Generalized holomorphic analytic torsion

José Ignacio Burgos Gil; Gerard Freixas i Montplet; Răzvan Liţcanu

Displaying similar documents to “Generalized holomorphic analytic torsion”

An extension theorem for separately holomorphic functions with analytic singularities

Marek Jarnicki, Peter Pflug (2003)

Annales Polonici Mathematici

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Let $D_{j} \subset ℂ^{k_{j}}$ be a pseudoconvex domain and let $A_{j} \subset D_{j}$ be a locally pluriregular set, j = 1,...,N. Put $X : = ⋃_{j = 1}^{N} A ₁ \times . . . \times A_{j - 1} \times D_{j} \times A_{j + 1} \times . . . \times A_{N} \subset ℂ^{k ₁ + . . . + k_{N}}$ . Let U be an open connected neighborhood of X and let M ⊊ U be an analytic subset. Then there exists an analytic subset M̂ of the “envelope of holomorphy” X̂ of X with M̂ ∩ X ⊂ M such that for every function f separately holomorphic on X∖M there exists an f̂ holomorphic on X̂∖M̂ with ${f ̂ |}_{X ∖ M} = f$ . The result generalizes special cases which were studied in [Ökt 1998], [Ökt 1999], [Sic 2001], and [Jar-Pfl 2001]. ...

Torsion points in families of Drinfeld modules

Dragos Ghioca, Liang-Chung Hsia (2013)

Acta Arithmetica

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Let $Φ^{λ}$ be an algebraic family of Drinfeld modules defined over a field K of characteristic p, and let a,b ∈ K[λ]. Assume that neither a(λ) nor b(λ) is a torsion point for $Φ^{λ}$ for all λ. If there exist infinitely many λ ∈ K̅ such that both a(λ) and b(λ) are torsion points for $Φ^{λ}$ , then we show that for each λ ∈ K̅, a(λ) is torsion for $Φ^{λ}$ if and only if b(λ) is torsion for $Φ^{λ}$ . In the case a,b ∈ K, we prove in addition that a and b must be $̅_{p}$ -linearly dependent.

On extensions of holomorphic functions satisfying a polynomial growth condition on algebraic varieties in $𝐂^{n}$

Jean Erik Björk (1974)

Annales de l'institut Fourier

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Let $V$ be an algebraic variety in $C^{n}$ and when $k \geq 0$ is an integer then $Pol (V, k)$ denotes all holomorphic functions $f (z)$ on $V$ satisfying $| f (z) | \leq C_{f} (1 + | z |)^{k}$ for all $z \in V$ and some constant $C_{f}$ . We estimate the least integer $ϵ (V, k)$ such that every $f \in Pol (V, k)$ admits an extension from $V$ into $C^{n}$ by a polynomial $P (z_{1}, ..., z_{n})$ , of degree $k + ϵ (V, k)$ at most. In particular ${lim}_{k > \infty} ϵ (V, k)$ is related to cohomology groups with coefficients in coherent analytic sheaves on $V$ . The existence of the finite integer $ϵ (V, k)$ is for example an easy consequence of Kodaira’s Vanishing Theorem. ...

The cuspidal torsion packet on hyperelliptic Fermat quotients

David Grant, Delphy Shaulis (2004)

Journal de Théorie des Nombres de Bordeaux

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Let $ℓ \geq 7$ be a prime, $C$ be the non-singular projective curve defined over $ℚ$ by the affine model $y (1 - y) = x^{ℓ}$ , $\infty$ the point of $C$ at infinity on this model, $J$ the Jacobian of $C$ , and $φ : C \to J$ the albanese embedding with $\infty$ as base point. Let $\bar{ℚ}$ be an algebraic closure of $ℚ$ . Taking care of a case not covered in [], we show that $φ (C) \cap J_{tors} (\bar{ℚ})$ consists only of the image under $φ$ of the Weierstrass points of $C$ and the points $(x, y) = (0, 0)$ and $(0, 1)$ , where $J_{tors}$ denotes the torsion points of $J$ .

The non-pluripolarity of compact sets in complex spaces and the property $(L B^{\infty})$ for the space of germs of holomorphic functions

Le Mau Hai, Tang Van Long (2002)

Studia Mathematica

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The aim of this paper is to establish the equivalence between the non-pluripolarity of a compact set in a complex space and the property $(L B^{\infty})$ for the dual space of the space of germs of holomorphic functions on that compact set.

On the geometry of the space of m-pairs of holomorphic subspaces of the n-dimensional projective space $P_{n}$ ove r an albebra $𝒜$

Е.М. Кузнецова (1986)

Trudy Geometriceskogo Seminara

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A note on the torsion of the Jacobians of superelliptic curves $y^{q} = x^{p} + a$

Tomasz Jędrzejak (2016)

Banach Center Publications

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This article is a short version of the paper published in J. Number Theory 145 (2014) but we add new results and a brief discussion about the Torsion Conjecture. Consider the family of superelliptic curves (over ℚ) $C_{q, p, a} : y^{q} = x^{p} + a$ , and its Jacobians $J_{q, p, a}$ , where 2 < q < p are primes. We give the full (resp. partial) characterization of the torsion part of $J_{3, 5, a} (ℚ)$ (resp. $J_{q, p, a} (ℚ)$ ). The main tools are computations of the zeta function of $C_{3, 5, a}$ (resp. $C_{q, p, a}$ ) over $_{l}$ for primes l ≡ 1,2,4,8,11 (mod 15) (resp. for primes l...

On the torsion of the Jacobians of the hyperelliptic curves y² = xⁿ + a and y² = x(xⁿ+a)

Tomasz Jędrzejak (2016)

Acta Arithmetica

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Consider two families of hyperelliptic curves (over ℚ), $C^{n, a} : y ² = x ⁿ + a$ and $C_{n, a} : y ² = x (x ⁿ + a)$ , and their respective Jacobians $J^{n, a}$ , $J_{n, a}$ . We give a partial characterization of the torsion part of $J^{n, a} (ℚ)$ and $J_{n, a} (ℚ)$ . More precisely, we show that the only prime factors of the orders of such groups are 2 and prime divisors of n (we also give upper bounds for the exponents). Moreover, we give a complete description of the torsion part of $J_{8, a} (ℚ)$ . Namely, we show that $J_{8, a} {(ℚ)}_{t o r s} = J_{8, a} (ℚ) [2]$ . In addition, we characterize the torsion parts of $J_{p, a} (ℚ)$ , where p is an odd...

Complete pluripolar graphs in $ℂ^{N}$

Nguyen Quang Dieu, Phung Van Manh (2014)

Annales Polonici Mathematici

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Let F be the Cartesian product of N closed sets in ℂ. We prove that there exists a function g which is continuous on F and holomorphic on the interior of F such that $Γ_{g} (F) : = (z, g (z)) : z \in F$ is complete pluripolar in $ℂ^{N + 1}$ . Using this result, we show that if D is an analytic polyhedron then there exists a bounded holomorphic function g such that $Γ_{g} (D)$ is complete pluripolar in $ℂ^{N + 1}$ . These results are high-dimensional analogs of the previous ones due to Edlund [Complete pluripolar curves and graphs, Ann. Polon. Math....

A characterization of Ext(G,ℤ) assuming (V = L)

Saharon Shelah, Lutz Strüngmann (2007)

Fundamenta Mathematicae

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We complete the characterization of Ext(G,ℤ) for any torsion-free abelian group G assuming Gödel’s axiom of constructibility plus there is no weakly compact cardinal. In particular, we prove in (V = L) that, for a singular cardinal ν of uncountable cofinality which is less than the first weakly compact cardinal and for every sequence $(ν_{p} : p \in Π)$ of cardinals satisfying $ν_{p} \leq 2^{ν}$ (where Π is the set of all primes), there is a torsion-free abelian group G of size ν such that $ν_{p}$ equals the p-rank of Ext(G,ℤ)...

Approximation of sets defined by polynomials with holomorphic coefficients

Marcin Bilski (2012)

Annales Polonici Mathematici

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Let X be an analytic set defined by polynomials whose coefficients $a ₁, . . ., a_{s}$ are holomorphic functions. We formulate conditions on sequences $a_{1, ν}, . . ., a_{s, ν}$ of holomorphic functions converging locally uniformly to $a ₁, . . ., a_{s}$ , respectively, such that the sequence $X_{ν}$ of sets obtained by replacing $a_{j}$ ’s by $a_{j, ν}$ ’s in the polynomials converges to X.

A result on extension of C.R. functions

Makhlouf Derridj, John Erik Fornaess (1983)

Annales de l'institut Fourier

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Let $Ω$ an open set in $C^{4}$ near $z_{0} \in \partial Ω$ , $λ$ a suitable holomorphic function near $z_{0}$ . If we know that we can solve the following problem (see [M. Derridj, Annali. Sci. Norm. Pisa, Série IV, vol. IX (1981)]) : $\overline{\partial} u = λ f$ , ( $f$ is a $(0, 1)$ form, $\overline{\partial}$ closed in $U (z_{0})$ in $U (z_{0})$ with supp $(u) \subset \overline{Ω} \cap U (z_{0})$ , then we deduce an extension result for $C . R .$ functions on $\partial Ω \cap U (z_{0})$ , as holomorphic fonctions in $Ω \cap V (z_{0})$ .

On spaces of holomorphic functions in ℂⁿ

Diana D. Jiménez S., Lino F. Reséndis O., Luis M. Tovar S. (2014)

Banach Center Publications

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Following the line of Ouyang et al. (1998) to study the $_{p}$ spaces of holomorphic functions in the unit ball of ℂⁿ, we present in this paper several results and relations among $_{p} (ₙ)$ , the α-Bloch, the Dirichlet $_{p}$ and the little $_{p, 0}$ spaces.