-1, the rigid divisors form a Zariski open subset in \(\mathrm{Div}^d\). The idea is for certain deformation classes \(\xi\) there exist some minimal supporting divisor \(D\) of degree \(d\) such that \(r(D)=0\). Then one may proceed to apply Lemma 2.2 of text to determine the dimension of the kernel of \( \cup \xi\).
In Section 4 the authors present semistable families of cyclic coverings of \(\mathbb{P}^1\) where \(K \supsetneqq \mathcal{U}\). The major result is Proposition 4.1. The statement is as follows.
Proposition. Let \(\pi: C \to \mathbb{P}^1\) be a simple cyclic cover of degree \(n\) with reduced branch divisor \(B=q_1+ \dots +q_m, n |m\), and \(g(C) \geq 2\). Assume \(f:\mathcal{C} \to \Delta\) be a deformation of \(C\) obtained by moving branch points \(q_1, \dots , q_k\) as defined above. If \(k <m/n\) then \(\text{rk}(K)=g-(n-1)k\) and \(\text{rk}(\mathcal{U}) \leq (g+1)/2\). It follows that if \(k <(g-1)/2(n-1)\ \) then \(\ \mathcal{U} \subsetneqq K\).
The proof sets \(p_i=\pi^{-1}(q_i)\) and defines \(D=(n-1)(p_1+ \dots p_k)\). The author show that \(D\) is a rigid divisor that supports \(f\) minimally. Then Lemma 3.3 proves the proposition, see also [B. Moonen, Doc. Math. 15, 793–819 (2010; Zbl 1236.11056)]. The application of the deformations of a curve of genus \(g\) defined by effective divisors on it to the numerical Hodge theoretic invariants is one of the interesting ideas of the paper which serves a value. Specially the method of of relating the Fujita decomposition to this part seems to be new from Hodge theory point of view. It could also be extended to more general cases.