arXiv subjects

Lázaro O. Rodríguez Díaz

Publications and source records attributed to Lázaro O. Rodríguez Díaz.

9 recordsLinked to original sources

arXiv2025

On the origin of the Jacobian conjecture

The Jacobian conjecture is thought to have been proposed by O. H. Keller in 1939. However, we have found that the statement of the conjecture is precisely the main result of a paper published by L. Kraus in 1884. Although the final step of Kraus's proof is flawed, the ideas he introduced anticipated approaches to the problem that would only emerge more than a century later. Interestingly, the root of Kraus's error remains the principal obstacle to algebro-geometric approaches: controlling the ramification at infinity.

math.AG↗

arXiv2024

A note on étale endomorphisms of normal schemes

We prove that under some extra hypothesis, given an étale endomorphism of a normal irreducible Noetherian and simply connected scheme, if the endomorphism is surjective then it is injective. The additional assumption concerns the possibility of constructing an étale cover out of a surjective étale morphism. We show that in some cases the surjectivity hypothesis can be removed if the intended étale cover is Galois.

math.AG↗

arXiv2024

On the integrality of étale extensions of polynomial rings

Motivated by a valuation theorem, recently obtained by Rangachev, we study the étale extensions $A\subset B$ of polynomial rings over an algebraically closed field of characteristic zero, such that the integral closure $\overline{A}$ is a primary $\overline{A}$-submodule of $B$. We prove that in this case $\overline{A}$ has infinite cyclic divisor class group, where the generator is a prime divisor equal to the complement of $\textrm{Spec}(B)$ in $\textrm{Spec}(\overline{A})$. Moreover, this prime divisor coincides with the ramification divisor of the finite extension $A\subset \overline{A}$. In this situation we carry out Wright's geometric approach for two-dimensional non-integral étale extensions. It follows from the work of Miyanishi that $\textrm{Spec}(\overline{A})$ is a smooth affine surface. We show that $\textrm{Spec}(\overline{A})$ is an $\mathbb{A}^{1}$-bundle over $\mathbb{P}^{1}$, more precisely a Danilov-Gizatullin surface of index three. Based on Wright's analysis of which of these affine surfaces can factorize an étale morphism of the complex affine plane and his description of its affine coordinate rings, we prove that under the strong assumption that $\overline{A}$ is always a primary $\overline{A}$-submodule of $B$, any two-dimensional complex étale extension is integral.

math.AG↗

arXiv2024

Étale extensions of polynomial rings are faithfully flat

We apply Ohi's criterion for faithfully flatness of extensions of commutative rings to prove that any étale extension $k[Y_1, \ldots, Y_n]\subseteq k[X_1, \ldots, X_n]$ of polynomial rings (each in $n$ indeterminates) over a commutative ring $k$ is faithfully flat. In particular, if $k$ is an algebraically closed field then any étale polynomial map $k^{n} \to k^{n}$ is surjective.

math.AC↗

arXiv2018

Gauge theory and G2-geometry on Calabi-Yau links

The $7$-dimensional link $K$ of a weighted homogeneous hypersurface on the round $9$-sphere in $\mathbb{C}^5$ has a nontrivial null Sasakian structure which is contact Calabi-Yau, in many cases. It admits a canonical co-closed $\rm G_2$-structure $φ$ induced by the Calabi-Yau $3$-orbifold basic geometry. We distinguish these pairs $(K,φ)$ by the Crowley-Nordström $\mathbb{Z}_{48}$-valued $ν$ invariant, for which we prove odd parity and provide an algorithmic formula. We describe moreover a natural Yang-Mills theory on such spaces, with many important features of the torsion-free case, such as a Chern-Simons formalism and topological energy bounds. In fact compatible $\rm G_2$-instantons on holomorphic Sasakian bundles over $K$ are exactly the transversely Hermitian Yang-Mills connections. As a proof of principle, we obtain $\rm G_2$-instantons over the Fermat quintic link from stable bundles over the smooth projective Fermat quintic, thus relating in a concrete example the Donaldson-Thomas theory of the quintic threefold with a conjectural $\rm G_2$-instanton count.

math.DG↗

arXiv2018

A note on Kirchhoff's theorem for almost complex spheres I

By a theorem of Kirchhoff if the six sphere admits an almost complex structure then the seven sphere is parallelizable, more crucial, he exhibited an explicit global frame constructed out of the given almost complex structure. This result implicitly equips the seven sphere with a definite H-space multiplication. We propose to address the existence problem of complex structures on the six sphere studying the associated parallelism-multiplications on the seven sphere. We ask to what extent the integrability condition of the almost complex structure amounts to the constancy of the structure functions of the global frame defining the parallelism, i.e, if this parallelism comes from a Lie group structure. At a more fundamental level we inquire if the integrability condition of the almost complex structure entails the homotopy associativity of the induced multiplication. A positive answer to these questions would rule out the six sphere of being a complex manifold since the seven sphere is not a Lie group, not even a homotopy associative H-space.

math.DG↗

arXiv2016

The $SW(3/2,2)$ superconformal algebra via a Quantum Hamiltonian Reduction of $osp(3|2)$

We prove that the family of non-linear $W$-algebras $SW(3/2,2)$ which are extensions of the $N=1$ superconformal algebra by a primary supercurrent of conformal weight $2$ can be realized as a quantum Hamiltonian reduction of the Lie superalgebra $osp(3|2)$. In consequence we obtain an explicit free field realization of the algebra in terms of the screening operators. At central charge $c=12$ the $SW(3/2,2)$ superconformal algebra corresponds to the superconformal algebra associated to sigma models based on eight-dimensional manifolds with special holonomy $Spin(7)$, i.e., the Shatashvili-Vafa $Spin(7)$ superconformal algebra.

math.QA↗

arXiv2016

$\rm G_2$ holonomy manifolds are superconformal

We study the chiral de Rham complex (CDR) over a manifold $M$ with holonomy $\rm G_2$. We prove that the vertex algebra of global sections of the CDR associated to $M$ contains two commuting copies of the Shatashvili-Vafa $\rm G_2$ superconformal algebra. Our proof is a tour de force, based on explicit computations.

math.QA↗

arXiv2014

The Shatashvili-Vafa $G_{2}$ superconformal algebra as a Quantum Hamiltonian Reduction of $D(2,1;α)$

We obtain the superconformal algebra associated to a sigma model with target a manifold with $G_{2}$ holonomy, i.e., the Shatashvili-Vafa $G_{2}$ algebra as a quantum Hamiltonian reduction of the exceptional Lie superalgebra $D(2,1;α)$ for $α=1$. We produce the complete family of $W$-algebras $SW(\frac{3}{2},\frac{3}{2}, 2)$ (extensions of the $N=1$ superconformal algebra by two primary supercurrents of conformal weight $\frac{3}{2}$ and $2$ respectively) as a quantum Hamiltonian reduction of $D(2,1;α)$. As a corollary we find a free field realization of the Shatashvili-Vafa $G_{2}$ algebra, and an explicit description of the screening operators.

math.QA↗