Let $f: X \to B$ be a relatively minimal fibration of maximal Albanese dimension from a variety X of dimension $n \ge 2$ to a curve B defined over an algebraically closed field of characteristic zero. We prove that $K_{X/B}^n \ge 2n! \chi _f$ . It verifies a conjectural formulation of Barja in [2]. Via the strategy outlined in [4], it also leads to a new proof of the Severi inequality for varieties of maximal Albanese dimension. Moreover, when the equality holds and $\chi _f> 0$ , we prove that the general fibre F of f has to satisfy the Severi equality that $K_F^{n-1} = 2(n-1)! \chi (F, \omega _F)$ . We also prove some sharper results of the same type under extra assumptions.

Let $X$ be a normal complex projective variety, $T\subseteq X$ a subvariety of dimension $m$ (possibly $T=X$) and $a:X\rightarrow A$ a morphism to an abelian variety such that $\text{Pic}^{0}(A)$ injects into $\text{Pic}^{0}(T)$; let $L$ be a line bundle on $X$ and $\unicode[STIX]{x1D6FC}\in \text{Pic}^{0}(A)$ a general element.

We introduce two new ingredients for the study of linear systems on $X$. First of all, we show the existence of a factorization of the map $a$, called the eventual map of $L$ on $T$, which controls the behavior of the linear systems $|L\otimes \unicode[STIX]{x1D6FC}|_{|T}$, asymptotically with respect to the pullbacks to the connected étale covers $X^{(d)}\rightarrow X$ induced by the $d$-th multiplication map of $A$.

Second, we define the so-called continuous rank function$x\mapsto h_{a}^{0}(X_{|T},L+xM)$, where $M$ is the pullback of an ample divisor of $A$. This function extends to a continuous function of $x\in \mathbb{R}$, which is differentiable except possibly at countably many points; when $X=T$ we compute the left derivative explicitly.

As an application, we give quick short proofs of a wide range of new Clifford–Severi inequalities, i.e., geographical bounds of the form

$$\begin{eqnarray}\displaystyle \text{vol}_{X|T}(L)\geqslant C(m)h_{a}^{0}(X_{|T},L), & & \displaystyle \nonumber\end{eqnarray}$$

$C(m)={\mathcal{O}}(m!)$

$X$

$L$

$a$

In this paper, we study the explicit geography problem of irregular Gorenstein minimal 3-folds of general type. We generalize the classical Noether–Castelnuovo type inequalities for irregular surfaces to irregular 3-folds according to the Albanese dimension.