Search

Let $$F:{\mathcal {V}}\rightarrow B$$ F : V → B be a smooth non-isotrivial $$1-$$ 1 - dimensional family of complex polarized abelian varieties and $$V_b=F^{-1}(b)$$ V b = F - 1 ( b ) be the general fiber. Let $${\mathcal {F}}^1\subset R^1F_*{\mathbb {C}}$$ F 1 ⊂ R 1 F ∗ C be the associated Hodge bundle filtration, $${\mathcal {F}}^1_b=H^{1.0}(V_b).$$ F b 1 = H 1.0 ( V b ) . Under the assumption that the Fujita decomposition for $${\mathcal {F}}^1$$ F 1 is non trivial, that is there is a non trivial flat sub-bundle $$0\ne {\mathbb {U}}\subset {\mathcal {F}}^1,$$ 0 ≠ U ⊂ F 1 , we show that $$V_b$$ V b has non-trivial endomorphism: $$End(V_b)\ne {\mathbb {Z}}.$$ E n d ( V b ) ≠ Z .

We study the subsets $$V_k(A)$$ V k ( A ) of a complex abelian variety A consisting in the collection of points $$x\in A$$ x ∈ A such that the zero-cycle $$\{x\}-\{0_A\}$$ { x } - { 0 A } is k-nilpotent with respect to the Pontryagin product in the Chow group. These sets were introduced recently by Voisin and she showed that $$\dim V_k(A) \le k-1$$ dim V k ( A ) ≤ k - 1 and $$\dim V_k(A)$$ dim V k ( A ) is countable for a very general abelian variety of dimension at least $$2k-1$$ 2 k - 1 . We study in particular the locus $${\mathcal {V}}_{g,2}$$ V g , 2 in the moduli space of abelian varieties of dimension g with a fixed polarization, where $$V_2(A)$$ V 2 ( A ) is positive dimensional. We prove that an irreducible subvariety $${\mathcal {Y}} \subset {\mathcal {V}}_{g,2}$$ Y ⊂ V g , 2 , $$g\ge 3$$ g ≥ 3 , such that for a very general $$y \in {\mathcal {Y}}$$ y ∈ Y there is a curve in $$V_2(A_y)$$ V 2 ( A y ) generating A satisfies $$\dim {\mathcal {Y}}\le 2g - 1.$$ dim Y ≤ 2 g - 1 . The hyperelliptic locus shows that this bound is sharp.

Let $$f:S\rightarrow B$$ be a fibration of curves and let $$f_*\omega _{S/B}={{\mathcal {U}}}\oplus {{\mathcal {A}}}$$ be the second Fujita decomposition of f. In this paper we study a kind of Massey products, which are defined as infinitesimal invariants by the cohomology of a curve, in relation to the monodromy of certain subbundles of $${{\mathcal {U}}}$$. The main result states that their vanishing on a general fibre of f implies that the monodromy group acts faithfully on a finite set of morphisms and is therefore finite. In the last part we apply our result in terms of the normal function induced by the Ceresa cycle. On the one hand, we prove that the monodromy group of the whole $${{\mathcal {U}}}$$ of hyperelliptic fibrations is finite (giving another proof of a result due to Luo and Zuo). On the other hand, we show that the normal function is non torsion if the monodromy is infinite (this happens e.g. in the examples shown by Catanese and Dettweiler).

We construct three new families of fibrations $\pi : S \to B$ where $S$ is an algebraic complex surface and $B$ a curve that violate Xiao's conjecture relating the relative irregularity and the genus of the general fiber. The fibers of $\pi$ are certain \'etale cyclic covers of hyperelliptic curves that give coverings of $P^1$ with dihedral monodromy. As an application, we also show the existence of big and nef effective divisors in the Brill-Noether range., Comment: 12 pages. Exposition improved. The last section has been expanded with more details of the proof. Accepted for publication in Ann. Mat. Pura Appl

We prove a new inequality for the Hodge number $$h^{1,1}$$ of irregular complex smooth projective surfaces of general type without irrational pencils of genus $$\ge $$ 2. More specifically we show that if the irregularity $$q$$ satisfies $$q=2^k+1$$ then $$h^{1,1}\ge 4q-3$$ . This generalizes results previously known for $$q=3$$ and $$q=5$$ .

We give some upper bounds on the dimension of the kernel of the cup product map \(H^{1}(X,\mathbb{C}) \otimes H^{1}(X,\mathbb{C}) \to H^{2}(X,\mathbb{C})\), where X is a compact Kahler variety without Albanese fibrations.

In this paper it is proved that a complete algebraic surface of general type with p g =q=3, without irrational pencil of genus bigger than one is birationally equivalent to the two-symmetric product of a curve of genus 3. This result completes the classification of the surfaces with p g =q=3. The main tools are the Lefschetz theorem and the use of the paracanonical system on the surface.

LetX be a smooth projective curve defined onC. The number of holomorphic maps from a fixedX to another curve, (both of genus bigger than or equal to two), is finite by the classical de Franchis theorem. In this paper we get an explicit bound for this number, depending on the genus ofX only. Our bound is better than all the previously given ones (by Howard-Sommese and Kani).

The general topological framework for testing the possible occurrence of anomalies in gauge theories can be constructed in terms of the theory of group actions on line bundles through the introduction of a suitable group cohomology. In this Letter, we generalize this construction in such a way that it can be applied to a larger class of theories, allowing for a noncontractible configuration space and a nonconnected ‘gauge’ group. This construction find applications to the problem of the lifts of principal group actions. As a physical application, we compare the mechanisms of the anomalies cancelation in gauge and string theories, through a geometrical splitting of local and global anomalies.

Si studiano le classi di Chern delle varieta determinantali in ipotesi di buone codimensioni. Si esplicitano in particolare i primi termini del carattere di Chern di tali varieta. Applicando tali risultati alle varieta W d r dei divisori speciali su una curva algebrica di tipo generale quando dim (W d r )=1,se ne calcula il genere.