2.1. Preliminary definitions

We begin by recalling the short exact sequence

(1) $$ \begin{align} 0\to f^*\omega_B\to \Omega^1_X\to \Omega^1_{X/B}\to 0, \end{align} $$

which defines the sheaf of holomorphic relative differentials $\Omega ^1_{X/B}$ . This sheaf, in general, is not locally free, and for this reason, it is often convenient to consider the logarithmic version of (1):

(2) $$ \begin{align} 0\to f^*\omega_B(E)\to \Omega^1_X(\log S)\to \Omega^1_{X/B}(\log)\to 0. \end{align} $$

We briefly recall that if S is locally given by $z_1z_2\ldots z_k=0$ in appropriate local coordinates, the sheaf $\Omega ^1_X(\log S)$ of logarithmic differentials is the locally free $\mathcal {O}_X$ -module generated by $dz_1/z_1,\ldots ,dz_k/z_k,dz_{k+1},\ldots ,dz_{n+1}$ .

In the case of a semistable fibration, (2) is an exact sequence of vector bundles and its p-wedge product, for $p=1,\dots , n$ , is

(3) $$ \begin{align} 0\to f^*\omega_B(E)\otimes \Omega^{p-1}_{X/B}(\log)\to \Omega^p_X(\log S)\to \Omega^p_{X/B}(\log)\to 0 \end{align} $$

and the determinant sheaf of $\Omega ^1_{X/B}(\log )$ is

$$ \begin{align*}\det(\Omega^1_{X/B}(\log))=\det(\Omega^1_X(\log S))\otimes f^*\omega_B(E)^\vee=\omega_X(S)\otimes f^*\omega_B(E)^\vee=\omega_X\otimes f^*\omega_B^\vee=:\omega_{X/B} \end{align*} $$

the relative dualizing sheaf of $f\colon X\to B$ .

The pushforward via f of Sequence (3) is a long exact sequence of locally free sheaves on B (see [Reference Steenbrink39, Theorem 2.11] cf. [Reference Kollár21, Lemma 2.11]), and we give the following definition.

Definition 2.1. We call $K^i$ the kernel of the connecting morphism

$$ \begin{align*}R^if_*\Omega^{n-i}_{X/B}(\log)\to \omega_B(E)\otimes R^{i+1}f_*\Omega^{n-i-1}_{X/B}(\log).\end{align*} $$

Remark 2.2. At general point $t\in B$ , the fiber of $K^i$ is the kernel of the homomorphism

$$ \begin{align*}H^i(X_t, \Omega^{n-i}_{X_t})\to T_{B,t}^\vee\otimes H^{i+1}(X_t, \Omega^{n-i-1}_{X_t}) \end{align*} $$

given by the cup product with the Kodaira–Spencer class $\xi _t\in H^1(X_t, T_{X_t})$ .

Following [Reference Rizzi and Zucconi35], we also associate to $f\colon X\to B$ a local system of vector spaces on B, that is, a sheaf on B which is locally isomorphic to a constant sheaf. Denote by $\Omega ^1_{X,d}$ the sheaf of de Rham closed holomorphic $1$ -forms on X, hence $\Omega ^1_{X,d}\subset \Omega ^1_{X}(\log S)$ and we consider the composition

$$ \begin{align*}f_*\Omega^1_{X,d}\hookrightarrow f_*\Omega^1_{X}(\log S)\to f_*\Omega^1_{X/B}(\log). \end{align*} $$

The image sheaf of $f_*\Omega _{X,d}^{1}\to f_*\Omega ^1_{X/B}(\log )$ is a local systems (see [Reference Rizzi and Zucconi35, Lemma 2.6]).

Equivalently, $\mathbb {D}^1$ fits into the following short exact sequence:

(4) $$ \begin{align} 0\to \omega_B\to f_*\Omega^1_{X,d}\to\mathbb{D}^1\to 0. \end{align} $$

Similarly, we denote by $\mathbb {D}^{n}$ the local system contained in $f_*\Omega ^{n}_{X/B}(\log )=f_*\omega _{X/B}$ and defined by the de Rham closed n-forms. As shown in [Reference Rizzi and Zucconi35, Theorem 3.7], $\mathbb {D}^{n}$ is actually the local system of the so-called second Fujita decomposition of $f_*\omega _{X/B}$ . In fact, as recalled in the Introduction, there is a decomposition

(5) $$ \begin{align} f_*\omega_{X/B}=\mathcal{U}\oplus\mathcal{A,} \end{align} $$

where $\mathcal {A}$ is an ample vector bundle (or $\mathcal {A}=0$ ) and $\mathcal {U}$ is a unitary flat vector bundle; $\mathbb {D}^{n}$ is such that $\mathcal {U}=\mathbb {D}^{n}\otimes _{\mathbb {C}} \mathcal {O}_B$ .

Finally, we point out that we have the inclusions $\mathbb {D}^n\subset \mathcal {U}\subseteq K^0\subseteq f_*\omega _{X/B}$ . The difference between the rank of $\mathcal {U}$ and the rank of $K^0$ is of interest (see, e.g., [Reference González-Alonso and Torelli11]).

2.2. Hermitian metrics on direct images

Consider the open subset $B\setminus E$ corresponding to the smooth fibers. The vector bundle $f_*\omega _{X/B}$ and, more in general, the bundles of primitive cohomology classes $\mathcal {P}^{n-i,i}\subset R^if_*\Omega ^{n-i}_{X/B}(\log )$ are endowed with a natural smooth Hermitian metric coming from the Hodge metric on the fibers (cf. [Reference Griffiths13], [Reference Griffiths and Tu15]). Actually, one can also consider the twisted cases $f_*(\omega _{X/B}\otimes L)$ and $R^if_*(\Omega ^{n-i}_{X/B}(\log )\otimes L),$ where L is a Hermitian line bundle on X whose curvature satisfies suitable properties. See [Reference Berndtsson1], [Reference Berndtsson2] for the case where L is semi-positive and [Reference Berndtsson, Păun and Wang3] for the case where L is semi-negative.

From now on, we fix a Kähler form $\Omega $ on X. We denote by $\omega _t$ the restriction of $\Omega $ to $X_t$ .

Following [Reference Berndtsson1], [Reference Berndtsson2], we first briefly recall the definition of the metric on $f_*\omega _{X/B}$ and some of its key properties. Consider u a local section of $f_*\omega _{X/B}$ over a disk $\Delta \subset B\setminus E$ . It can be seen as a function that maps a general point $t\in \Delta $ to a global holomorphic $(n,0)$ -form $u_t$ on the corresponding fiber $X_t$ . The Hermitian norm is then defined as

(6) $$ \begin{align} \lVert u_t\lVert^2_t=c_n\int_{X_t} u_t\wedge\bar{u}_t, \end{align} $$

where $c_n=i^{{n}^2}$ is a unimodular constant.

Key computations on this metric strongly rely on the notion of a good representative of a section. A representative of u, denoted by $\mathbf {u}$ , is a smooth $(n,0)$ -form on $f^{-1}(\Delta )\subset X$ which restricts to $u_t$ on the fibers.

The key point is that we have an explicit expression for the Chern connection $D = D'+ D"$ and curvature $\Theta $ in terms of the representative $\mathbf {u}$ . Indeed, we can write

$$ \begin{align*}\partial\mathbf{u}=\mu\wedge dt, \end{align*} $$

and, since $\bar {\partial }\mathbf {u}$ is trivial along the fibers, we can write

$$ \begin{align*}\bar{\partial}\mathbf{u}=\nu\wedge d\bar{t}+\eta\wedge dt, \end{align*} $$

where $\nu $ and $\mu $ are relative forms of bidegree $(n,0)$ and $\eta $ is of bidegree $(n-1,1)$ . We have that

(7) $$ \begin{align} (D"u)_t=\nu_t d\bar{t} \end{align} $$

and

(8) $$ \begin{align} (D'u)_t=P(\mu_t)dt, \end{align} $$

where P is the orthogonal projection of $(n, 0)$ -forms on the space of holomorphic $(n, 0)$ -forms. The $(n-1,1)$ -form $\eta _t$ has a neat interpretation in terms of the cup product with the Kodaira–Spencer class $\xi _t$ : it is a representative of the cohomology class of the cup product $\xi _t\cup u_t\in H^{n-1,1}(X_t)$ . See [Reference Berndtsson1] for all the details on these formulas.

Of course, the choice of the representative $\mathbf {u}$ is not unique; two such choices differ by a term of the form $v\wedge dt$ , but everything mentioned so far is independent from this choice.

Nevertheless, if u is a holomorphic section of $f_*\omega _{X/B}$ and t is a point in $\Delta $ , by [Reference Berndtsson1], [Reference Berndtsson2], there is a preferred choice of $\mathbf {u}$ such that $\mu _t$ is holomorphic (i.e., $P(\mu _t)=\mu _t$ ) and $\eta _t$ is primitive. The key advantage of this choice is that we get $(D'u)_t=\mu _t dt$ and $-c_n\int _{X_t}\eta _t\wedge \bar {\eta }_t=\lVert \eta _t\lVert ^2$ . This choice of representative allows to compute the Chern curvature formula for the Hermitian metric as follows:

(9) $$ \begin{align} \langle\Theta u_t,u_t \rangle_t=\lVert\eta_t\lVert^2. \end{align} $$

See [Reference Griffiths and Tu15] for the original proof. Berndtsson in [Reference Berndtsson2, §3] proves that $\eta _t$ is the unique harmonic representative of the cohomology class $\xi _t\cup u_t\in H^{n-1,1}(X_t)$ , hence we can also write this formula as

(10) $$ \begin{align} \langle\Theta u_t,u_t \rangle_t=\lVert\xi_t\cup u_t\lVert^2. \end{align} $$

In the case of $\mathcal {P}^{n-i,i}$ , the situation is similar, but we have to be a little more careful since a section $[u]$ is a function that maps t to a primitive cohomology class of $(n-i,i)$ -forms on the corresponding fiber $X_t$ . For this part, we follow [Reference Berndtsson, Păun and Wang3].

We denote by $u_{t,h}$ the harmonic representative of this class. By a fundamental result of Kodaira–Spencer, the variation of $u_{t,h}$ with respect to t is smooth. The norm is

(11) $$ \begin{align} \lVert[u_t]\lVert^2_t=(-1)^ic_n\int_{X_t} u_{t,h}\wedge \bar{u}_{t,h}. \end{align} $$

As in the $(n,0)$ case, we consider a representative $\mathbf {u}$ of $[u]$ , that is, a smooth $(n-i,i)$ -form on $f^{-1}(\Delta )$ which is $\bar {\partial }$ -closed on the fibers of f and whose restriction to each $X_t$ belongs to the cohomology class $[u_t]$ . In this case, the preferred choice of such a representative is given by the so-called vertical representative.

The vertical representative is unique, see [Reference Berndtsson, Păun and Wang3, Proposition 5.1]; we consider $\mathbf {u}$ the vertical representative of $[u]$ . Similarly as before, we write

$$ \begin{align*}\bar{\partial}\mathbf{u}=\nu\wedge d\bar{t}+\eta\wedge dt \end{align*} $$

and

$$ \begin{align*}\partial\mathbf{u}=\zeta \wedge d\bar{t}+\mu\wedge dt. \end{align*} $$

Then, $\nu $ and $\mu $ are relative forms of bidegree $(n-i,i)$ , $\eta $ is of bidegree $(n-i-1,i+1),$ and $\zeta $ is of bidegree $(n-i+1,i-1)$ ; they are primitive on the fibers (see [Reference Berndtsson, Păun and Wang3, §7]). As before, we can explicitly compute the Chern connection as

(12) $$ \begin{align} (D"[u])_t=[\nu_t] d\bar{t} \end{align} $$

and

(13) $$ \begin{align} (D'[u])_t=[\mu_t] dt. \end{align} $$

As a representative of the Kodaira–Spencer class $\xi _t$ , we choose the (0,1)-form with coefficients in the tangent space $T_X$ given by $\bar {\partial }V$ , where V is the horizontal lift of the $(1,0)$ vector $\partial /\partial t$ on $\Delta $ . That is, on $X_t$ , we denote $k_t:=\bar {\partial }V_{|X_t}$ and $\xi _t=[k_t]\in H^1(X_t,T_{X_t})$ . It turns out that with this choice

(14) $$ \begin{align} \eta_t=k_t\cup u_t \end{align} $$

and

(15) $$ \begin{align} \zeta_t=\bar{k}_t\cup u_t \end{align} $$

if $u_t$ is harmonic.

Remark 2.5. In terms of cohomology classes, this means that

$$ \begin{align*}[\eta_t]=\xi_t\cup [u_t] \end{align*} $$

and

$$ \begin{align*}[\zeta_t]=\overline{(\xi_t\cup\overline{[u_t]})} \end{align*} $$

since the contraction with $\bar {k}_t$ corresponds to the composition

$$ \begin{align*}H^{p,q}\cong H^{q,p}\xrightarrow{\xi_t\cup\ } H^{q-1,p+1}\cong H^{p+1,q-1}, \end{align*} $$

where the isomorphisms are given by complex conjugation.

The curvature formula generalizing (9) is

(16) $$ \begin{align} \langle\Theta [u_t],[u_t] \rangle_t=\lVert\eta_{t,h}\lVert^2-\lVert\zeta_{t,h}\lVert^2, \end{align} $$

where by $\eta _{t,h}$ and $\zeta _{t,h}$ , we mean the harmonic part of $\eta _t$ and $\zeta _t$ , respectively (see [Reference Berndtsson, Păun and Wang3]).

One of the immediate consequences of the curvature formulas (9) and (16) is that the bundles $K^i_{\text {prim}}:=K^i\cap \mathcal {P}^{n-i,i}$ are semi-negatively curved since by definition they are the primitive part of the kernel of the cup product with the Kodaira–Spencer class. For example, if u is a section of $K^0_{\text {prim}}=K^0$ , then by (9), we have that $\langle \Theta u_t,u_t \rangle _t=0$ and for $[u],$ a section of $K^i_{\text {prim}}$ , $i\geq 1$ , by (16), we have that $\langle \Theta [u_t],[u_t] \rangle _t=-\lVert \zeta _{t,h}\lVert ^2\leq 0$ . In both cases, since curvature decreases in subbundles, we have that $K^i_{\text {prim}}$ is semi-negatively curved on $B\setminus E$ for $i\geq 0$ .

Actually, there is an extension across the singularities. Indeed, $\mathcal {P}^{n-i,i}$ and $K^i_{\text {prim}}$ are firstly defined on $B\setminus E$ , but since they are subsheaves of $R^if_*\Omega ^{n-i}_{X/B}(\log )$ , which is defined over the whole base B, they have a natural extension across E. Furthermore, by [Reference Berndtsson, Păun and Wang3, §9], the chosen smooth Hermitian metric on $B\setminus E$ extends to a singular Hermitian metric. This singular metric is semi-negatively curved on $K^i_{\text {prim}}$ in the sense of singular metrics.

2.3. Metric interpretation

The Chern connection D on the vector bundle $f_*\omega _{X/B}$ is compatible with the Gauss–Manin connection. This is shown, for example, in [Reference Berndtsson2, §2.3]. Here, we rephrase it using the objects introduced in the beginning of this section. Using the same notation as above, we denote by $\Theta $ the Chern curvature of $f_*\omega _{X/B}$ .

3.1. Strongly non-isotrivial fibrations

We give the following definition.

Definition 3.1. We denote by $\phi _i\colon f_*\omega _{X/B}\to \omega _B(E)^{\otimes i}\otimes R^if_*\Omega ^{n-i}_{X/B}(\log )$ the composition of the connecting morphisms obtained from the direct images of (3):

$$ \begin{align*} \phi_i\colon f_*\omega_{X/B}\to \omega_B(E)\otimes R^1f_*\Omega^{n-1}_{X/B}(\log)\to \dots\to \omega_B(E)^{\otimes i}\otimes R^if_*\Omega^{n-i}_{X/B}(\log). \end{align*} $$

Similarly, we denote by $\varphi _i\colon f_*\omega _{X/B}\otimes T_B(-E)^{\otimes i}\to R^if_*\Omega ^{n-i}_{X/B}(\log )$ the morphism obtained from $\phi _i$ by tensoring with $T_B(-E)^{\otimes i}$ .

In the literature, they are referred to as the Griffiths–Yukawa coupling or as iterated Higgs maps.

These morphisms are given on smooth fibers by iterated cup product with the Kodaira–Spencer class. From the Introduction, we recall that if $\phi _n$ is not zero (equivalently, $\varphi _n$ is not zero), then f is strongly non-isotrivial. By the Fujita decomposition (5), $ f_*\omega _{X/B}=\mathcal {U}\oplus \mathcal {A}$ and, since $\mathcal {U}$ is contained in $K^0=\ker \phi _1$ , we look at the image of $\mathcal {A}$ .

In particular, note that if $\mathcal {A}$ is not contained in $K^0$ (i.e., $\mathcal {A}\neq 0$ ), and also $\phi _i(\mathcal {A})$ is not contained in $K^i\otimes \omega _B(E)^{\otimes i}$ , for $i=1,\dots , n-1$ , then the image $\phi _n(\mathcal {A})$ is not zero and f is strongly non-isotrivial. Actually, in our setting, we are dealing with primitive forms. In fact, the holomorphic $(n,0)$ -forms on the fibers are primitive and hence the images $\phi _i(\mathcal {A})$ are bundles of primitive classes, since the cup product of a primitive form with the Kodaira–Spencer class is again primitive.

3.1.1. Curvature conditions for strong non-isotriviality

A possible condition that ensures that $\phi _i(\mathcal {A})$ is not contained in $K^i_{\text {prim}}\otimes \omega _B(E)^{\otimes i}$ , $i=1,\dots , n-1$ , since $\mathcal {A}$ is ample, is that $K^i_{\text {prim}}\otimes \omega _B(E)^{\otimes i}$ induces a metric with semi-negative curvature on $\phi _i(\mathcal {A})\cap (K^i_{\text {prim}}\otimes \omega _B(E)^{\otimes i})$ .

We have seen in Section 2 how to define a smooth metric on $\mathcal {P}^{n-i,i}|_{B\setminus E}$ . Given a metric on $\omega _{B}(E)$ of local weight $\psi $ and curvature denoted by $c(\psi )$ , we consider the tensor metric on the bundle $\mathcal {P}^{n-i,i}\otimes \omega _{B}(E)^{\otimes i}|_{B\setminus E}$ . The following result gives a curvature formula for this metric. We write an element of $\mathcal {P}^{n-i,i}\otimes \omega _{B}(E)^{\otimes i}$ at the point t as $[u_t]\otimes l_t$ , where $l_t$ is of norm 1 and $u_t$ is harmonic on $X_t$ .

Proposition 3.2. The curvature $\Theta _i$ of $\mathcal {P}^{n-i,i}\otimes \omega _{B}(E)^{\otimes i}|_{B\setminus E}$ satisfies

(17) $$ \begin{align} \langle\Theta_i ([u_t]\otimes l_t),[u_t]\otimes l_t\rangle=\lVert\eta_{t,h}\lVert^2-\lVert\zeta_{t,h}\lVert^2+ic(\psi)\lVert [u_t]\lVert^2, \end{align} $$

where $\eta _{t,h}$ and $\zeta _{t,h}$ are defined as in (16).

Proof. This is the formula for the curvature of a tensor product. Call for simplicity $F=\mathcal {P}^{n-i,i}$ and $L= \omega _{B}(E)^{\otimes i}$ . Then, $\Theta _{F\otimes L}=\Theta _F\otimes I_L+I_F\otimes \Theta _L$ and

$$ \begin{align*}\langle\Theta_{F\otimes L} [u_t]\otimes l_t,[u_t]\otimes l_t\rangle=\langle\Theta_F [u_t],[u_t]\rangle\lVert l_t\lVert^2+\langle\Theta_L l_t,l_t\rangle\lVert [u_t]\lVert^2 \end{align*} $$

and we are done using that $\lVert l_t\lVert =1$ and Equation (16).

We prove the following proposition.

Proposition 3.3. Assume that there exists a smooth metric on $\omega _B(E)$ of local weight $\psi $ such that on $B\setminus E,$ we have $\lVert (\bar {k}_t\cup u_t)_h\lVert ^2\geq ic(\psi )\lVert [u_t]\lVert ^2$ for every section $[u]$ of $\varphi ^i(\mathcal {A}\otimes T_B(-E)^{\otimes i})\cap K^i_{\text {prim}}$ . Then, $\phi _i(\mathcal {A})\cap (K^i_{\text {prim}}\otimes \omega _B(E)^{\otimes i})$ admits a semi-negative singular Hermitian metric.

Proof. As recalled in Section 2, by [Reference Berndtsson, Păun and Wang3], $K^i_{\text {prim}}$ has a semi-negatively curved singular Hermitian metric. We consider on $K^{i}_{\text {prim}}\otimes \omega _B(E)^{\otimes i}$ the singular metric given by the tensor product and on $\phi _i(\mathcal {A})\cap (K^i_{\text {prim}}\otimes \omega _B(E)^{\otimes i})$ its restriction.

On $B\setminus E$ , these metrics are smooth and coincide with the restriction of the metric on $\mathcal {P}^{n-i,i}\otimes \omega _{B}(E)^{\otimes i}|_{B\setminus E}$ discussed above.

For the sections of $K^{i}_{\text {prim}}\otimes \omega _B(E)^{\otimes i}$ , we have by (15) and (17),

(18) $$ \begin{align} \langle\Theta_i ([u_t]\otimes l_t),[u_t]\otimes l_t\rangle=-\lVert\zeta_{t,h}\lVert^2+ic(\psi)\lVert [u_t]\lVert^2=-\lVert(\bar{k}_t\cup u_t)_h\lVert^2+ic(\psi)\lVert [u_t]\lVert^2. \end{align} $$

By our hypothesis, the metric on $\phi _i(\mathcal {A})\cap (K^i_{\text {prim}}\otimes \omega _B(E)^{\otimes i})$ is then semi-negatively curved on $B\setminus E$ , since curvature decreases in subbundles.

By the results in [Reference Berndtsson, Păun and Wang3], it is not difficult to see that it is semi-negative everywhere in the sense of singular metrics.

So, under the hypotheses of Proposition 3.3, the image $\phi _i(\mathcal {A})$ is not contained in $K^i_{\text {prim}}\otimes \omega _B(E)^{\otimes i}$ , $i=1,\dots , n-1$ , since $\mathcal {A}$ is ample. Hence, we conclude the following theorem.

Theorem 3.4. Let $f\colon X\to B$ be a semistable fibration and assume that there exists a smooth metric on $\omega _B(E)$ of local weight $\psi $ such that on $B\setminus E,$ we have $\lVert (\bar {k}_t\cup u_t)_h\lVert ^2\geq ic(\psi )\lVert [u_t]\lVert ^2$ for every section $[u]$ of $\varphi ^i(\mathcal {A}\otimes T_B(-E)^{\otimes i})\cap K^i_{\text {prim}}$ , $i=1,\dots ,n-1$ . Then, either $f_*\omega _{X/B}$ is unitary flat or the fibration is strongly non-isotrivial.

Note that a parallel approach is given by the study of the semi-positivity of $\mathcal {A}\otimes T_B(-E)^{\otimes i}$ for $i=1,\dots ,n-1$ . Similarly as before, this condition shows that $\varphi _i (\mathcal {A}\otimes T_B(-E)^{\otimes i})$ is not contained in $K^i_{\text {prim}}$ for $i=1,\dots ,n-1$ and hence $\varphi _n$ is not zero and f is strongly non-isotrivial (or $\mathcal {A}$ is zero). We skip the computations since they are similar to the previous ones. We note that we only need the semi-positivity of $\mathcal {A}\otimes T_B(-E)^{\otimes n-1}$ since it implies the semi-positivity of $\mathcal {A}\otimes T_B(-E)^{\otimes i}$ , $i=1,\dots ,n-2$ . This semi-positivity is given on $B\setminus E$ by the inequality $\lVert ({k}_t\cup u_t)_h\lVert ^2\geq (n-1)c(\psi )\lVert u_t\lVert ^2$ .

Theorem 3.5. Let $f\colon X\to B$ be a semistable fibration on a smooth projective curve B and assume that there exists a smooth metric on $\omega _B(E)$ of local weight $\psi $ such that on $B\setminus E,$ we have $\lVert ({k}_t\cup u_t)_h\lVert ^2\geq (n-1)c(\psi )\lVert u_t\lVert ^2$ for all u in $\mathcal {A}$ . Then, either $f_*\omega _{X/B}$ is unitary flat or the fibration is strongly non-isotrivial.

Remark 3.6. In principle, there is no correlation between the inequalities of Theorems 3.4 and 3.5, except when $n=2$ . In fact, in this case, note that the condition $\lVert ({k}_t\cup u_t)_h\lVert ^2\geq c(\psi )\lVert u_t\lVert ^2$ of Theorem 3.5 together with the inequality $\lVert (\bar {k}_t\cup ({k}_t\cup u_t)_h)_h\lVert ^2\leq \lVert ({k}_t\cup u_t)_h\lVert \lVert u_t\lVert $ coming from Cauchy formula together with the fact that $k_t$ and $\bar {k_t}$ are adjoint (see Proposition 3.7 below), give the condition $\lVert (\bar {k}_t\cup ({k}_t\cup u_t)_h)_h\lVert ^2\geq c(\psi )\lVert ({k}_t\cup u_t)_h\lVert ^2$ . This is the condition required in Theorem 3.4 since ${k}_t\cup u_t$ gives, while $u_t$ varies in $\mathcal {A}$ , all the elements in $\varphi ^i(\mathcal {A}\otimes T_B(-E)^{\otimes i})\cap K^i_{\text {prim}}$ .

As a final note, everything in this section can actually be generalized for fibrations over a higher-dimensional base. In fact, even when B is not a curve, by [Reference Catanese and Kawamata6], the direct image $f_*\omega _{X/B}$ has a decomposition $f_*\omega _{X/B}=\mathcal {U}\oplus \hat {\mathcal {A,}}$ where $\mathcal {U}$ is unitary flat and $\hat {\mathcal {A}}$ is generically ample, that is, $\hat {\mathcal {A}}$ restricts to an ample vector bundle on the general complete intersection smooth curve in B. Hence, it is possible to consider the generalization of the morphism $\phi _i$ and $\varphi _i$ as $\phi _i\colon f_*\omega _{X/B}\to \text {Sym}^i\Omega ^1_B(\log E)\otimes R^if_*\Omega ^{n-i}_{X/B}(\log )$ and $\varphi _i\colon f_*\omega _{X/B}\otimes \text {Sym}^iT_B(-\log E)\to R^if_*\Omega ^{n-i}_{X/B}(\log )$ and argue similarly to obtain similar conditions involving the curvature of the symmetric bundles.

3.1.2. A case on surfaces

Now, consider again the composition $\phi _n$

$$ \begin{align*} \phi_n\colon f_*\omega_{X/B}\to \omega_B(E)\otimes R^1f_*\Omega^{n-1}_{X/B}(\log)\to\dots\to \omega_B(E)^{\otimes n}\otimes R^nf_*\mathcal{O}_{X} \end{align*} $$

and restrict it to the general smooth fiber $X_t$ as

(19) $$ \begin{align} \phi_{n,t}\colon H^{n,0}(X_t)\to T_{B,t}^\vee\otimes H^{n-1,1}(X_t)\to\dots\to (T_{B,t}^\vee)^{\otimes n}\otimes H^{0,n}(X_t). \end{align} $$

Since $\dim B=1$ , choosing a local coordinate around t, we identify $T_{B,t}^\vee \cong \mathbb {C}$ and, with a little abuse of notation, we denote by $\xi _t\cup $ every map

$$ \begin{align*}\xi_t\cup \colon H^{p,q}(X_t)\to H^{p-1,q+1}(X_t) \end{align*} $$

since they are all defined by the cup product with the Kodaira–Spencer class $\xi _t$ .

Proposition 3.7. Let $[u]\in H^{p,q}_{\text {prim}}(X_t)$ and $[v]\in H^{p-1,q+1}_{\text {prim}}(X_t)$ , $p+q=n$ . Then,

$$ \begin{align*}\langle\xi_t\cup [u],[v]\rangle=\langle [u],\overline{\xi_t\cup \overline{[v]}}\rangle. \end{align*} $$

Proof. We choose harmonic representatives $u_h$ and $v_h$ of the classes $[u]$ and $[v]$ , respectively. Also, we represent the class $\xi _t$ by its representative $k_t$ as above.

We have

$$ \begin{align*}\langle\xi_t\cup [u],[v]\rangle &=(-1)^{q+1}c_n\int_{X_t}(\xi_t\cup [u])_h\wedge\bar{v}_h \\ &=(-1)^{q+1}c_n\int_{X_t}(k_t\cup u_h)_h\wedge\bar{v}_h=(-1)^{q+1}c_n\int_{X_t}( k_t\cup u_h)\wedge\bar{v}_h \end{align*} $$

and

$$ \begin{align*}\langle [u],\overline{\xi_t\cup \overline{[v]}}\rangle &=(-1)^{q}c_n\int_{X_t} u_h\wedge( {\xi_t\cup \overline{[v]}})_h\\ &=(-1)^{q}c_n\int_{X_t} u_h\wedge({k_t\cup \bar{v}_h})_h=(-1)^{q}c_n\int_{X_t} u_h\wedge({k_t\cup \bar{v}_h}). \end{align*} $$

It is now easy to see, for example, writing locally $u_h,v_h$ , and $k_t$ , that $(-1)^{q+1}c_n\int _{X_t}( k_t\cup u_h)\wedge \bar {v}_h=(-1)^{q}c_n\int _{X_t} u_h\wedge ({k_t\cup \bar {v}_h})$ . This can, for example, be verified on decomposable forms, since everything is linear.

Remark 3.8. From this formula, we easily get the condition $K^q_{\text {prim},t}\subset \overline {\xi _t(H^{q+1,p-1}_{\text {prim}}(X_t))}^\perp $ .

Denote by $\xi ^q_t$ the composition of the cup product with $\xi _t$ taken q times,

$$ \begin{align*}\xi_t^q\colon H^{n,0}(X_t)\to H^{n-q,q}(X_t). \end{align*} $$

Hence, we have the following condition for strong non-isotriviality.

Proposition 3.9. If $\xi _t^{n-1}(H^{n,0}(X_t))\cap \overline {\xi _t(H^{n,0}(X_t))}\neq \{0\}$ , then the fibration is strongly non-isotrivial.

Proof. This comes from the fact that all the elements of $K^{n-1}_{\text {prim},t}$ are contained in $\overline {\xi _t(H^{n,0}_{\text {prim}}(X_t))}^\perp $ by the previous remark. Hence, $\phi _{n,t}=\xi ^n_t\neq 0$ .

In the case of $n=2$ , this condition suggests an interesting numerical bound concerning families of surfaces with positive index.

Theorem 3.10. Let $f\colon X\to B$ be a semistable fibration and denote by r the rank of $K^0$ . Assume that the general fiber $X_t$ is a surface satisfying $K^2_{X_t}> 8\chi (\mathcal {O}_{X_t})+2r+1$ , then f is strongly non-isotrivial.

Proof. By Proposition 3.9, it is enough to show that the intersection $\xi _t(H^{2,0}(X_t))\cap \overline {\xi _t(H^{2,0}(X_t))}$ is nontrivial in $H^{1,1}_{\text {prim}}(X_t)$ . This is true if $\dim \xi _t(H^{2,0}(X_t))>h^{1,1}_{\text {prim}}/2$ , that is, $p_g-r>h^{1,1}_{\text {prim}}/2$ . Now since $c_2=2-4q+2p_g+h^{1,1}$ and $h^{1,1}_{\text {prim}}\leq h^{1,1}-1$ , by Noether’s formula $12\chi (\mathcal {O}_{X_t})=K_{X_t}^2+c_2$ , the above condition is implied by

$$ \begin{align*}2p_g-2r>-2p_g+4q-2+12\chi(\mathcal{O}_{X_t})-K_{X_t}^2-1, \end{align*} $$

which we can write as

$$ \begin{align*}K^2_{X_t}> 8\chi(\mathcal{O}_{X_t})+2r+1. \end{align*} $$

Remark 3.11. Note that since $9\chi (\mathcal {O}_{X_t})\leq K^2_{X_t}$ by the Miyaoka–Yau inequality [Reference Miyaoka25], [Reference Yau43], our assumption can hold only if $r\leq \frac {p_g-q}{2}$ .

3.2. Conditions for triviality of a fibration

In this section, we look at the opposite problem and give conditions for the triviality of $f\colon X\to B$ over a Zariski open subset of B. So far, we have considered the direct image of the relative dualizing sheaf $f_*\omega _{X/B}$ , here, we will work on the direct image of the relative pluricanonical bundles $f_*(\omega _{X/B}^{\otimes m})$ . Hence, we briefly recall how to construct a Hermitian metric on the vector bundle associated with $f_*(\omega _{X/B}\otimes L)$ , where L is a line bundle on X endowed with a singular metric assumed to be smooth when restricted to the general fiber. We denote by $\psi $ the local weight of this metric and assume that L is semi-positively curved.

Without covering all the details, we just highlight the differences with the untwisted case of Section 2. In particular, on an appropriate open subset of B, the metric in (6) is replaced by

(20) $$ \begin{align} \lVert u_t\lVert^2_t=\int_{X_t} c_n u_t\wedge\bar{u}_te^{-\psi} \end{align} $$

and the curvature in (9) is replaced by

(21) $$ \begin{align} \langle\Theta u_t,u_t \rangle_t=f_*(c_ni\partial\bar{\partial}\psi\wedge\mathbf{u}\wedge\bar{\mathbf{u}}e^{-\psi})/dV_t+\lVert\eta_t\lVert^2, \end{align} $$

where $dV_t=idt\wedge d\bar {t}$ .

For our purposes, we take $L=\omega _{X/B}^{\otimes {m-1}}$ and we prove the following theorem which complements the constancy result proved in [Reference Xie and Yuan42].

Proof. By [Reference Iwai18], [Reference Lombardi and Schnell23], the direct image $f_*(\omega _{X/B}^{\otimes m})$ has a so-called Catanese–Fujita–Kawamata decomposition

$$ \begin{align*}f_*(\omega_{X/B}^{\otimes m})=\mathcal{U}\oplus \mathcal{A}, \end{align*} $$

where $\mathcal {U}$ is unitary flat and $\mathcal {A}$ is ample (or zero). On the other hand, we also get an injective morphism $f_*(\omega _{X/B}^{\otimes m})\hookrightarrow p_*((\Omega ^n_{V\times B/B})^{\otimes m})=H^0(V, (\Omega ^n_V)^{\otimes m})\otimes \mathcal {O}_B$ . From this, it is not difficult to see that $\mathcal {A}=0$ and $\mathcal {U}=\mathcal {O}_B^l$ is trivial.

Now, we write $f_*(\omega _{X/B}^{\otimes m})=f_*(\omega _{X/B}\otimes \omega _{X/B}^{\otimes m-1})$ and take $L=\omega _{X/B}^{\otimes m-1}$ . L admits a singular metric which is smooth and with strictly positive curvature when restricted to the smooth fibers (cf. [Reference Schumacher37]). By [Reference Păun and Takayama27, Theorem 3.3.5], the vector bundle $f_*(\omega _{X/B}^{\otimes m})$ admits a singular Hermitian metric with semi-positive curvature. But by the fact that $f_*(\omega _{X/B}^{\otimes m})$ is trivial, the curvature of this metric cannot be strictly positive definite; actually, it is zero, see, for example, [Reference Hacon, Popa and Schnell16, Lemma 13.2]. On the open subset where f is smooth, this metric coincides with the one recalled above in (20) and by the explicit curvature formula given in [Reference Berndtsson2, Theorem 1.2] and following remarks, the Kodaira–Spencer class vanishes at every point of this open subset.

Moreover, by the argument in [Reference Berndtsson2, §4.1, p. 1216], the Kodaira–Spencer class $\xi _t$ is represented by the form $\bar {\partial }{V_\psi }_{|X_t}$ , where $V_\psi $ is a (in principle only smooth) vector field which turns out to be holomorphic since the curvature of $f_*(\omega _{X/B}^{\otimes m})$ is not strictly positive. The fibration f is then analytically locally trivial.

To show that f is trivial over a Zariski open subset U of B, we can proceed then as in [Reference Xie and Yuan42, p. 22].

4.1. A curvature formula on Massey products

In this section, we consider a disk $\Delta \subset B\setminus E$ . Using the same notation as above, take sections $\eta _1,\dots ,\eta _{n+1}\in \Gamma (\Delta ,K_{\partial })$ and $\alpha $ their Massey product. The notion of Massey triviality, see Definition 4.2, means that $\alpha $ is a section of the $\mathcal {O}_B$ -submodule $\mathcal {W}=\langle \alpha _{1},\dots ,\alpha _{n+1}\rangle \subseteq {f_*\omega _{X/B}}_{|\Delta }$ , hence it is natural to study the quotient $\mathcal {Q}:={f_*\omega _{X/B}}_{|\Delta }/\mathcal {W}$ . Indeed, the class $[\alpha ]$ does not depend on the choice of liftings $s_i$ of $\eta _i$ , while the section $\alpha $ does. Following this idea, we assume that $\mathcal {W}$ is a non-trivial subbundle of ${f_*\omega _{X/B}}_{|\Delta }$ and we compute the curvature of $\mathcal {Q}$ on the section $[\alpha ]$ and relate it to the cup product with the Kodaira–Spencer class and the pointwise liftability of $\alpha $ . The result is a version of Formula (10), the difference is that working on the quotient $\mathcal {Q}$ makes the key role of the local system $\mathbb {D}^1$ more clear.

In this section, let $\mathcal {E}:={f_*\omega _{X/B}}_{|\Delta }$ and, to avoid confusion, here we denote by $\langle ,\rangle _{\mathcal {E}}$ its metric introduced in Section 2 and by $\langle ,\rangle _{\mathcal {Q}}$ the quotient metric on $\mathcal {Q}$ .

By the above discussion, the case to be studied is when $\alpha _t$ is not contained in the vector space $\mathcal {W}_t=\langle \alpha _{1,t},\dots ,\alpha _{n+1,t}\rangle $ . Without loss of generality, we assume that, by a standard orthogonalization process, $\alpha _{1,t},\dots ,\alpha _{m,t},\alpha _t$ , $m\leq n+1$ , are a unitary basis of $\langle \mathcal {W}_t,\alpha _t\rangle $ (note that $\alpha _{1,t},\dots ,\alpha _{n+1,t}$ are not necessarily linearly independent). We can extend to a $\mathcal {C}^\infty $ unitary frame of $\mathcal {E}$ , denoted by $\alpha ^{\prime }1,\dots ,\alpha ^{\prime }{m},\alpha ',\tau _1,\dots ,\tau _k$ , such that $\alpha ^{\prime }1,\dots ,\alpha ^{\prime }{m}$ is a smooth frame for $\mathcal {W}$ , $\alpha ',\tau _1,\dots ,\tau _k$ is a smooth frame for $\mathcal {W}^\perp \cong \mathcal {Q}$ , and $\alpha ^{\prime }1,\dots ,\alpha ^{\prime }{m},\alpha '$ restrict to $\alpha _{1,t},\dots ,\alpha _{m,t},\alpha _t$ on $X_t$ .

We compute $\langle \Theta _{\mathcal {Q}} [\alpha _t],[\alpha _t]\rangle _{\mathcal {Q}}$ by means of the second fundamental form $\mathfrak {S}$ of $\mathcal {Q}$ in $\mathcal {E}$ . Following the notation of [Reference Griffiths and Harris14], the second fundamental form $\mathfrak {S}\colon \mathcal {A}^0(\mathcal {Q})\to \mathcal {A}^1(\mathcal {W})$ is defined as the composition of the connection D on $\mathcal {E}$ restricted to $\mathcal {Q}\cong \mathcal {W}^\perp $

$$ \begin{align*}D_{|\mathcal{Q}}\colon \mathcal{A}^0(\mathcal{Q})\to \mathcal{A}^1(\mathcal{E}) \end{align*} $$

and the projection to $\mathcal {W}$

$$ \begin{align*}\mathcal{A}^1(\mathcal{E}) \to \mathcal{A}^1(\mathcal{W}). \end{align*} $$

We have

$$ \begin{align*}\langle\Theta_{\mathcal{Q}} [\alpha_t],[\alpha_t]\rangle_{\mathcal{Q}}=\langle\Theta_{\mathcal{E}} \alpha_t,\alpha_t\rangle_{\mathcal{E}}-\langle \mathfrak{S} \alpha_t,\mathfrak{S}\alpha_t\rangle_{\mathcal{E}}. \end{align*} $$

The first of these summands is given by (10), that is,

$$ \begin{align*}\langle\Theta_{\mathcal{E}} \alpha_t,\alpha_t\rangle_{\mathcal{E}}=\lVert\xi_t\cup {\alpha}_t\lVert^2 \end{align*} $$

hence we are left with the second summand.

In our chosen unitary frame, we write $\mathfrak {S}\alpha _t=\lambda _1\alpha _{1,t}+\dots +\lambda _{m}\alpha _{m,t}$ , for certain $(0,1)$ -forms $\lambda _i$ , [Reference Kobayashi20, Proposition 1.6.6]. Explicitly, since $\mathcal {W}$ is a holomorphic subbundle of $\mathcal {E}$ ,

$$ \begin{align*}\lambda_i=\langle(D\alpha')_t,\alpha_{i,t}\rangle_{\mathcal{E}}=-\langle\alpha_t,(D\alpha^{\prime}{i})_t\rangle_{\mathcal{E}}=-\left(\int_{X_b}c_n\alpha_t\wedge \overline{P(\mu^{\prime}i)}\right)d\bar t, \end{align*} $$

where we use the same notation as Section 2, Equation (8).

Proposition 4.5. The following formula computes the curvature of $\mathcal {Q}$ on the class $[\alpha _t]$ :

(24) $$ \begin{align} \langle\Theta_{\mathcal{Q}} [\alpha_t],[\alpha_t]\rangle_{\mathcal{Q}}=\lVert\xi_t\cup {\alpha}_t\lVert^2+\sum_i\Big|\int_{X_b}c_n\alpha_t\wedge \overline{P(\mu^{\prime}i)}\Big|^2. \end{align} $$

As pointed out above, the case where the sections $\eta _{i}$ are in the local system $\mathbb {D}^1$ is of particular interest. The following corollary indeed specifies what happens in this case.

Another interesting point not fully analyzed in the literature is when the Massey product is a section of $\mathcal {A}$ , the ample part of the Fujita decomposition. In this case, we can relax the assumption on the $\eta _{i}$ and obtain the same conclusion.

The above corollaries give a metric interpretation of the liftability of Massey products. Based on our experience concerning the case of trivial Massey products, we expect these results to be useful in the future.