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Riccardo Re

Publications and source records attributed to Riccardo Re.

14 recordsLinked to original sources
arXiv2022

The Universal de Rham/Spencer Double Complex on a Supermanifold

The universal Spencer and de Rham complexes of sheaves over a smooth or analytical manifold are well known to play a basic role in the theory of $\mathcal{D}$-modules. In this article we consider a double complex of sheaves generalizing both complexes for an arbitrary supermanifold, and we use it to unify the notions of differential and integral forms on real, complex and algebraic supermanifolds. The associated spectral sequences give the de Rham complex of differential forms and the complex of integral forms at page one. For real and complex supermanifolds both spectral sequences converge at page two to the locally constant sheaf. We use this fact to show that the cohomology of differential forms is isomorphic to the cohomology of integral forms, and they both compute the de Rham cohomology of the reduced manifold. Furthermore, we show that, in contrast with the case of ordinary complex manifolds, the Hodge-to-de Rham (or Frölicher) spectral sequence of supermanifolds with Kähler reduced manifold does not converge in general at page one.

math.AG↗
arXiv2020

A Note on Super Koszul Complex and the Berezinian

We construct the super Koszul complex of a free supercommutative $A$-module $V$ of rank $p|q$ and prove that its homology is concentrated in a single degree and it yields an exact resolution of $A$. We then study the dual of the super Koszul complex and show that its homology is concentrated in a single degree as well and isomorphic to $Π^{p+q} A$, with $Π$ the parity changing functor. Finally, we show that, given an automorphism of $V$, the induced transformation on the only non-trivial homology class of the dual of the super Koszul complex is given by the multiplication by the Berezinian of the automorphism, thus relating this homology group with the Berezinian module of $V$.

math.AG↗
arXiv2020

Cohomology of normal bundles of special rational varieties

We give a new method for calculating the cohomology of the normal bundles over rational varieties which are smooth projections of Veronese embeddings. The method can be used also when the projections are not smooth, in this case it provides information about the critical locus of maps between projective spaces.

math.AG↗
arXiv2019

Non Projected Calabi-Yau Supermanifolds over $\mathbb{P}^2$

We start a systematic study of non-projected supermanifolds, concentrating on supermanifolds with fermionic dimension 2 and with the reduced manifold a complex projective space. We show that all the non-projected supermanifolds of dimension $2|2$ over $\mathbb{P}^2$ are completely characterised by a non-zero 1-form $ω$ and by a locally free sheaf $\mathcal{F}$ of rank $0|2$, satisfying $Sym^2 \mathcal{F} \cong K_{\mathbb{P}^2}$. Denoting such supermanifolds with $\mathbb{P}^{2}_ω(\mathcal{F})$, we show that all of them are Calabi-Yau supermanifolds and, when $ω\neq 0$, they are non-projective, that is they cannot be embedded into any projective superspace $\mathbb{P}^{n|m}$. Instead, we show that every non-projected supermanifolds over $\mathbb{P}^2$ admits an embedding into a super Grassmannian. By contrast, we give an example of a supermanifold $\mathbb P^{2}_ω(\mathcal F)$ that cannot be embedded in any of the $Π$-projective superspaces $\mathbb P^{n}_Π$ introduced by Manin and Deligne. However, we also show that when $\mathcal F$ is the cotangent bundle over $\mathbb{P}^2$, then the non-projected $\mathbb{P}^2_ω(\mathcal F)$ and the $Π$-projective plane $\mathbb P^{2}_Π$ do coincide.

math.AG↗
arXiv2018

Complete Intersections of Quadrics and the Weak Lefschetz Property

We consider artinian algebras $A=\mathbb{C}[x_0,\ldots,x_m]/I$, with $I$ generated by a regular sequence of homogeneous forms of the same degree $d\geq 2$. We show that the multiplication by a general linear form from $A_{d-1}$ to $A_d$ is injective. We prove that the Weak Lefschetz Property holds for artinian complete intersection algebras as above, with $d=2$ and $m\leq 4$. Apparently, this was previously known only for $m\leq 3$. Although we are proposing only very limited progress towards the WLP conjecture for complete intersections, we hope that the methods of the present article can illustrate some geometrical aspects of the general problem.

math.AC↗
arXiv2017

One-Dimensional Super Calabi-Yau Manifolds and their Mirrors

We apply a definition of generalised super Calabi-Yau variety (SCY) to supermanifolds of complex dimension one. One of our results is that there are two SCY's having reduced manifold equal to $\mathbb{P}^1$, namely the projective super space $\mathbb{P}^{1|2} $ and the weighted projective super space $\mathbb{WP}^{1|1}_{(2)}$. Then we compute the corresponding sheaf cohomology of superforms, showing that the cohomology with picture number one is infinite dimensional, while the de Rham cohomology, which is what matters from a physical point of view, remains finite dimensional. Moreover, we provide the complete real and holomorphic de Rham cohomology for generic projective super spaces $\mathbb P^{n|m}$. We also determine the automorphism groups: these always match the dimension of the projective super group with the only exception of $\mathbb{P}^{1|2} $, whose automorphism group turns out to be larger than the projective general linear supergroup. By considering the cohomology of the super tangent sheaf, we compute the deformations of $\mathbb{P}^{1|m}$, discovering that the presence of a fermionic structure allows for deformations even if the reduced manifold is rigid. Finally, we show that $\mathbb{P}^{1|2} $ is self-mirror, whereas $\mathbb{WP} ^{1|1}_{(2)}$ has a zero dimensional mirror. Also, the mirror map for $\mathbb{P}^{1|2}$ naturally endows it with a structure of $N=2$ super Riemann surface.

hep-th↗
arXiv2016

Irreducible Components of Hilbert Schemes of Rational Curves with given Normal Bundle

We develop a new general method for computing the decomposition type of the normal bundle to a projective rational curve. This method is then used to detect and explain an example of a Hilbert scheme that parametrizes all the rational curves in $\mathbb{P}^s$ with a given decomposition type of the normal bundle and that has exactly two irreducible components. This gives a negative answer to the very old question whether such Hilbert schemes are always irreducible. We also characterize smooth non-degenerate rational curves contained in rational normal scroll surfaces in terms of the splitting type of their restricted tangent bundles, compute their normal bundles and show how to construct these curves as suitable projections of a rational normal curve.

math.AG↗
arXiv2014

PGL(2) actions on Grassmannians and projective construction of rational curves with given restricted tangent bundle

We give an explicit parametrization of the Hilbert schemes of rational curves C in P^n having a given splitting type of the restricted tangent bundle from P^n to C. The adopted technique uses the description of such curves as projections of a rational normal curve from a suitable linear vertex and a classification of those vertices that correspond to the required splitting type of the restricted tangent bundle. This classification involves the study of a suitable PGL(2) action on the relevant Grassmannian variety.

math.AG↗
arXiv2011

Principal part bundles on $\PP^n$ and quiver representations

We study the principal parts bundles $P^k (L)$ of the degree $d$ line bundle $L$ on the $n$ dimensional projective space as homogeneous bundles and we describe their associated quiver representations. We use this approach to show that if $n$ is greater or equal that 2, and $0\leq d<k$, then there exists an invariant splitting $P^k(L)=Q\oplus (S^dV\otimes \OO_{\PP^n})$ with $Q$ a stable homogeneous vector bundle. The splitting properties of such bundles were previously known only for n=1 or $k\leq d$ or $d<0$. Moreover we show that for any $d$ and any $h<k$ the canonical map from $P^k(L)$ to $P^h(L)$ always induces a linear map on the spaces of global sections which has maximal rank.

math.AG↗
arXiv2010

Multiplications of Maximal Rank in the Cohomology of P^1\times P ^1

We show that the linear map defined by multiplication with a general bi-homogeneous form between two bi-graduated pieces of the first cohomology of a nonsingular quadric in the projective space is of maximal rank. This is the first non trivial case of a more general open problem on natural linear maps between vector spaces of tensors defined in terms of multiplications and contractions. An interpretation in terms of bi-homogeneous linear differential operators with polynomial coefficients is also given.

math.AG↗
arXiv2010

On the maximum rank of a real binary form

We show that a real homogeneous polynomial f(x,y) with distinct roots and degree d greater or equal than 3 has d real roots if and only if for any (a,b) not equal to (0,0) the polynomial af_x+bf_y has d-1 real roots. This answers to a question posed by P. Comon and G. Ottaviani, and shows that the interior part of the locus of degree d binary real binary forms of rank equal to d is given exactly by the forms with d real roots.

math.AG↗
arXiv2010

Groupoid Quantales: a non étale setting

It is well known that if G is an étale topological groupoid then its topology can be recovered as the sup-lattice generated by G-sets, i.e. by the images of local bisections. This topology has a natural structure of unital involutive quantale. We present the analogous construction for any non étale groupoid with sober unit space G_0. We associate a canonical unital involutive quantale with any inverse semigroup of G-sets which is also a sheaf over G_0. We introduce axiomatically the class of quantales so obtained, and revert the construction mentioned above by proving a representability theorem for this class of quantales, under a natural spatiality condition.

math.QA↗