Related papers: Severi inequality for varieties of maximal Albanes…

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!…

We prove the so-called Severi inequality, stating that the invariants of a minimal smooth complex projective surface of maximal Albanese dimension satisfy: K^2_S >= 4\chi(S).

Let $X$ be a minimal surface of general type and maximal Albanese dimension with irregularity $q\geq 2$. We show that $K_X^2\geq 4\chi(\mathcal O_X)+4(q-2)$ if $K_X^2<\frac92\chi(\mathcal O_X)$, and also obtain the characterization of the…

Let S be a minimal complex surface of general type and of maximal Albanese dimension; by the Severi inequality one has $K^2_S\geq 4\chi(\mathcal O_S)$. We prove that the equality $K^2_S=4\chi(\mathcal O_S)$ holds if and only if $q(S):=…

Let S be a smooth minimal complex projective surface of maximal Albanese dimension. Under the assumption that the canonical class of S is ample and the irregularity of S, q(S), is greater or equal to 5 we show that K^2>=…

Let $X$ be a minimal surface of general type over an algebraically closed field $\mathbf{k}$ of $\mathrm{char}.(\mathbf{k})=p\ge 0$. If the Albanese morphism $a_X:X\to \mathrm{Alb}_X$ is generically finite onto its image, we formulate a…

We study Severi type inequalities for big line bundles on irregular varieties via cohomological rank functions. We show that these Severi type inequalities on an irregular variety $X$ are related to some natural defined birational…

We give a sharp lower bound for the selfintersection of a nef line bundle $L$ on an irregular variety $X$ in terms of its continuous global sections and the Albanese dimension of $X$, which we call the Generalized Clifford-Severi…

Let $X$ be a surface of general type with maximal Albanese dimension: if $K_X^2<\frac{9}{2}\chi(\mathcal{O}_X)$, one has $K_X^2\geq 4\chi(\mathcal{O}_X)+4(q-2)$. We give a complete classification of surfaces for which equality holds for…

Let $X$ be a surface of general type with maximal Albanese dimension over an algebraically closed field of characteristic greater than two: we prove that if $K_X^2<\frac{9}{2}\chi(\mathcal{O}_X)$, one has $K_X^2\geq…

We continue our study on smooth complex projective varieties $X$ of maximal Albanese dimension and of general type satisfying $\chi(X, \omega_X)=0$. We formulate a conjectural characterization of such varieties and prove this conjecture…

We show that a smooth proper weakly ordinary variety $X$ of maximal Albanese dimension satisfies $\chi(X, \omega_X) \geq 0$. We also show that if $X$ is not of general type, then $\chi(X, \omega_X) = 0$ and the Albanese image of $X$ is…

Let $f: X\to Y$ be a surjective morphism of smooth $n$-dimensional projective varieties, with $Y$ of maximal Albanese dimension. Hacon and Pardini studied the structure of $f$ assuming $P_m(X)=P_m(Y)$ for some $m\geq 2$. We extend their…

We show that if $X$ is a smooth complex projective variety with Kodaira dimension $0$ then the Kodaira dimension of a general fiber of its Albanese map is at most $h^0(\Omega ^1 _X)$.

In this Thesis we study surfaces of general type with maximal Albanese dimension for which the quantity $K_X^2-4\chi(\mathcal{O}_X)-4(q-2)$ vanishes or is "small", that is surfaces close to the Severi lines. Over the complex numbers, it is…

In this short article we show that if $(X, B)$ is a compact K\"ahler klt pair of maximal Albanese dimension, then it has a good minimal model, i.e. there is a bimeromorphic contraction $\phi:X\dashrightarrow X'$ such that $K_{X'}+B'$ is…

We prove that every smooth projective variety with maximal Albanese dimension has a good minimal model. We also treat Ueno's problem on subvarieties of Abelian varieties.

Let $X$ be a Gorenstein minimal $3$-fold of general type. We prove the optimal inequality: $$K_X^{3}\geq \frac{4}{3}\chi(\omega_X)-2,$$ where $\chi(\omega_X)$ is the Euler-Poincar$\acute{\text{e}}$ characteristic of the dualizing sheaf…

Given a smooth complex projective variety X, a line bundle L of X an element v of H^1(O_X) and a section s in H^0(L) that deforms to first order in the direction v, we give a sufficient condition on v in terms of Koszul cohomology for this…

Let $X$ be a minimal projective Gorenstein 3-fold of general type. We give two applications of an inequality between $\chi (\omega_X)$ and $p_g(X)$: 1) Assume that the canonical map $\Phi_{|K_X|}$ is of fiber type. Let $F$ be a smooth model…