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Giuseppe Ancona

Publications and source records attributed to Giuseppe Ancona.

14 recordsLinked to original sources
arXiv2025

Relative and absolute Lefschetz standard conjectures for some Lagrangian fibrations

We show that the hyper-Kähler varieties of OG10-type constructed by Laza-Saccà-Voisin (LSV) verify the Lefschetz standard conjecture. This is an application of a more general result, stating that certain Lagrangian fibrations verify this conjecture. The main technical assumption of this general result is that the Lagrangian fibration satisfies the hypotheses of Ngô's support theorem. Verifying that the LSV tenfolds do satisfy those hypotheses is of independent interest. Another point of independent interest of the paper is the definition and the study of the Lefschetz standard conjecture in the relative setting, and its relation to the classical absolute case.

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arXiv2025

Ramified periods and field of definition

Let $L/K$ be an extension of number fields that is ramified above $p$. We give a new obstruction to the descent to $K$ of smooth projective varieties defined over $L$. The obstruction is a matrix of $p$-adic numbers that we call ``ramified periods'' arising from the comparison isomorphism between de Rham cohomology and crystalline cohomology. As an application, we give simple examples of hyperelliptic curves over $\mathbb{Q}(\sqrt p)$ that are isomorphic to their Galois conjugates but such that their Jacobians do not descend to $\mathbb{Q}$ even up to isogeny.

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arXiv2025

An Artin--Mumford criterion for conic bundles in characteristic two

We prove a characteristic two version of the famous criterion of Artin and Mumford for irrationality of conic bundles. On the one hand, combined with the pathological behaviour of conic bundles in characteristic two, this allows us to construct easier and more explicit examples of irrational conic bundles. On the other hand, degeneration techniques à la Voisin allow to deduce irrationality results in characteristic zero.

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arXiv2025

The Hilbert symbol in the Hodge standard conjecture

We study the Hodge standard conjecture for varieties over finite fields admitting a CM lifting, such as abelian varieties or products of K3 surfaces. For those varieties we show that the signature predicted by the conjecture holds true modulo $4$. This amounts to determining the discriminant and the Hilbert symbol of the intersection product. The first is obtained by $\ell$-adic arguments whereas the second needs a careful computation in $p$-adic Hodge theory.

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arXiv2024

Ngô support theorem and polarizability of quasi-projective commutative group schemes

We prove that any commutative group scheme over an arbitrary base scheme of finite type over a field with connected fibers and admitting a relatively ample line bundle is polarizable in the sense of Ngô. This extends the applicability of Ngô's support theorem to new cases, for example to Lagrangian fibrations with integral fibers and has consequences to the construction of algebraic classes.

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arXiv2024

Algebraic classes in mixed characteristic and André's p-adic periods

Motivated by the study of algebraic classes in mixed characteristic we define a countable subalgebra of $\bar{\mathbb{Q}}_p$ which we call the algebra of André's $p$-adic periods. We construct a tannakian framework to study these periods. In particular, we bound their transcendence degree and formulate the analog of the Grothendieck period conjecture. We exhibit several examples where special values of classical $p$-adic functions appear as André's $p$-adic periods and we relate these new conjectures to some classical problems on algebraic classes.

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arXiv2023

Some arithmetic and geometric aspects of algebraic cycles and motives

This is my habilitation thesis. As the tradition wants, I tried to give an introduction of my field of research. I post it on the ArXiv with the hope it can be useful to young researchers looking for a short and friendly text on cohomologies of algebraic varieties, periods, algebraic cycles and motives. I might one day find the energy to expand these notes and maybe translate them in English. In the meantime, please feel free to ask questions. The first sections of this text propose an introduction to the theory, I tried to present points of view and examples which are not always stressed in the literature. A leitmotif of the thesis is the difference between phenomena in characteristic zero and those in positive characteristic. The main results are: the standard conjecture of Hodge type for abelian foufolds; the construction of a class of $p$-adic periods controlled by motivic Galois groups and for which it is possible to formulate a $p$-adic analog of the Grothendieck period conjecture; the standard conjecture of Lefschetz type for the Laza--Saccà--Voisin varieties and the Chow-Künneth conjecture for commutative group schemes. The techniques are based on $p$-adic Hodge theory, representation theory, perverse sheaves and motivic complexes à la Voevodsky.

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arXiv2020

Standard conjectures for abelian fourfolds

Let A be an abelian fourfold. We prove the Standard Conjecture of Hodge type for A. By combining this result with a theorem of Clozel we deduce that numerical equivalence on A coincides with l-adic homological equivalence on A for infinitely many l. The approach consists in reformulating this question into a p-adic problem and then using p-adic Hodge Theory to solve it.

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arXiv2016

On the motive of a commutative algebraic group

We prove a canonical Kunneth decomposition for the motive of a commutative group scheme over a field. Moreover, we show that this decomposition behaves under the group law just as in cohomology. We also deduce applications of the decomposition to the existence of a weight filtration, computation of any Weil cohomology theory and study of 1-motives.

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arXiv2016

On the relative motive of a commutative group scheme

We prove a canonical Kuenneth decomposition of the relative motive with rational coefficients of a smooth commutative group scheme over a noetherian finite dimensional base. This paper is a follow-up of "On the motive of a commutative algebraic group" arXiv:1312.4171

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arXiv2016

Numerical functors on Voevodsky's motives

We study mixed versions of the classical quotient functor from Chow motives to numerical motives. We compare two natural definitions, which turn out to be very different. We investigate fullness, conservativity and exactness of these two functors.

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arXiv2014

Decomposition de motifs abeliens

Let A be an abelian variety and let us fix a Weil cohomology with coefficients in F. Let $H^1(A,F)$ be the first cohomology group of A and $Lef(A) \subset GL(H^1(A,F))$ be its Lefschetz group, i.e. the sub-group of $GL(H^1(A,F))$ of linear applications commuting with endomorphisms of A and respecting the pairing induced by a polarization. We give an explicit presentation of a $\mathbb{Q}$-algebra of correspondences $B_{i,r}$ such that the cycle class map induces an isomorphism $cl_{|_{B_{i,r}}}: B_{i,r} \otimes_{\mathbb{Q}} F \cong End_{Lef(A)}(H^i(A^r,F)).$ We also give relative versions of this result. We deduce in particular the following fact. Let $S=S_K(G,\mathcal{X})$ be a Shimura variety of PEL type. Then the functor \textit{canonical construction} ${μ: Rep (G) \rightarrow VHS(S(\mathbb{C}))}$ lifts to a functor ${\tildeμ: Rep (G) \rightarrow CHM(S)_{\mathbb{Q}}}$, where $CHM(S)_{\mathbb{Q}}$ is the category of relative Chow motives.

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