In 1932 F. Severi claimed, with an incorrect proof, that every smooth minimal projective surface $S$ such that the bundle $Ω^1_S$ is generically generated by global sections satisfies the topological inequality $2c_1^2(S)\ge c_2(S)$. According to Enriques-Kodaira classification, the above inequality is easily verified when the Kodaira dimension of the surface is $\le 1$, while for surfaces of general type it is still an open problem known as Severi conjecture. In this paper we prove Severi conjecture under the additional mild hypothesis that $S$ has ample canonical bundle. Moreover, under the same assumption, we prove that $2c_1^2(S)=c_2(S)$ if and only if $S$ is a double cover of an abelian surface.
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 $K_{X/B}^n \ge 2n! χ_f$, which was conjectured by Barja in [2]. Via the strategy outlined in [5], it also leads to a new proof of the Severi inequality for varieties of maximal Albanese dimension. Moreover, when the equality holds and $χ_f > 0$, we prove that the general fiber $F$ of $f$ has to satisfy the Severi equality that $K_F^{n-1} = 2(n-1)! χ(F, ω_F)$. We also prove some …
We prove the equivalence between Clifford-Severi inequalities for good classes of varieties of maximal Albanese dimension and Slope Inequalities for fibrations of such varieties over curves. This provides a big set of new Slope Inequalities and characterizes the limit cases. It also gives a machinery to automatically obtain other higher dimensional Slope and Clifford-Severi inequalities from inequalities in low dimension. For this, we construct a continuous version of Xiao's method for irregular fibrations.
Let $X$ be a surface of general type with maximal Albanese dimension: if $K_X^2<\frac{9}{2}χ(\mathcal{O}_X)$, one has $K_X^2\geq 4χ(\mathcal{O}_X)+4(q-2)$. We give a complete classification of surfaces for which equality holds for $q(X)\geq 3$: these are surfaces whose canonical model is a double cover of a product elliptic surface branched over an ample divisor with at most negligible singularities which intersects the elliptic fibre twice. We also prove, in the same hypothesis, that a surface $X$ with $K_X^2\neq 4χ(\mathcal{O}_X)+4(q-2)$ satisfies $K_X^2\geq 4χ(\mathcal{O}_X)+8(q-2)$ and we …
We prove a certain transcendence property of the unipotent Albanese map of a smooth variety, conditional on the Ax-Schanuel conjecture for variations of mixed Hodge structure. We show that this property allows the Chabauty-Kim method to be generalized to higher-dimensional varieties. In particular, we conditionally generalize several of the main Diophantine finiteness results in Chabauty-Kim theory to arbitrary number fields.
Let $X$ be a product of smooth projective curves over a finite unramified extension $k$ of $\mathbb{Q}_p$. Suppose that the Albanese variety of $X$ has good reduction and that $X$ has a $k$-rational point. We propose the following conjecture. The kernel of the Albanese map $CH_0(X)^0\rightarrow\text{Alb}_X(k)$ is $p$-divisible. When $p$ is an odd prime, we prove this conjecture for a large family of products of elliptic curves and certain principal homogeneous spaces of abelian varieties. Using this, we provide some evidence for a local-to-global conjecture for zero-cycles of Colliot-Thélène a…
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 $K_X^2<\frac{9}{2}χ(\mathcal{O}_X)$, one has $K_X^2\geq 4χ(\mathcal{O}_X)+4(q-2)$. Moreover we give a complete classification of surfaces for which equality holds for $q(X)\geq 3$: these are surfaces whose canonical model is a double cover of a product elliptic surface branched over an ample divisor with at most negligible singularities which intersects the elliptic fibre twice. In addition we expose a similar partial result over algebraic…
Let X be a complex surface with no nontrivial 2-forms. Then we show that Bloch's conjecture is true (i.e. the Albanese map in this case is injective) if and only if any homologically trivial idempotent in the ring of correspondences vanishes. Furthermore the cube of the ideal of homologically trivial correspondences is zero if these equivalent conditions are satisfied (e.g. if X is not of general type).
We give a new proof of Kollár's conjecture on the pushforward of the dualizing sheaf twisted by a variation of Hodge structure. This conjecture was settled by M. Saito via mixed Hodge modules and has applications in the investigation of Albanese maps. Our technique is the $L^2$-method and we give a concrete construction and proofs of the conjecture. The $L^2$ point of view allows us to generalize Kollár's conjecture to the context of non-abelian Hodge theory.
In this Thesis we study surfaces of general type with maximal Albanese dimension for which the quantity $K_X^2-4χ(\mathcal{O}_X)-4(q-2)$ vanishes or is "small", that is surfaces close to the Severi lines. Over the complex numbers, it is known that a surface $X$, provided that $K_X^2<\frac{9}{2}χ(\mathcal{O}_X)$, has to satisfy the inequality $K_X^2-4χ(\mathcal{O}_X)-4(q-2)\geq 0$. We give a constructive and complete classification of surfaces for which equality holds: these are surfaces whose canonical model is a double cover of an Abelian surface ($q=2$) or of a product elliptic surface ($q\g…
Bloch's conjecture and valences of correspondences for K3 surfaces
代数闭域$k$上的曲面$X$的Bloch猜想指出，每个同调平凡对应$Γ$在Albanese核$T(X_Ω)$上充当0，其中$Ω$是包含$k$的泛域。在这里，我们证明了，对于一个复杂的K3曲面$X$，Bloch猜想是等价于存在的每一个对应价。我们也给出了这个结果的应用程序的情况下，对应的有穷级的自同构和存在的恒定循环曲线的$X$。最后证明了K. O 'Grady提出的Franchetta猜想对亏格为$g$的极化K3曲面族成立，如果$ 3 \le g \le 6$
J. Stix proved that a curve of positive genus over $\mathbb{Q}$ which maps to a non-trivial Brauer-Severi variety satisfies the section conjecture. We prove that, if $X$ is a curve of positive genus over a number field $k$ and the Weil restriction $R_{k/\mathbb{Q}}X$ admits a rational map to a non-trivial Brauer-Severi variety, then $X$ satisfies the section conjecture. As a consequence, if $X$ maps to a Brauer-Severi variety $P$ such that the corestriction $\operatorname{cor}_{k/\mathbb{Q}}([P])\in\operatorname{Br}(\mathbb{Q})$ is non-trivial, then $X$ satisfies the section conjecture.
We prove the following generalization of Severi's Theorem: Let $X$ be a fixed complex variety. Then there exist, up to birational equivalence, only finitely many complex varieties $Y$ of general type of dimension at most three which admit a dominant rational map $f$ from $X$ to Y$.
We study symplectic varieties defined over fields of positive characteristics, especially the supersingular ones, generalizing the theory of supersingular K3 surfaces. In this work, we are mainly interested in the following two types of symplectic varieties over an algebraically closed field of positive characteristic, under natural numerical conditions: (1) smooth moduli spaces of sheaves on K3 surfaces and (2) smooth Albanese fibers of moduli spaces of sheaves on abelian surfaces. Several natural definitions of the supersingularity for symplectic varieties are discussed, which are proved to …
