2.1 $\varepsilon $ for divisors

Let V be a smooth variety of dimension $n> 0$ , and let L be a ${\mathbb {Q}}$ -divisor on V. For any big divisor M on V with $|M|$ base point free, take the smallest integer $\lambda _M> 0$ so that the divisor $\lambda _M M - L$ is pseudo-effective. When $n \ge 2$ , we define

$$ \begin{align*}\varepsilon(V, L, M):=(\lambda_M + 1)^{n-1} M^n. \end{align*} $$

When $n=1$ , we simply set

$$ \begin{align*}\varepsilon(V, L, M) = 1. \end{align*} $$

For any $n> 0$ , define

$$ \begin{align*}\varepsilon(V, L) := \inf_M \varepsilon(V, L, M), \end{align*} $$

where the infimum is taken over all divisors M on V chosen as above. In particular, when $n=1$ , we have

$$ \begin{align*}\varepsilon(V, L) = 1. \end{align*} $$

It is straightforward to check that

2.2 A Clifford type inequality

The main result in this section is the following one, which will be used later in the proof of Theorem 1.2.

Theorem 2.2. Let V be a smooth variety of dimension $n> 0$ . Suppose that L is a ${\mathbb {Q}}$ -divisor on V such that $K_V-L$ is pseudo-effective. Then

$$ \begin{align*} h^0(V, {\lfloor L \rfloor}) \le \frac{1}{2n!} {\mathrm{Vol}} (L) + n \varepsilon(V, L). \end{align*} $$

Proof. By [Reference Yuan and Zhang24, Theorem 1.2], which was stated only for integral divisors, we have

$$ \begin{align*} h^0(V, {\lfloor L \rfloor}) \le \frac{1}{2n!} {\mathrm{Vol}} ({\lfloor L \rfloor}) + n \varepsilon(V, {\lfloor L \rfloor}). \end{align*} $$

Note that ${\mathrm {Vol}}({\lfloor L \rfloor }) \le {\mathrm {Vol}}(L)$ , and by Proposition 2.1, $\varepsilon (V, {\lfloor L \rfloor }) \le \varepsilon (V, L)$ . Thus the result follows easily.

3.1 A filtration for nef divisors

Let $f: V \to B$ be a fibration from a smooth variety V of dimension n to a smooth curve B with a general fibre F. Let L be a nef divisor on V. We first recall the following theorem.

Theorem 3.1 [Reference Zhang26, Theorem 4.1]

Let $f: V \to B$ , F and L be as above. Then there is a birational morphism $\sigma : V_L \to V$ and a sequence of triples

$$ \begin{align*} \{(L_i, Z_i, a_i) | i=0, 1, \cdots, N\} \end{align*} $$

on $V_L$ with the following properties:

• ○ $(L_0, Z_0, a_0)=(\sigma ^*L, 0, {\mathrm {int}}_{f_L}(L_0))$ , where $f_L: V_L \stackrel {\sigma }{\to } V \stackrel {f}{\to } B$ is the induced fibration.

• ○ For any $i=0, \cdots , N-1$ , there is a decomposition

  $$ \begin{align*} |L_i-a_iF_L|= |L_{i+1}| + Z_{i+1} \end{align*} $$
  such that $Z_{i+1} \ge 0$ is the fixed part of $|L_i-a_iF_L|$ and the movable part $|L_{i+1}|$ of $|L_i-a_iF_L|$ is base point free. Here $F_L = \sigma ^*F$ denotes a general fibre of $f_L$ , and $a_{i} = {\mathrm {int}}_{f_L}(L_{i})$ .

• ○ We have $h^0(V_L, L_N-a_NF_L)=0$ .

In the above theorem, for any $0 \le i \le N$ , the number ${\mathrm {int}}_{f_L}(L_i)$ is defined by

$$ \begin{align*}{\mathrm{int}}_{f_L}(L_i) := \min \{a \in {\mathbb{Z}}| L_i - aF_L \,\mbox{is not nef}\}. \end{align*} $$

Thus via Theorem 3.1, we obtain a filtration

$$ \begin{align*}\sigma^* L = L_0> L_1 > \cdots > L_N \ge 0 \end{align*} $$

of nef divisors on a birational model $V_L$ of V. For simplicity, we denote by F a general fibre of $f_L: V_L \to B$ in the rest of this section.

Proposition 3.2 [Reference Yuan and Zhang24, Proposition 2.2]

We have the following two inequalities:

$$ \begin{align*} h^0(V, L) & \le \sum_{i=0}^{N} a_i h^0(F, L_i|_F); \\ L^n & \ge n \sum_{i=0}^{N} a_i (L_i|_F)^{n-1} - n(L_0|_F)^{n-1}. \end{align*} $$

Proposition 3.3 [Reference Yuan and Zhang24, Lemma 2.3]

We have

$$ \begin{align*} L_0^n \ge \left(\sum_{i=0}^{N} a_i - 1 \right) (L_0|_F)^{n-1}. \end{align*} $$

3.2 Sharper bound involving subcanonicity

Let V be a smooth variety of dimension $n> 0$ with the Kodaira dimension $\kappa (V) \ge 0$ , and let L be a ${\mathbb {Q}}$ -divisor on V. Let M be a big divisor on V such that $|M|$ is base point free. We recall that the numerical subcanonicity of L with respect to M is defined in [Reference Barja, Pardini and Stoppino6, Definition 5.1] as follows:

$$ \begin{align*}r(L, M):= \frac{LM^{n-1}}{K_VM^{n-1}}. \end{align*} $$

When $n=1$ , set $r(L, M) = \frac {\deg L}{\deg K_V}$ . When $K_VM^{n-1} = 0$ , we have $\kappa (V) = 0$ . In this case, we set $r(L, M) = + \infty $ . Define a function $\delta $ as follows:

$$ \begin{align*}\delta(x) = \left\{ \begin{array} {ll} 2, & x \le 1; \\ \frac{2x}{2x-1}, & x> 1. \end{array} \right. \end{align*} $$

Theorem 3.4. Let L and M be as above, and write $r = r(L, M)$ . Then

$$ \begin{align*} h^0(V, {\lfloor L \rfloor}) \le \frac{1}{\delta(r)n!} {\mathrm{Vol}}(L) + n \varepsilon(V, L, M). \end{align*} $$

Proof. The proof is by induction, and we present it in several steps.

Notice that the required inequality holds trivially if $h^0(V,{\lfloor L \rfloor })=0$ . We may make assumption $h^0(V, {\lfloor L \rfloor })>0$ from now on.

Step 1: Reduce to the case when L is nef.

In fact, by replacing V by an appropriate blowing up, we may assume that

$$ \begin{align*} L = L' + Z, \end{align*} $$

where $L'$ is the movable part of $|{\lfloor L \rfloor }|$ and Z is its fixed part. It is clear that

$$ \begin{align*} r(L, M) \ge r(L', M), \quad {\mathrm{Vol}}(L) \ge {\mathrm{Vol}}(L'), \quad \varepsilon(V, L, M) \ge \varepsilon(V, L', M). \end{align*} $$

Thus it suffices to prove Theorem 3.4 for $L'$ .

From now on, we assume that L is a nef divisor.

Step 2: The $n=1$ case.

When $n=1$ , Theorem 3.4 is straightforward. If $h^1(V, L) \ne 0$ , the classical Clifford inequality implies Theorem 3.4. Otherwise, by the Riemann-Roch theorem,

$$ \begin{align*} h^0(V, L) = \deg L - \frac{1}{2} \deg K_V = \left(1 - \frac{1}{2r}\right) \deg L. \end{align*} $$

Thus the proof is completed.

Step 3: The proof when $L^n> 0$ .

Now we assume that Theorem 3.4 holds for dimension $k < n$ . Choose a general pencil in $|M|$ , and blow up the indeterminacies of this pencil, denoted by $\pi : V_0 \to V$ . We get a fibration

$$ \begin{align*} f: V_0 \to {\mathbb{P}}^1 \end{align*} $$

such that the general fibre F of f is isomorphic to a general member of the chosen pencil. By the adjunction, $\kappa (F) \ge 0$ . Write $M_0 = \pi ^*M$ and $L_0 = \pi ^*L$ . It follows that

$$ \begin{align*} r(L, M) = \frac{L_0M_0^{n-2}F}{(\pi^*K_V)M_0^{n-2}F} \ge r(L_0|_F, M_0|_F), \end{align*} $$

where the last inequality follows from the adjunction.

Apply Theorem 3.1 to f and $L_0$ . Replacing $V_0$ by a further blowing up if necessary, we get triples

$$ \begin{align*} (L_i, Z_i, a_i) \quad (i=0, \ldots, N) \end{align*} $$

on $V_0$ , and $L_i$ and $a_i$ satisfy the inequalities in Proposition 3.2 and 3.3. Note that by the definition of $r(L, M)$ , we see that

$$ \begin{align*} r(L_i|_F, M_0|_F) \le r(L_0|_F, M_0|_F) = r. \end{align*} $$

By induction and using the fact that the function $\delta $ is nonincreasing, we have

$$ \begin{align*} h^0(F, L_i|_F) \le \frac{1}{\delta(r)(n-1)!} (L_i|_F)^{n-1} + (n-1) \varepsilon(F, L_i|_F, M_0|_F). \end{align*} $$

Combine this with Proposition 3.2. It follows that

$$ \begin{align*} h^0(V_0, L_0) - \frac{1}{\delta(r)n!} L_0^n \le (n-1) \sum_{i=0}^{N} a_i\varepsilon(F, L_i|_F, M_0|_F) + \frac{1}{(n-1)!}(L_0|_F)^{n-1}. \end{align*} $$

To estimate the right-hand side of the above inequality, let $\lambda $ be the smallest integer such that $\lambda M-L$ is pseudo-effective. Note that $L^n> 0$ .

1. (1) It implies that $L^n \le \lambda L^{n-1}M = \lambda (L_0|_F)^{n-1}$ . In particular, $(L_0|_F)^{n-1}> 0$ . Thus, by Proposition 3.3,

   $$ \begin{align*} \sum_{i=0}^{N} a_i \le \frac{L_0^n}{(L_0|_F)^n} + 1 \le \lambda + 1. \end{align*} $$

2. (2) By Proposition 2.1 (1),

   $$ \begin{align*} \varepsilon(F, L_i|_F, M_0|_F) \le \varepsilon(F, L_0|_F, M_0|_F). \end{align*} $$
   Moreover, since $\lambda M_0|_F - L_0|_F$ is also pseudo-effective, we have
   $$ \begin{align*} \varepsilon(F, L_0|_F, M_0|_F) \le (\lambda+1)^{n-2}(M_0|_F)^{n-1} = (\lambda+1)^{n-2}M^n. \end{align*} $$

3. (3) We have

   $$ \begin{align*} (L_0|_F)^{n-1} = L^{n-1}M \le \lambda L^{n-2}M^2 \le \cdots \le \lambda^{n-1}M^n. \end{align*} $$

Combining all of the above inequalities, it follows that

$$ \begin{align*} h^0(V_0, L_0) - \frac{1}{\delta(r)n!} L_0^n & \le (n-1)(\lambda+1)^{n-1}M^n + \frac{1}{(n-1)!} \lambda^{n-1}M^n \\ & \le n \varepsilon(V, L, M). \end{align*} $$

Thus the proof in this case is completed.

Step 4. The proof when $L^n = 0$ .

In this case, the proof is easier. Since L is not big, we know that

$$ \begin{align*}h^0(V, L-M) = 0. \end{align*} $$

Take W to be a general member in $|M|$ , and we have

$$ \begin{align*}h^0(V, L) \le h^0(W, L|_W). \end{align*} $$

Therefore, by induction, we deduce that

$$ \begin{align*}h^0(V, L) \le \frac{1}{(n-1)!} (L|_W)^{n-1} + (n-1)\varepsilon(W, L|_W, M|_W). \end{align*} $$

Let $\lambda $ be the smallest integer such that $\lambda M - L$ is pseudo-effective. Similar to Step 3, we have

1. (1) $(L|_W)^{n-1} = L^{n-1}M \le \lambda ^{n-1}M^n$ ;

2. (2) $\varepsilon (W, L|_W, M|_W) \le (\lambda +1)^{n-2}M^n$ .

Combining the above inequalities, it follows that

$$ \begin{align*}h^0(V, L) \le \frac{1}{(n-1)!} \lambda^{n-1}M^n + (n-1) (\lambda+1)^{n-2}M^n \le n \varepsilon(V, L, M). \end{align*} $$

Thus the whole proof is completed.

3.3 Sharper bound involving the mapping degree

Let V be a smooth variety of dimension $n \ge 2$ , and let L be a ${\mathbb {Q}}$ -divisor on V such that $K_V-L$ is pseudo-effective. Instead of the subcanonicity, we suppose that

$$ \begin{align*}a: V \to \Sigma \end{align*} $$

is a generically finite morphism onto a (possibly singular) variety $\Sigma $ . Let H be a sufficiently ample divisor on $\Sigma $ , and write $M = a^*H$ . The assumption will be used until the end of this section.

3.3.1 Preparation

We first assume that V is a surface and $|L|$ is base point free. Although this assumption looks simple, all the results we need can be reduced to this setting.

Lemma 3.5. If $h^0(V, L-M) = 0$ , then

$$ \begin{align*} h^0(L) \le \frac{1}{2}LM + 1 \le \varepsilon(V, L, M). \end{align*} $$

Proof. Choose a general curve $C \in a^*|H|$ . By Bertini’s theorem, we may assume that C is smooth. The assumption $h^0(V, L-M) = 0$ just tells us that $h^0(V, L) \le h^0(C, L|_C)$ . Thus the first inequality is just a combination of the Clifford inequality and the Riemann-Roch theorem again.

The second inequality is directly from the definition of $\varepsilon $ . Let $\lambda $ be the smallest integer such that $\lambda M - L$ is pseudo-effective. Then

$$ \begin{align*} \frac{1}{2}LM + 1 \le \frac{\lambda}{2} M^2 + 1 \le (\lambda + 1) M^2 = \varepsilon(V, L, M). \end{align*} $$

The proof is completed.

Now suppose that $h^0(V, L-M)> 0$ . Let

$$ \begin{align*}\gamma : = \max\{i \in {\mathbb{Z}}| h^0(V, L-iM)> 0 \}. \end{align*} $$

Obviously, $\gamma \ge 1$ .

Lemma 3.6. If $h^0(V, L-M)> 0$ , then

$$ \begin{align*} L^2 \ge h^0(V, 2L) - h^0(V, L) - 1. \end{align*} $$

Proof. Take a general member $D \in |L|$ . By assumption, D is big. Thus we may assume that D is smooth and irreducible. Consider the following exact sequence:

$$ \begin{align*} 0 \to H^0(V, L) \to H^0(V, 2L) \to H^0(D, 2L|_D). \end{align*} $$

Since $K_V - L$ is pseudo-effective, we know that $\deg (2L|_D) \le \deg (K_V|_D + L|_D) = \deg K_D$ : that is, $K_D - 2L|_D$ is pseudo-effective. Apply the Clifford inequality (when $h^1(D, 2L|_D)> 0$ ) or the Riemann-Roch theorem (when $h^1(D, 2L|_D) = 0$ ) for $2L|_D$ , and it follows that

$$ \begin{align*} L^2 = \frac{1}{2} \deg(2L|_D) \ge h^0(D, 2L|_D) - 1 \ge h^0(V, 2L) - h^0(V, L) - 1. \end{align*} $$

The proof is completed.

Let $C \in a^*|H|$ be a general member, hence smooth. Consider the two restriction maps

$$ \begin{align*}\mathrm{res}_{1, i}: H^0(V, L-iM) \to H^0(C, L|_C - iM|_C) \end{align*} $$

and

$$ \begin{align*}\mathrm{res}_{2, j}: H^0(V, 2L-jM) \to H^0(C, 2L|_C - jM|_C). \end{align*} $$

The kernels of the above two maps are $H^0(V, L-(i+1)M)$ and $H^0(V, 2L-(j+1)M)$ , respectively.

Let $V_{1, i}$ (respectively, $V_{2, j}$ ) denote the image of $H^0(V, L-iM)$ (respectively, $H^0(V, 2L-jM)$ ) under $\mathrm {res}_{1, i}$ (respectively, $\mathrm {res}_{2, j}$ ).

Lemma 3.7. We have

$$ \begin{align*} h^0(V, L) & = \sum_{i=0}^{\gamma-1} \dim V_{1, i} + h^0(V, L-\gamma M) \\ h^0(V, 2L) & = \sum_{j=0}^{2\gamma-1} \dim V_{2, j} + h^0(V, 2L-2\gamma M) \ge 2\sum_{i=0}^{\gamma-1} \dim V_{2, 2i} - \dim V_{2, 0}. \end{align*} $$

Proof. The two equalities are obvious. The last inequality in the second formula holds simply because $h^0(V, 2L-2\gamma M)> 0$ and $\dim V_{2, 2i-1} \ge \dim V_{2, 2i}$ for any $1 \le i \le \gamma -1$ .

Let $|N_i|$ denote the movable part of $|L-iM|$ . Note that the base locus of $|N_i|$ is either empty or of dimension zero. We deduce that $N_i$ is nef. Also, we have

$$ \begin{align*}\dim V_{1, i} = \dim |N_i||_C+1. \end{align*} $$

Lemma 3.8. For $0 \le i \le \gamma $ , we have

$$ \begin{align*} \dim V_{2, 2i} \ge 2 \dim V_{1, i} - 1. \end{align*} $$

If, moreover, the linear system $|N_i||_C$ induces a birational map on C, then

$$ \begin{align*} \dim V_{2, 2i} \ge 3 (\dim V_{1, i} - 1). \end{align*} $$

Proof. This is just [Reference Barja, Pardini and Stoppino6, Lemma 5.3] for $k=2$ .

In the following, we will apply the above results to deduce more inequalities subject to the degree of the map a. The notation here will be used frequently.

3.3.2 $\deg a = 1$

We first consider the case when a is birational.

Theorem 3.9. Suppose that $\deg a = 1$ , and $K_V - L$ is pseudo-effective. Then we have

$$ \begin{align*} h^0(V, {\lfloor L \rfloor}) \le \frac{2}{5 n!} {\mathrm{Vol}}(L) + n \varepsilon(V, L, M). \end{align*} $$

Similar to the proof of Theorem 3.4, we may assume that L is nef. We may even assume that $|L|$ is base point free. Moreover, we only need to prove Theorem 3.9 when $n = 2$ (i.e., Lemma 3.10), and the general result follows by an inductive argument almost identical to Step 3 and Step 4 in the proof of Theorem 3.4.

One little difference is that, instead of choosing a general pencil in $|M|$ as in Step 3 of the proof of Theorem 3.4, here we choose a general pencil in the sublinear system $a^*|H| \subseteq |M|$ . Since $a^*|H|$ is also base point free, the smoothness of a general member in it is guaranteed by Bertini’s theorem. This adjustment will be used until the end of this section. Note that the restriction of a on a general member of $a^*|H|$ has degree one. This is the key point for us to use the induction.

With this adjustment and by Lemma 3.5, we eventually reduce Theorem 3.9 to the following lemma.

Lemma 3.10. Theorem 3.9 holds when $n=2$ , $|L|$ is base point free and $h^0(V, L-M)> 0$ .

Proof. We claim that

(3.1) $$ \begin{align} h^0(V, 2L) - 6h^0(V, L) \ge -8LM - 7. \end{align} $$

Suppose the claim holds. Together with Lemma 3.6, we deduce that

$$ \begin{align*} h^0(V, L) \le \frac{1}{5} L^2 + \frac{8}{5}(LM + 1), \end{align*} $$

and the proof will be completed by noting that

$$ \begin{align*} \frac{8}{5}(LM + 1) < 2\varepsilon(V, L, M) \end{align*} $$

just as in the proof of Lemma 3.5.

To prove the claim, let C, $\gamma $ , $V_{1, i}$ , $V_{2, j}$ be the same as in Section 3.3.1. For $0 \le i \le \gamma -1$ , $|M|$ is a sublinear system of $|L-iM|$ , which means $|M||_C$ is a sublinear system of $V_{1, i}$ . Note that $|M||_C$ induces a birational map from C. We deduce that the map induced by $V_{1, i} \ (0 \le i \le \gamma -1)$ is birational. Thus it follows from Lemma 3.7 and the second inequality in Lemma 3.8 that

$$ \begin{align*} h^0(V, 2L) - 6h^0(V, L) \ge - 6 \left(\gamma + h^0(V, L-\gamma M)\right) - \dim V_{2,0}. \end{align*} $$

Let us estimate the right-hand side of the above inequality.

1. (1) Since $h^0(V, L-(\gamma +1)M) = 0$ , by Lemma 3.5, we have

   $$ \begin{align*} h^0(V, L-\gamma M) \le \frac{1}{2}(LM - \gamma M^2) + 1 \le LM - \gamma M^2 + 1. \end{align*} $$
   In particular,
   $$ \begin{align*} h^0(V, L-\gamma M) + \gamma \le LM + 1. \end{align*} $$

2. (2) Note that $\dim V_{2, 0} \le h^0(C, 2L|_C)$ . By the Clifford inequality and the Riemann-Roch theorem, similar to before, we simply deduce that

   $$ \begin{align*} \dim V_{2, 0} \le h^0(C, 2L|_C) \le \deg(2L|_C) + 1 = 2LM + 1. \end{align*} $$

Combining the above two inequalities, we prove the claim.

3.3.3 $a$ is not composed with an involution

Second, we consider the case when a is not composed with an involution. That is, there is no generically finite map $V \dashrightarrow V'$ of degree two through which a factors birationally.

Theorem 3.11. Suppose that a is not composed with an involution and $K_V - L$ is pseudo-effective. Then we have

$$ \begin{align*} h^0(V, {\lfloor L \rfloor}) \le \frac{4}{9 n!} {\mathrm{Vol}}(L) + n \varepsilon(V, L, M). \end{align*} $$

Similar to what we did for Theorem 3.9, we may assume that $n=2$ , $|L|$ is base point free and $h^0(V, L-M)> 0$ . For general n, we use the induction. Note that by our assumption, the restriction of a on a general member of $a^*|H|$ is not composed with an involution either. See [Reference Barja, Pardini and Stoppino6, Proposition 2.8], for example. This guarantees that the inductive argument also works in this situation. Therefore, Theorem 3.11 boils down to the following lemma.

Lemma 3.12. Theorem 3.11 holds when $n=2$ , $|L|$ is base point free and $h^0(V, L-M)> 0$ .

Proof. We sketch the proof here since it is similar to that of Lemma 3.10.

Let C, $\gamma $ , $V_{1, i}$ , $V_{2, j}$ , $N_i$ be identical to those in Section 3.3.1. Let

$$ \begin{align*} i_0 = \min \{0 \le i \le \gamma-1 | V_{1, i} \,\mbox{does not induce a birational map on } C\}. \end{align*} $$

With this notation, using the same strategy as for proving equation (3.1), we deduce that

$$ \begin{align*} h^0(V, 2L) - 6h^0(V, L) \ge - 8LM - 7 - 2\sum_{i=i_0}^{\gamma-1} \dim V_{1, i}. \end{align*} $$

Comparing to the proof of equation (3.1), the only modification we make here is that, for $i \ge i_0$ , we have to use the first inequality in Lemma 3.8 to compare $\dim V_{2, 2i}$ with $\dim V_{1, i}$ , which is the reason for having an extra term $-2\sum _{i=i_0}^{\gamma -1} \dim V_{1, i}$ on the right-hand side.

Combining this inequality with Lemma 3.6, it follows that

(3.2) $$ \begin{align} L^2 \ge 5h^0(V, L) - 2\sum_{i=i_0}^{\gamma-1} \dim V_{1, i} - 8 (LM +1). \end{align} $$

On the other hand, recall that for any $0 \le i \le \gamma -1$ , $N_i$ is nef and

$$ \begin{align*} \dim V_{1, i} = \dim |N_i||_C+1. \end{align*} $$

Note that in the current setting, $N_0 = L$ and $|N_{i+1}|$ is also the movable part of $|N_i-M|$ .

For any $i> 0$ , we have

(3.3) $$ \begin{align} N_{i-1}^2 - N_i^2 \ge (N_{i-1} + N_i) M \ge 2 N_iM \ge 4 \dim V_{1, i} - 4, \end{align} $$

where the last inequality follows from the fact that $K_C - (L|_C - iM|_C)$ is pseudo-effective. When $i \ge i_0$ , $V_{1, i}$ induces a map on C of degree at least three. Otherwise, the map $\phi _{|L-iM|}$ induced by the linear system $|L-iM|$ would factor through a degree two map from V, and a would factor through $\phi _{|L-iM|}$ , which is a contradiction. Let

$$ \begin{align*} \phi_i: C \to C^{\prime}_i \end{align*} $$

be the morphism induced by the movable part of $V_{1, i}$ . Then $\deg \phi _i \ge 3$ . Since $\phi _i$ factor through the normalisation of $C^{\prime }_i$ , we may assume that the curve $C^{\prime }_i$ is normal, hence smooth. Then

$$ \begin{align*} |N_i||_C = \phi_i^*|L^{\prime}_i| + Z^{\prime}_i, \end{align*} $$

where $L^{\prime }_i$ and $Z^{\prime }_i$ are effective divisors on $C'$ . Since

$$ \begin{align*} \dim V_{1, i} \, \le \, h^0(C^{\prime}_i, L^{\prime}_i) \le \deg L^{\prime}_i + 1 \le \frac{1}{\deg \phi_i} N_iM + 1, \end{align*} $$

similar to equation (3.3), we deduce that for $i \ge \max \{1, i_0\}$ ,

(3.4) $$ \begin{align} N_{i-1}^2 - N_i^2 \ge 2N_i M \ge 6 \dim V_{1, i} - 6. \end{align} $$

Note that we also have

$$ \begin{align*} \dim V_{1, 0} \le \left\{ \begin{array} {ll} \frac{1}{2}LM + 1, & i_0> 0; \\ & \\ \frac{1}{3}LM + 1, & i_0 = 0. \end{array} \right. \end{align*} $$

Together with equations (3.3) and (3.4) for all $i> 0$ , we deduce that

(3.5) $$ \begin{align} L^2 & = \sum_{i=1}^{\gamma-1} (N_{i-1}^2 - N_i^2) + N_{\gamma-1}^2 \nonumber \\ & \ge 4 \sum_{i=0}^{i_0-1} \dim V_{1, i} + 6 \sum_{i=i_0}^{\gamma-1} \dim V_{1, i} - 2LM - 6\gamma + N_{\gamma-1}^2 \nonumber\\ & \ge 4h^0(V, L) + 2 \sum_{i=i_0}^{\gamma-1} \dim V_{1, i} - 4h^0(V, L-\gamma M) - 2LM - 6\gamma \nonumber \\ & \ge 4h^0(V, L) + 2 \sum_{i=i_0}^{\gamma-1} \dim V_{1, i} - 4LM - 6 \gamma - 4. \end{align} $$

The third inequality here is due to Lemma 3.7. For the last inequality, by Lemma 3.5 and the definition of $\gamma $ , we have

$$ \begin{align*} h^0(V, L-\gamma M) \le \frac{1}{2} (LM - \gamma M^2) + 1. \end{align*} $$

Then it is easy to deduce that

$$ \begin{align*} 4h^0(V, L-\gamma M) \le 2LM - 2\gamma M^2 + 4. \end{align*} $$

Thus equation (3.5) is verified.

Now, adding equations (3.2) and (3.5), it follows that

$$ \begin{align*} 2L^2 \ge 9h^0(V, L) - 12 LM - 6\gamma - 12, \end{align*} $$

that is,

$$ \begin{align*} h^0(V, L) \le \frac{2}{9} L^2 + \frac{4}{3} LM + \frac{2}{3} \gamma + \frac{4}{3}. \end{align*} $$

Finally, let $\lambda $ be the smallest integer such that $\lambda M-L$ is pseudo-effective. Noting that $\gamma \le \lambda $ , we deduce that

$$ \begin{align*} \frac{4}{3} LM + \frac{2}{3} \gamma + \frac{4}{3} \le \frac{4}{3} \lambda M^2 + \frac{2}{3} \lambda + \frac{4}{3} \le 2(\lambda+1)M^2 = 2 \varepsilon(V, L, M). \end{align*} $$

Thus the whole proof of this lemma is completed.

3.3.4 $a$ is composed with an involution and $\kappa (\Sigma )> 0$

Finally, we consider the case when a is composed with an involution and $\Sigma $ is birational to a smooth projective variety of positive Kodaira dimension. Let $\pi : \Sigma ' \to \Sigma $ be a resolution of singularities of $\Sigma $ . Then $\kappa (\Sigma ')> 0$ . Set

$$ \begin{align*}r'(L, M, \Sigma') := \frac{LM^{n-1}}{2K_{\Sigma'}(\pi^*H)^{n-1}.} \end{align*} $$

By the assumption, $K_{\Sigma '}(\pi ^*H)^{n-1}> 0$ . Thus $r'(L, M, \Sigma ') < \infty $ .

Theorem 3.13. Let the notation be as above. Write $r' = r'(L, M, \Sigma ')$ . Suppose that $K_V-L$ is pseudo-effective. Then we have

$$ \begin{align*}h^0(V, {\lfloor L \rfloor}) \le \frac{2 \delta(r') - 1}{(5 \delta(r') - 3) n!} {\mathrm{Vol}}(L) + n \varepsilon(V, L, M). \end{align*} $$

Moreover, for any ${\mathbb {Q}}$ -divisor $L_1 \le L$ , we have

$$ \begin{align*}h^0(V, {\lfloor L_1 \rfloor}) \le \frac{2 \delta(r') - 1}{(5 \delta(r') - 3) n!} {\mathrm{Vol}}(L_1) + n \varepsilon(V, L, M). \end{align*} $$

Here the function $\delta (x)$ is the same as that in Theorem 3.4. Note that under this setting, $\delta (r')> 1$ . Moreover, since $r^{\prime }_1:=r'(L_1, M, \Sigma ') \le r'$ , we have $\delta (r^{\prime }_1) \ge \delta (r')$ and $\frac {2 \delta (r^{\prime }_1) - 1}{5 \delta (r^{\prime }_1) - 3} \le \frac {2 \delta (r') - 1}{5 \delta (r') - 3}$ . Therefore, the second inequality in Theorem 3.13 can be deduced from the first one for $L_1$ .

Note that the restriction of a on a general member of $a^*|H|$ is composed with an involution. Furthermore, by the adjunction, a smooth model of a general member of $|H|$ has positive Kodaira dimension. Thus the induction method works here, and Theorem 3.13 is finally reduced to the following result.

Lemma 3.14. Theorem 3.13 holds when $n=2$ , $|L|$ is base point free and $h^0(V, L-M)> 0$ .

Proof. The proof is just a modification of the proof of Lemma 3.12. We sketch it and leave the details to the interested reader.

Let C, $\gamma $ , $V_{1, i}$ , $V_{2, j}$ , $N_i$ and $i_0$ be identical to those in the proof of Lemma 3.12. Then it is easy to see that equation (3.2) still holds here: that is,

(3.6) $$ \begin{align} L^2 \ge 5h^0(V, L) - 2\sum_{i=i_0}^{\gamma-1} \dim V_{1, i} - 8 (LM +1). \end{align} $$

For any $1 \le i \le \gamma -1$ , equation (3.3) also holds here: that is,

(3.7) $$ \begin{align} \dim V_{1, i} \le \frac{1}{4} (N_{i-1}^2 - N_i^2) + 1. \end{align} $$

The major modification is a replacement of equation (3.4). For $i_0 \le i \le \gamma - 1$ , $V_{1, i}$ induces a map on C of degree at least two. Let $\phi _i: C \to C^{\prime }_i$ , $L^{\prime }_i$ and $Z^{\prime }_i$ be as in the proof of Lemma 3.12. We may further assume that the curve $C^{\prime }_i$ is normal. By Theorem 3.4 and the fact that $\deg \phi _i \ge 2$ , we deduce that

$$ \begin{align*} \dim V_{1, i} \, \le \, h^0(C^{\prime}_i, L^{\prime}_i) \le \frac{1}{\delta(r^{\prime}_i)} \deg L^{\prime}_i + 1 \le \frac{1}{2\delta(r^{\prime}_i)} N_iM + 1, \end{align*} $$

where $r^{\prime }_i = \frac {\deg L^{\prime }_i}{\deg K_{C^{\prime }_i}}$ . Now we claim that

$$ \begin{align*} \delta(r^{\prime}_i) \ge \delta(r') \end{align*} $$

for any $i \ge i_0$ as above. With this claim, we deduce that for $i \ge \max \{1, i_0\}$ ,

(3.8) $$ \begin{align} \dim V_{1, i} \le \frac{1}{4\delta(r')} (N_{i-1}^2 - N_i^2) + 1. \end{align} $$

To prove the claim, we only need to prove that $r^{\prime }_i \le r'$ . Since we already have $\deg L^{\prime }_i \le \frac {1}{2} LM$ as above, it suffices to prove that $\deg K_{C^{\prime }_i} \ge K_{\Sigma '}(\pi ^*H)$ . This is rather obvious. The key is to note that $a|_C$ factors through $\phi _i$ . Via this factorisation, $C^{\prime }_i$ maps to a general curve in $|H|$ on $\Sigma $ . Since $\pi ^*|H|$ is base point free, by Bertini’s theorem, a general member of $\pi ^*|H|$ is smooth. Moreover, the aforementioned map on $C^{\prime }_i$ lifts to a map from $C^{\prime }_i$ to a general member $C''\in \pi ^*|H|$ . Therefore, by the Hurwitz formula and the adjunction formula,

$$ \begin{align*} \deg K_{C^{\prime}_i} \ge \deg K_{C''} = K_{\Sigma'}(\pi^*H) + (\pi^*H)^2> K_{\Sigma'}(\pi^*H). \end{align*} $$

Thus the claim is verified, and equation (3.8) is established.

Having the above modification, we can proceed the proof as before. Sum up equations (3.7) and (3.8) over all the above $i> 0$ . Note that

$$ \begin{align*} \dim V_{1, 0} \le \left\{ \begin{array} {ll} \frac{1}{2}LM + 1, & i_0> 0; \\ & \\ \frac{1}{2 \delta(r')}LM + 1, & i_0 = 0. \end{array} \right. \end{align*} $$

It follows that

$$ \begin{align*} L^2 & \ge 4 \sum_{i=0}^{i_0-1} \dim V_{1, i} + 4\delta(r') \sum_{i=i_0}^{\gamma-1} \dim V_{1, i} - 2LM - 4 \delta(r')\gamma + N_{\gamma-1}^2 \\ & \ge 4h^0(V, L) + 4(\delta(r') - 1) \sum_{i=i_0}^{\gamma-1} \dim V_{1, i} - 4h^0(V, L-\gamma M) \\ & \quad - 2LM - 4 \delta(r') \gamma. \end{align*} $$

Using the argument for proving equation (3.5), we can similarly deduce that

$$ \begin{align*} 4h^0(V, L-\gamma M) + 2LM + 4 \delta(r') \gamma \le 4LM + 4 \delta(r')\gamma + 4. \end{align*} $$

The above two inequalities imply that

(3.9) $$ \begin{align} L^2 \ge 4h^0(V, L) + 4(\delta(r') - 1) \sum_{i=i_0}^{\gamma-1} \dim V_{1, i} - 4LM - 4 \delta(r') \gamma - 4. \end{align} $$

For simplicity, we write $\delta = \delta (r')$ . As before, we use equations (3.6) and (3.9) to eliminate $\sum _{i=i_0}^{\gamma -1} \dim V_{1, i}$ . It follows that

$$ \begin{align*} (2 \delta- 1) L^2 \ge (10 \delta- 6) h^0(V, L) - (16 \delta - 12)LM - 4 \delta \gamma - (16 \delta - 12), \end{align*} $$

that is,

$$ \begin{align*} h^0(V, L) \le \frac{2 \delta - 1}{10 \delta- 6} L^2 + \frac{8 \delta - 6}{5 \delta - 3} LM + \frac{2 \delta}{5 \delta - 3} \gamma + \frac{8 \delta - 6}{5 \delta - 3}. \end{align*} $$

Since $1 < \delta \le 2$ , it is straightforward to check that the above inequality implies that

(3.10) $$ \begin{align} h^0(V, L) \le \frac{2 \delta - 1}{10 \delta - 6} L^2 + \frac{10}{7}LM + \gamma + \frac{10}{7}. \end{align} $$

Once again, let $\lambda $ be the smallest integer such that $\lambda M - L$ is pseudo-effective. Since $M^2 = (\deg a)H^2 \ge 2$ and $\gamma \le \lambda $ , we deduce that

$$ \begin{align*} \frac{10}{7} LM + \gamma + \frac{10}{7} \le \frac{10}{7} \lambda M^2 + \frac{1}{2} \lambda M^2 + \frac{10}{7} < 2(\lambda+1)M^2 = 2 \varepsilon(V, L, M). \end{align*} $$

Thus the whole proof is completed.

4.1 $\chi _f$ equals the degree of a twisted Hodge bundle

The following result relates $\chi _f$ to the degree of a twisted Hodge bundle.

Proposition 4.1. With the above notation, we have

$$ \begin{align*} \chi_f = \deg f_*(\omega_{X/B} \otimes {\mathcal{P}}), \end{align*} $$

where ${\mathcal {P}}$ is a general torsion element in ${\mathrm {Pic}}^0(X)$ .Footnote ⁴

Proof. This result has been proved by Hacon and Pardini [Reference Hacon and Pardini12, Theorem 2.4] assuming $g(B) \ge 2$ . In fact, this assumption can be removed. Here we give a slightly different proof that works for any curve B.

By the assumption, $a|_F: F \to A$ is generically finite onto its image. Let ${\mathcal {P}} \in {\mathrm {Pic}}^0(X)$ be a general torsion element. Applying exactly the proof of [Reference Hacon and Pardini12, Corollary 2.3], we conclude that $f_* (\omega _{X/B} \otimes {\mathcal {P}})$ is torsion free, hence a locally free sheaf on B of rank $r = \chi (F, \omega _F)$ . Still, by [Reference Hacon and Pardini12, Corollary 2.3], for any $i> 0$ ,

$$ \begin{align*} R^i f_* (\omega_{X/B} \otimes {\mathcal{P}}) = 0. \end{align*} $$

Together with the Leray spectral sequence, we know that for any $i \ge 0$ ,

$$ \begin{align*} h^i(X, \omega_X \otimes {\mathcal{P}}) = h^i(B, f_* (\omega_X \otimes {\mathcal{P}}) ). \end{align*} $$

In particular,

$$ \begin{align*} \chi(X, \omega_X) = \chi(X, \omega_X \otimes {\mathcal{P}}) = \chi(B, f_*(\omega_X \otimes {\mathcal{P}})). \end{align*} $$

Combine all of the above, and apply the Riemann-Roch theorem for $f_* (\omega _X \otimes {\mathcal {P}})$ . It follows that

$$ \begin{align*} \deg f_*(\omega_{X/B} \otimes {\mathcal{P}}) & = \deg f_*(\omega_X \otimes {\mathcal{P}}) - 2 \chi(F, \omega_F) \chi(B, \omega_B) \\ & = \chi (B, f_* (\omega_X \otimes {\mathcal{P}}) ) - \chi(F, \omega_F) \chi(B, \omega_B) \\\phantom{\deg f_*(\omega_{X/B} \otimes {\mathcal{P}})} & = \chi(X, \omega_X) - \chi(F, \omega_F) \chi(B, \omega_B) \\ & = \chi_f. \end{align*} $$

Thus the proof is completed.

4.2 The degree of the Hodge bundle under étale covers

In this subsection, we assume that $g> 0$ . Thus X itself is of maximal Albanese dimension.

Let $\mu _m: A \to A$ be the multiplication-by-m map of A. Let $X_m = X \times _{\mu _m} A$ . Since a is the Albanese map, $X_m$ is irreducible. Let $J(B)$ be the Jacobian variety of B. By the abuse of notation, let $\mu _m: J(B) \to J(B)$ also denote the multiplication-by-m map of $J(B)$ , and let $B_m=B \times _{\mu _m} J(B)$ . Thus we have the following commutative diagram:

Now we claim that if m is a sufficiently large prime number, the morphism

$$ \begin{align*}f_m: X_m \to B_m \end{align*} $$

is always a fibration: that is, it has connected fibres. To see this, let $A_F = \ker h$ , which is also an abelian variety. We may assume that up to a translation by a point in $J(B)$ , $a(F)$ generates $A_F$ . Thus the kernel of the map $(a|_F)^*: {\mathrm {Pic}}^0(A_F) \to {\mathrm {Pic}}^0(F)$ is finite. Thus for any integer m coprime to the cardinality of this kernel, the general fibre of $f_m$ is irreducible.

Proposition 4.2. With the above notation, we have

$$ \begin{align*} \lim\limits_{m \, \mathrm{prime}, m \to \infty} \frac{\deg {f_m}_* \omega_{X_m/B_m}}{m^{2q}} = \chi_f. \end{align*} $$

Proof. From the above construction, we know that for any $m> 0$ , the morphism $\sigma _m: B_m \to B$ is étale. By the projection formula,

$$ \begin{align*} {\sigma_m}_* {\mathcal{O}}_{B_m} = \bigoplus_{\mathcal{P} \in T_m(B)} {\mathcal{P}}, \end{align*} $$

where $T_m(B) \subset {\mathrm {Pic}}^0(B)$ is the subgroup of all m-torsion line bundles on B. There is a natural injective group homomorphism

$$ \begin{align*} f^*: T_m(B) \to T_m(X) \end{align*} $$

given by the pull-back of f, where $T_m(X) \subset {\mathrm {Pic}}^0(X)$ is the subgroup of all m-torsion line bundles on X. Let m be a sufficiently large prime number, and let $X^{\prime }_m = X \times _B B_m$ . Then we have the following commutative diagram:

It is clear that $\nu ^{\prime }_m: X_m \to X^{\prime }_m$ is a Galois cover with ${\mathrm {Gal}}(\nu ^{\prime }_m) \simeq \frac {T_m(X)}{f^*T_m(B)}$ . Thus, by the projection formula,

$$ \begin{align*}{\nu^{\prime}_m}_* \omega_{X_m/B_m} = \bigoplus_{{\mathcal{Q}} + f^*T_m(B)} \omega_{X^{\prime}_m/B_m} \otimes (\sigma^{\prime}_m)^*{\mathcal{Q}}. \end{align*} $$

Here the summation runs over all cosets of $f^*T_m(B)$ in $T_m(X)$ (whose cardinality equals $m^{2q-2g}$ ), and ${\mathcal {Q}}$ is any representative in each corresponding coset. Thus we have the following splitting:

$$ \begin{align*} {f_m}_* \omega_{X_m/B_m} = {f^{\prime}_m}_* \left( {\nu^{\prime}_m}_* \omega_{X_m/B_m} \right) = \bigoplus_{{\mathcal{Q}} + f^*T_m(B)} \sigma_m^*(f_* (\omega_{X/B} \otimes {\mathcal{Q}})). \end{align*} $$

All the above imply particularly that

$$ \begin{align*} \deg {f_m}_* \omega_{X_m/B_m} & = \deg {\sigma_m}_* \left({f_m}_* \omega_{X_m/B_m} \right) \\ & = \deg \left({\sigma_m}_* \left( {f_m}_* \omega_{X_m} \right) \otimes \omega_B^{-1} \right) \\ & = \deg \left( f_* \left({\nu_m}_* \omega_{X_m} \right) \otimes \omega_B^{-1} \right). \end{align*} $$

On the other hand, by the projection formula,

$$ \begin{align*} {\nu_m}_* \omega_{X_m} = \bigoplus_{{\mathcal{P}} \in T_m(X)} \omega_X \otimes {\mathcal{P}}. \end{align*} $$

Thus it follows that

$$ \begin{align*} \deg {f_m}_* \omega_{X_m/B_m} = \sum_{{\mathcal{P}} \in T_m(X)} \deg f_* ( \omega_{X/B} \otimes {\mathcal{P}}). \end{align*} $$

Let $S_m(X) = \{{\mathcal {P}} \in T_m(X) | \deg f_*(\omega _{X/B} \otimes {\mathcal {P}}) = \chi _f \}$ be the subset of $T_m(X)$ . By Proposition 4.1, we know that the set

$$ \begin{align*} \bigcup_{m \in {\mathbb{Z}}} \left(T_m(X) \backslash S_m(X)\right) \end{align*} $$

is contained in a proper subvariety of ${\mathrm {Pic}}^0(X)$ . In particular,

$$ \begin{align*} \lim\limits_{m \to \infty} \frac{\# S_m(X)}{\# T_m(X)} = \lim\limits_{m \to \infty} \frac{\# S_m(X)}{m^{2q}} = 1. \end{align*} $$

Note that $\deg f_*(\omega _{X/B} \otimes {\mathcal {P}})$ is always nonnegative (e.g., see [Reference Hacon and Pardini12]) and bounded from above independent of m. We deduce that

$$ \begin{align*} \lim\limits_{m \to \infty} \frac{\deg {f_m}_* \omega_{X_m/B_m}}{m^{2q}} = \chi_f. \end{align*} $$

Thus the proof is completed.

5.1 Xiao’s method

Here we review Xiao’s method and list some inequalities deduced from it. Most of the following facts can be found in [Reference Xiao23] when $n=2$ and in [Reference Ohno19, Reference Konno15, Reference Barja and Stoppino4] for general $n \ge 2$ .

Let L be a nef ${\mathbb {Q}}$ -divisor on X. Let

$$ \begin{align*}0= {\mathcal{E}}_0 \subsetneq {\mathcal{E}}_1 \subsetneq \cdots \subsetneq {\mathcal{E}}_m = f_* {\mathcal{O}}_X({\lfloor L \rfloor}) \end{align*} $$

be the Harder-Narasimhan filtration of $f_* {\mathcal {O}}_X({\lfloor L \rfloor })$ . For any $0 \le i \le m$ , set

$$ \begin{align*}r_i= {\mathrm{rank}} {\mathcal{E}}_i, \quad \mu_i =\frac{\deg({\mathcal{E}}_i/{\mathcal{E}}_{i-1})}{{\mathrm{rank}} ({\mathcal{E}}_i / {\mathcal{E}}_{i-1})}. \end{align*} $$

Then we have

$$ \begin{align*}\mu_1> \mu_2 > \cdots > \mu_m \end{align*} $$

as well as

(5.1) $$ \begin{align} \deg {\mathcal{E}}_k = \sum_{i=1}^{k-1} r_i (\mu_i - \mu_{i+1}) +r_k \mu_{k} \end{align} $$

for each $1\le k\le m$ .

For each $1 \le i \le m$ , consider the rational map $\phi _i: X \dashrightarrow \mathbb {P}_B(\mathcal {E}_i)$ associated to the evaluation morphism $f^*{\mathcal {E}}_i \to {\mathcal {O}}_X({\lfloor L \rfloor })$ . We may choose a common blowing up $\sigma : Y \to X$ , which resolves all indeterminacies of $\phi _i$ . Denote by $F_1$ a general fibre of $f \circ \sigma : Y \to B$ . Applying Xiao’s method, we obtain a sequence of nef $\mathbb {Q}$ -Cartier divisors

$$ \begin{align*}N_1\le N_2 \le \cdots \le N_m \le N_{m+1}:= \sigma^*L \end{align*} $$

on Y. Here $N_i = (\phi _i \circ \sigma )^*H_{{\mathcal {E}}_i} - \mu _iF_1$ , where $H_{{\mathcal {E}}_i}$ is a hyperplane section of ${\mathbb {P}}_{B}({\mathcal {E}}_i)$ . For each $1 \le i \le m$ , $N_i|_{F_1}$ is Cartier, $h^0(F_1, N_i|_{F_1})=r_i$ ,

$$ \begin{align*}N_{i+1}\ge N_i+(\mu_i-\mu_{i+1})F_1, \end{align*} $$

and

$$ \begin{align*}\sigma^*L\ge N_i+\mu_i F_1. \end{align*} $$

In particular, $\sigma ^*L - \mu _1 F_1$ is pseudo-effective, and for $1\le i\le m-1$ , we have

$$ \begin{align*} N_{i+1}^n\ge (N_i+(\mu_i-\mu_{i+1})F_1)^n\ge N_i^n+n(\mu_i-\mu_{i+1})(N_i|_{F_1})^{n-1}. \end{align*} $$

Thus the following lemma follows easily by induction.

Lemma 5.1. Keep the same notation as above. Suppose that for some $1\le i\le m$ , we have $\mu _i\ge 0$ . Let $k:=\max \{i|\ 1\le i\le m \ \text {and}\ \mu _i \ge 0 \}$ . Then we have

$$ \begin{align*} L^n\ge n\sum_{i=1}^{k-1}(\mu_i-\mu_{i+1})(N_i|_{F_1})^{n-1}+n\mu_k(N_k|_{F_1})^{n-1}. \end{align*} $$

Proof. Inductively using the above estimate, we have

$$ \begin{align*} N_k^n \ge N_1^n + n\sum_{i=1}^{k-1}(\mu_i-\mu_{i+1})(N_i|_{F_1})^{n-1}\ge n\sum_{i=1}^{k-1}(\mu_i-\mu_{i+1})(N_i|_{F_1})^{n-1}. \end{align*} $$

The last inequality holds since $N_1$ is nef. Notice that $\sigma ^*L\ge N_k+\mu _k F_1$ and $\mu _k\ge 0$ . We have

$$ \begin{align*} L^n \ge (N_k+\mu_k F_1)^n = N_k^n+n\mu_k (N_k|_{F_1})^{n-1}. \end{align*} $$

Thus the proof is completed by combining the above estimates.

5.2 A basic slope inequality

We have the following result.

Proposition 5.2. Let $f: X \to B$ and F be as before. Suppose that L is a nef ${\mathbb {Q}}$ -divisor on X such that $L|_F$ is big and $K_F - L|_F$ is pseudo-effective. Then we have

$$ \begin{align*} \left(1 + \frac{2n!(n-1) \varepsilon(F, L|_F)}{(L|_F)^{n-1}}\right) L^n \ge 2n! \deg f_* {\mathcal{O}}_X({\lfloor L \rfloor}). \end{align*} $$

Proof. The inequality holds trivially when $\deg f_*{\mathcal {O}}_X({\lfloor L \rfloor })\le 0$ . Thus we may assume that $\deg f_*{\mathcal {O}}_X({\lfloor L \rfloor })>0$ .

Let

$$ \begin{align*} 0= {\mathcal{E}}_0 \subsetneq {\mathcal{E}}_1 \subsetneq \cdots \subsetneq {\mathcal{E}}_m = f_* {\mathcal{O}}_X({\lfloor L \rfloor}) \end{align*} $$

be the Harder-Narasimhan filtration of $f_* {\mathcal {O}}_X({\lfloor L \rfloor })$ . Keep the same notation as in Section 5.1. Since $\deg f_*{\mathcal {O}}_X({\lfloor L \rfloor })>0$ , we have $\mu _i>0$ for some $1\le i\le m$ . Let $k:=\max \{i|\ 1 \le i \le m \ \text {and}\ \mu _i\ge 0 \}$ . We have

(5.2) $$ \begin{align} \deg {\mathcal{E}}_k\ge \deg f_*{\mathcal{O}}_X({\lfloor L \rfloor}). \end{align} $$

By equation (5.1) and Lemma 5.1, we have the following two inequalities:

$$ \begin{align*} L^n & \ge n \sum_{i=1}^{k-1} (\mu_i-\mu_{i+1})(N_i|_{F_1})^{n-1}+n\mu_k(N_k|_{F_1})^{n-1}, \\ \deg {\mathcal{E}}_k & = \sum_{i=1}^{k-1} r_i (\mu_i - \mu_{i+1})+r_k\mu_k. \end{align*} $$

On the other hand, note that $N_i|_{F_1} \le \sigma ^*L|_{F_1}$ for any $1 \le i \le m$ and $K_{F_1} -\sigma ^*L|_{F_1} \ge \sigma ^*(K_F - L|_F)$ is pseudo-effective. By Theorem 2.2 and Proposition 2.1, we have

(5.3) $$ \begin{align} r_i \le \frac{1}{2(n-1)!} (N_i|_{F_1})^{n-1} + (n-1) \varepsilon(F, L|_F). \end{align} $$

Combine the above three (in)equalities. We deduce that

$$ \begin{align*} L^n & \ge 2n! \deg {\mathcal{E}}_k - 2n!(n-1) \varepsilon(F, L|_F) (\sum_{i=1}^{k-1} (\mu_i - \mu_{i+1})+\mu_{k}) \\ & = 2n! \deg {\mathcal{E}}_k - 2n!(n-1) \varepsilon(F, L|_F) \mu_1\\ &\ge 2n!\deg f_*{\mathcal{O}}_X({\lfloor L \rfloor})-2n!(n-1) \varepsilon(F, L|_F) \mu_1, \end{align*} $$

where the last inequality follows by equation (5.2).

What is left to us is to estimate $\mu _1$ . Note that $\sigma ^*L - \mu _1 F_1$ is pseudo-effective. Thus

$$ \begin{align*} L^n = (\sigma^*L)^n \ge \mu_1 (\sigma^*L|_{F_1})^{n-1} = \mu_1 (L|_F)^{n-1}. \end{align*} $$

As a result, we deduce that

$$ \begin{align*} \left(1 + \frac{2n!(n-1) \varepsilon(F, L|_F)}{(L|_F)^{n-1}}\right) L^n \ge 2n! \deg f_* {\mathcal{O}}_X({\lfloor L \rfloor}). \end{align*} $$

Thus the proof is completed.

Before going further, we would like to remark that the inequality in Proposition 5.2 is by no means sharp. For example, when $n=2$ , f is a relatively minimal fibration by curves of genus $g \ge 2$ , and $L = K_{X/B}$ (in this case $\varepsilon (F, L|_F) = 1$ ), Proposition 5.2 yields

$$ \begin{align*}K_{X/B}^2 \ge \left(\frac{4g-4}{g+1}\right) \deg f_* \omega_{X/B}, \end{align*} $$

which is weaker than the optimal slope inequality with the slope $\frac {4g-4}{g}$ . This is because our estimate is not as delicate as Xiao’s original version in [Reference Xiao23], which also considers the intersection number contributed by the horizontal part $N_i|_{F_1}-N_{i+1}|_{F_1}$ . See the proof of [Reference Xiao23, Lemma 2] for details. In other words, we have not employed Xiao’s method in its full strength. However, Proposition 5.2 is already enough to deduce Theorem 1.2. Moreover, instead of using Theorem 1.2, Proposition 5.2 is sufficient for us to run the argument as in [Reference Barja and Stoppino4, Proposition 4.4] to deduce the absolute Severi inequality.

5.3 Sharper slope inequalities

In the following, we assume that

$$ \begin{align*}a: F \to \Sigma \end{align*} $$

is a generically finite map onto a projective variety $\Sigma $ . Let H be a sufficiently ample divisor on $\Sigma $ . Let $M = a^*H$ .

6.1 Preparation when $g(B)>0$

Before proving the results, we list some notations that will be used throughout the section. We first assume that $g(B)>0$ . Note that in this case, X itself is of maximal Albanese dimension.

Let $\pi : Y \to X$ be a resolution of singularities of X. Thus Y is also of maximal Albanese dimension. Let

$$ \begin{align*}f': = f \circ \pi: Y \to B \end{align*} $$

be the induced fibration with a general fibre $F'$ , and let

$$ \begin{align*}b: Y \to A \end{align*} $$

be the Albanese map of Y.

Let m be a sufficiently large prime number. Similar to Section 4.2 but adding Y to it, we have the following commutative diagram:

Here $\mu _m$ still denotes the multiplication-by-m map of A or $J(B)$ , the Jacobian variety of B, $X_m$ and $f_m$ are identical to those in Section 4.2, $Y_m = Y \times _{\mu _m} A$ and

$$ \begin{align*}f^{\prime}_m: Y_m \to B_m \end{align*} $$

is the Stein factorisation of the morphism $Y_m \to Y \to B$ . Clearly, $X_m$ has at worst terminal singularities, and $\pi _m: Y_m \to X_m$ is also a resolution of singularities of $X_m$ . Denote by $F^{\prime }_m$ a general fibre of $f^{\prime }_m$ . Moreover, we will fix a sufficiently ample divisor H on A. By [Reference Birkenhake and Lange7, Proposition 2.3.5],

(6.1) $$ \begin{align} m^2H \equiv \mu_m^*H. \end{align} $$

6.2 Proof of Theorem 1.2

We divide the proof into two cases.

6.2.1 Case I: $g(B)> 0$

We first prove Theorem 1.2 when $g:=g(B)> 0$ .

If F is not of general type, neither is $F'$ . In this case, for a general torsion element ${\mathcal {P}} \in {\mathrm {Pic}}^0(Y)$ , $f_* (\omega _{Y/B} \otimes {\mathcal {P}})$ is of rank $\chi (F', \omega _{F'}) = 0$ . We deduce that $f_* (\omega _{Y/B} \otimes {\mathcal {P}}) = 0$ . By Proposition 4.1, $\chi _f = \chi _{f'} = 0$ . Thus equation (1.1) holds trivially.

From now on, we will always assume that F is of general type. Set

$$ \begin{align*}L:=\pi^*K_{X/B}, \quad L_m:= \nu_m^*L = \pi_m^* K_{X_m/B_m}. \end{align*} $$

Clearly, $L_m$ is nef, and $L_m|_{F^{\prime }_m}$ is big. Since X has at worst terminal singularities, $K_Y - \pi ^*K_X$ is effective. Thus $K_{F^{\prime }_m} - L_m|_{F^{\prime }_m}$ is pseudo-effective. Moreover, since

$$ \begin{align*}{f^{\prime}_m}_* {\mathcal{O}}_{Y_m}({\lfloor L_m \rfloor}) = {f_m}_* \omega_{X_m/B_m} = {f^{\prime}_m}_* \omega_{Y_m/B_m}, \end{align*} $$

by [Reference Fujita11, Main Theorem], we deduce that ${f^{\prime }_m}_* {\mathcal {O}}_{Y_m}({\lfloor L_m \rfloor })$ is semi-positive.

Since $\deg \nu _m = m^{2q}$ , we have

(6.2) $$ \begin{align} L_m^n = m^{2q}L^n. \end{align} $$

There is a natural restriction morphism

$$ \begin{align*}\nu_m|_{F^{\prime}_m}: F^{\prime}_m \to F'. \end{align*} $$

It is an étale morphism and $\deg \nu _m|_{F^{\prime }_m} = m^{2q-2g}$ . Therefore, we deduce that

(6.3) $$ \begin{align} (L_m|_{F^{\prime}_m})^{n-1} = m^{2q-2g} (L|_{F'})^{n-1} = m^{2q-2g} K_F^{n-1}. \end{align} $$

Moreover, we claim that

(6.4) $$ \begin{align} \varepsilon(F^{\prime}_m, L_m|_{F^{\prime}_m}) \sim O(m^{2q-2g-2}). \end{align} $$

In fact, we may assume that $b^*H-L$ is pseudo-effective. By equation (6.1), $m^2(b_m^* H)-L_m$ is also pseudo-effective. Thus

$$ \begin{align*} \varepsilon(F^{\prime}_m, L_m|_{F^{\prime}_m}, (b^*_m H)|_{F^{\prime}_m}) & \le (m^2+1)^{n-2}\left((b_m^*H)|_{F^{\prime}_m}\right)^{n-1} \\ & \le \frac{2^{n-2}}{m^2} \left((b_m^*(\mu_m^*H))|_{F^{\prime}_m}\right)^{n-1} \\ & = \frac{2^{n-2}}{m^2} \left((\nu_m^*(b^*H)))|_{F^{\prime}_m}\right)^{n-1} \\ & = 2^{n-2} m^{2q-2g-2} \left((b^*H)|_{F'}\right)^{n-1}. \end{align*} $$

Thus the claim is verified.

Now, applying Proposition 5.2 to $f^{\prime }_m$ and $L_m$ , we deduce that

(6.5) $$ \begin{align} \left(1 + \frac{2n!(n-1) \varepsilon(F^{\prime}_m, L_m|_{F^{\prime}_m})} {(L_m|_{F^{\prime}_m})^{n-1}} \right) L_m^n \ge 2n! \deg {f^{\prime}_m}_* {\mathcal{O}}_{Y_m}({\lfloor L_m \rfloor}). \end{align} $$

Recall that

$$ \begin{align*}{f^{\prime}_m}_* {\mathcal{O}}_{Y_m}({\lfloor L_m \rfloor}) = {f^{\prime}_m}_* \omega_{Y_m/B_m}. \end{align*} $$

Together with equations (6.2), (6.3) and (6.4), the above inequality in equation (6.5) implies that

(6.6) $$ \begin{align} \left(1 + O(m^{-2}) \right) K_{X/B}^n \ge 2n! \left(\frac{\deg {f^{\prime}_m}_* \omega_{Y_m/B_m}}{m^{2q}} \right). \end{align} $$

Let $m \to \infty $ . The left-hand side of equation (6.6) clearly tends to $K_{X/B}^n$ . By Proposition 4.2, the right-hand side tends to $\chi _{f'} = \chi (Y, \omega _Y) - \chi (F', \omega _{F'}) \chi (B, \omega _B)$ , which is nothing but $\chi _f$ . Thus the proof for $g>0$ is completed.

6.2.2 Case II: $g(B) = 0$

Now we prove Theorem 1.2 when $g(B) = 0$ . It is easy to see that the argument for $g(B)> 0$ does not directly apply here. However, we can reduce this case to the previous one via a base change.

Choose four general distinct closed points $P_1$ , …, $P_4$ on B. Let $\sigma : C \to B$ be a double cover branched along $P_1$ , …, $P_4$ . By the Hurwitz formula, $g(C) = 1$ . Let $Y=X \times _B C$ and

$$ \begin{align*}f': Y \to C \end{align*} $$

be the induced fibration. Thus we have the following commutative diagram:

Since f is relatively of maximal Albanese dimension, so is $f'$ . As $g(C) = 1$ , Y itself is of maximal Albanese dimension. Since $P_1$ , …, $P_4$ are general, we deduce that Y is normal. Moreover, we claim that Y has at worst terminal singularities. In fact, let $\mu : X' \to X$ be a resolution of singularities of X. Then $X' \times _B C \to Y$ is just a resolution of singularities of Y, and the claim is an easy consequence of the adjunction.

Since $K_{Y/C} = \pi ^*K_{X/B}$ , $f'$ is also relatively minimal, and we have

(6.7) $$ \begin{align} K_{Y/C}^n = 2K_{X/B}^n. \end{align} $$

We also have

$$ \begin{align*}\pi_* \omega_Y = \omega_X \oplus \left(\omega_X \otimes {\mathcal{O}}_X(2F)\right) \end{align*} $$

from the above double cover. Thus, from the adjunction formula, we deduce that

(6.8) $$ \begin{align} \chi_{f'} & = \chi(Y, \omega_Y) - \chi(C, \omega_C) \chi(F, \omega_F) \nonumber \\ & = \chi(X, \omega_X) + \chi(X, \omega_X \otimes {\mathcal{O}}_X(2F)) \\ & = 2 \chi(X, \omega_X) + 2 \chi(F, \omega_F) \nonumber \\ & = 2 \chi_f. \nonumber \end{align} $$

Now that $g(C) = 1> 0$ , we have

$$ \begin{align*}K_{Y/C}^n \ge 2n! \chi_{f'} \end{align*} $$

as in Section 6.2.1. Together with equations (6.7) and (6.8), it implies that

$$ \begin{align*}K_{X/B}^n \ge 2n! \chi_f. \end{align*} $$

Thus the whole proof of Theorem 1.2 is completed.

Remark 6.1. With this framework, it is easy to see that in order to get inequalities of the same type as equation (1.1) with various slopes, we only need to (up to a base change to the $g(B)> 0$ case) replace equation (6.5) by a corresponding explicit estimate with the same slope, and the same argument will give rise to the desired results. This is a crucial observation to us.

6.3 Sharper inequalities

As an example of the above remark, we can easily obtain the following result.

Proof. By our assumption, $K(V) \simeq \frac {K(W)(t)}{(f(t))}$ , where $f(t)$ is an irreducible polynomial of degree d with coefficients in $K(W)$ . Using Galois theory, we can find a variety U and a generically finite map $\beta : U \to V$ such that $K(U)$ is the splitting field of $f(t)$ . Thus $K(U)/K(W)$ is a Galois extension. Write

$$ \begin{align*} H = {\mathrm{Gal}}(K(U)/K(W)). \end{align*} $$

Then H is a subgroup of $S_d$ . In particular, $|H|$ divides $d!$ . Since $p> d$ and G is a p-group, we have $(|G|, |H|) = 1$ .

Let $U_p = U \times _V V_p$ . We claim that $U_p$ is irreducible. Otherwise, let $U^{\prime }_p$ be an irreducible component of $U_p$ . Now the morphism $U^{\prime }_p \to W$ has two factorisations $U^{\prime }_p \to U \to W$ and $U^{\prime }_p \to W_p \to W$ . Thus both $|H|$ and $|G|$ divide $[K(U^{\prime }_p) : K(W)]$ . Since $(|G|, |H|) = 1$ , we have

$$ \begin{align*} [K(U^{\prime}_p) : K(W)] \ge |G| |H|. \end{align*} $$

On the other hand, since the degree of the map $U^{\prime }_p \to V_p$ is strictly less than $\deg \beta $ , we have

$$ \begin{align*} [K(U^{\prime}_p) : K(W)] = [K(U^{\prime}_p):K(W_p)][K(U):K(W)] < |G| |H|. \end{align*} $$

This is a contradiction. As a result, $U_p$ is irreducible. In particular, the natural morphism $U_p \to U$ is also a Galois cover and

$$ \begin{align*} G = {\mathrm{Gal}}(K(U_p)/K(U)). \end{align*} $$

We claim that the extension $K(U_p)/K(W)$ is also Galois. Write

$$ \begin{align*} G_p = {\mathrm{Aut}}(K(U_p)/K(W)). \end{align*} $$

It is clear that

$$ \begin{align*} |G_p| \le [K(U_p):K(W)] = [K(U_p):K(U)][K(U):K(W)] = |G||H|. \end{align*} $$

On the other hand, since $H = {\mathrm {Gal}}(K(U_p)/K(W_p))$ , we may view both G and H as subgroups of $G_p$ . Since $(|G|, |H|) = 1$ , we deduce that

$$ \begin{align*} |G_p| \ge |G| |H|. \end{align*} $$

Therefore, $|G_p| = |G| |H|$ , and the claim is verified. As a consequence of this claim, G is a normal subgroup in $G_p$ .

Now suppose that $\alpha _p$ is composed with an involution. This means there exists a variety $V^{\prime }_p$ such that $K(V_p) \supset K(V^{\prime }_p) \supseteq K(W_p)$ and

$$ \begin{align*} [K(V_p):K(V^{\prime}_p)] = 2. \end{align*} $$

Write $H_1={\mathrm {Aut}}(K(U_p)/K(V_p))$ and $H^{\prime }_1={\mathrm {Aut}}(K(U_p)/K(V^{\prime }_p))$ . Then the fundamental theorem of Galois theory tells us that $H_1 \subset H^{\prime }_1$ are both subgroups of $G_p$ and

$$ \begin{align*} [H^{\prime}_1:H_1] = 2. \end{align*} $$

Since G is normal, we consider another two subgroups $H_1G \subset H^{\prime }_1G$ of $G_p$ . Then we still have

$$ \begin{align*} [H^{\prime}_1G :H_1G] = 2. \end{align*} $$

Note that $K(U_p)^{H_1G} = K(V)$ . Again by the fundamental theorem of Galois theory, $K(U_p)^{H^{\prime }_1G}$ is a subfield of $K(V)$ and

$$ \begin{align*} [K(V):K(U_p)^{H^{\prime}_1G}] = 2. \end{align*} $$

This implies that $\alpha $ is composed with an involution. However, this is absurd. Thus the proof is completed.

6.4 An example

We provide an example showing that equation (1.1) is sharp.

Let $Y:=B \times A$ be a product of a smooth curve B of genus g and an abelian variety A of dimension $n-1$ , with two natural projections $p_1: Y \to B$ and $p_2: Y \to A$ . Take two sufficiently ample divisors $L_1$ on B and $L_2$ on A, respectively. Denote $L = p_1^*L_1 + p_2^*L_2$ . Choose a smooth divisor $D \in |2L|$ on Y. Let $\pi : X \to Y$ be a double cover branched along D. It is easy to see that

$$ \begin{align*}f: X \to B \end{align*} $$

is a relatively minimal fibration whose general fibre F is a double cover of A branched along $L_2$ and thus is of general type. Moreover, f is relatively minimal of maximal Albanese dimension.

Since $K_{X/B} \sim \pi ^*L$ , we have

$$ \begin{align*}K_{X/B}^n = (\pi^*L)^n = 2L^n = 2n (\deg L_1)L_2^{n-1}. \end{align*} $$

On the other hand, since

$$ \begin{align*}\pi_* \omega_X = {\mathcal{O}}_Y(p_1^*K_B) \oplus {\mathcal{O}}_Y(L + p_1^*K_B) \end{align*} $$

and

$$ \begin{align*}\pi_* \omega_F = {\mathcal{O}}_A \oplus {\mathcal{O}}_A(L_2), \end{align*} $$

by the Künneth formula, we have

$$ \begin{align*} \chi(X, \omega_X) & = \chi(Y, {\mathcal{O}}_Y(p_1^*K_B)) + \chi(Y, {\mathcal{O}}_Y(L + p_1^*K_B)) \\ & = \chi(B, \omega_B) \chi(A, {\mathcal{O}}_A) + \chi(B, {\mathcal{O}}_B(L_1+K_B)) \chi(A, {\mathcal{O}}_A(L_2)) \\ & = \chi(B, {\mathcal{O}}_B(L_1+K_B)) \chi(A, {\mathcal{O}}_A(L_2)), \end{align*} $$

and

$$ \begin{align*}\chi(F, \omega_F) = \chi(A, {\mathcal{O}}_A) + \chi(A, {\mathcal{O}}_A(L_2)) = \chi(A, {\mathcal{O}}_A(L_2)). \end{align*} $$

It follows that

$$ \begin{align*}\chi_f = \chi(A, {\mathcal{O}}_A(L_2)) \left( \chi(B, {\mathcal{O}}_B(L_1+K_B)) - \chi(B, \omega_B) \right) = \frac{L_2^{n-1}}{(n-1)!} \deg L_1. \end{align*} $$

Thus for this fibration f, we have $K_{X/B}^n = 2n! \chi _f> 0$ .

6.5 Proof of Theorem 1.3

Since the result is either known or trivial when $n = 2$ , in the following, we assume that $n \ge 3$ and

$$ \begin{align*}K_{X/B}^n = 2n! \chi_f> 0. \end{align*} $$

We first prove Theorem 1.3 (1). Via a base change argument as in Section 6.2.2, we may assume that $g(B)> 0$ . Thus we are under the setting of Section 6.1. Moreover, by Theorem 1.4, we know that $a|_F$ is composed with an involution.

Resume all notation in Section 6.1. Write $\Sigma = a(F)$ . Then $\Sigma $ is a subvariety of an abelian variety $A_F$ , a general fibre of $A \to J(B)$ of dimension $q - g(B)$ , and $\Sigma $ generates $A_F$ . To show that $\Sigma = A_F$ , we only need to show that the smooth model of $\Sigma $ has Kodaira dimension zero.

Let $\sigma : \Sigma ' \to \Sigma $ be a resolution of singularities of $\Sigma $ . Let $\Sigma _m = a_m(F_m)$ and $\Sigma ^{\prime }_m = \Sigma _m \times _{\sigma } \Sigma '$ . Then $\sigma _m: \Sigma ^{\prime }_m \to \Sigma _m$ is also a resolution of singularities of $\Sigma _m$ . Let $\nu _m: \Sigma ^{\prime }_m \to \Sigma '$ be the induced étale map. Thus we have the following diagram:

Denote

$$ \begin{align*}r' = r'(L|_{F'}, (b^*H)|_{F'}, \Sigma'), \quad r^{\prime}_m = r'(L_m|_{F^{\prime}_m}, (b^*_m H)|_{F^{\prime}_m}, \Sigma^{\prime}_m). \end{align*} $$

With this notation, by equation (6.1), we have

$$ \begin{align*}r':= \frac{(L|_{F'}) \left((b^*H)|_{F'} \right)^{n-2}}{K_{\Sigma'} \left( \sigma^*(H|_\Sigma) \right)^{n-2}} = \frac{(L_m|_{F^{\prime}_m}) \left((b_m^*H)|_{F^{\prime}_m} \right)^{n-2}}{K_{\Sigma^{\prime}_m} \left(\sigma_m^*(H|_{\Sigma_m})\right)^{n-2}} =: r^{\prime}_m. \end{align*} $$

It simply implies that

$$ \begin{align*}\delta(r') = \delta(r^{\prime}_m). \end{align*} $$

Now we use the framework of the proof of Theorem 1.2 again and replace equation (6.5) by the one in Proposition 5.3 (3). Together with the above equality, we deduce that

$$ \begin{align*}K_{X/B}^n \ge \frac{(5\delta(r') - 3) n!}{2 \delta(r') - 1} \chi_f. \end{align*} $$

However, if $\kappa (\Sigma ')> 0$ , we would have $\delta (r')> 1$ and thus $\frac {5\delta (r') - 3}{2 \delta (r') - 1}> 2$ . This is a contradiction. As a result, $\kappa (\Sigma ') = 0$ and $\Sigma = A_F$ .

Now we prove Theorem 1.3 (2). Note that $K_{X/B}^n> 0$ implies that $K_{X/B}$ is also big. In particular, a general fibre F of f is a minimal variety of general type. By [Reference Kawamata, Matsuda and Matsuki14, Theorem 1-2-5], we have

$$ \begin{align*}R^if_*\omega_{X/B}^{[l]} = R^if_*\omega_X^{[l]} \otimes \omega_B^{\otimes (-l)} =0 \end{align*} $$

for any $i>0$ and $l \ge 2$ . Thus for any $l \ge 2$ , we have

$$ \begin{align*}\chi(B, f_*\omega_{X/B}^{[l]})=\chi(X, \omega_{X/B}^{[l]}). \end{align*} $$

Let $P_l(F)$ denote the $l^{\mathrm {th}}$ plurigenus of F. Then we have

$$ \begin{align*} \deg f_*\omega_{X/B}^{[l]} & = \chi(B, f_*\omega_{X/B}^{[l]}) - P_l(F) \chi(B, {\mathcal{O}}_B) \\ & = \chi(X, \omega_{X/B}^{[l]}) - P_l(F) \chi(B, {\mathcal{O}}_B) \\ & = \frac{l^n K_{X/B}^n}{n!} + o(l^n). \end{align*} $$

In particular, for $l \gg 0$ , $\det f_*\omega _{X/B}^{[l]}$ is an ample line bundle on B. By [Reference Viehweg22, Proposition 4.6], we know that for $l \gg 0$ , the vector bundle $f_*\omega _{X/B}^{[l]}$ is ample. Thus, by [Reference Ohno19, Theorem 1.4], $mK_{X/B} - F$ is nef for a sufficiently large $m \in {\mathbb {Z}}$ . Replacing B by one of its cyclic covers of degree m, which is either étale (if $g(B)> 0$ ) or ramified at general points (if $g(B) = 0$ ), and replacing $f: X \to B$ by the fibration induced by this base change, we may assume that $K_{X/B} - F$ is nef. Similar to Section 6.2.2, we know that this induced fibration is also relatively minimal and of maximal Albanese dimension. Moreover, we still have

$$ \begin{align*}K_{X/B}^n = 2n! \chi_f> 0 \end{align*} $$

for this new fibration f.

Using the same strategy as in the proof of Theorem 1.2, but replacing $K_{X/B}$ by $K_{X/B}-F$ , we deduce that

$$ \begin{align*}(K_{X/B}-F)^n\ge 2n!\deg f_*(\omega_{X/B} \otimes {\mathcal{P}} \otimes {\mathcal{O}}_X (-F)), \end{align*} $$

where ${\mathcal {P}}\in {\mathrm {Pic}}^0(X)$ is a general torsion element. That is,

$$ \begin{align*}K_{X/B}^n - nK_F^{n-1} \ge 2n!(\chi_f - \chi(F, \omega_F)). \end{align*} $$

By the assumption that $K_{X/B}^n = 2n! \chi _f$ , we have

$$ \begin{align*}K_F^{n-1} \le 2(n-1)! \chi(F, \omega_F). \end{align*} $$

Since F is minimal of maximal Albanese dimension, together with the absolute Severi inequality for F, we deduce that

$$ \begin{align*}K_F^{n-1}=2(n-1)!\chi(F, \omega_F). \end{align*} $$

Thus the proof is completed.