arXiv subjects

Silvia Sabatini

Publications and source records attributed to Silvia Sabatini.

At least 19 recordsLinked to original sources

arXiv2026

Finiteness and boundedness of positive monotone Hamiltonian GKM$_3$ spaces

In this paper, we establish three finiteness and boundedness theorems for compact positive monotone symplectic manifolds endowed with special actions, called GKM$_3$, which generalize smooth toric varieties. Specifically, we prove that, for fixed dimension and Euler characteristic, there are only finitely many complex cobordism classes of such spaces. Moreover, modulo lattice transformations, the moment map image can be embedded into a box of explicitly bounded size, and all Chern numbers satisfy quantitative bounds. In particular, this yields a bound on the volume of the underlying symplectic manifold, analogous to the one obtained by Kollár-Miyaoka-Mori for Fano varieties.

math.SG↗

arXiv2024

On a symplectic generalization of a Hirzebruch problem

Motivated by a problem of Hirzebruch, we study $8$-dimensional, closed, symplectic manifolds having a Hamiltonian torus action with isolated fixed points and second Betti number equal to $1$. Such manifolds are automatically positive monotone. Our main result concerns those endowed with a Hamiltonian $T^2$-action and fourth Betti number equal to $2$. We classify their isotropy data, (equivariant) cohomology rings and (equivariant) Chern classes, and prove that they agree with those of certain explicit Fano $4$-folds with torus actions. Moreover, under more general assumptions, we prove several finiteness results concerning Betti and Chern numbers of $8$-dimensional, positive monotone symplectic manifolds with a Hamiltonian torus action.

math.SG↗

arXiv2023

Compact monotone tall complexity one $T$-spaces

In this paper we study compact monotone tall complexity one $T$-spaces. We use the classification of Karshon and Tolman, and the monotone condition, to prove that any two such spaces are isomorphic if and only if they have equal Duistermaat-Heckman measures. Moreover, we show that the moment polytope is Delzant and reflexive, and provide a complete description of the possible Duistermaat-Heckman measures. Whence we obtain a finiteness result that is analogous to that for compact monotone symplectic toric manifolds. Furthermore, we show that any such $T$-action can be extended to a toric $(T \times S^1)$-action. Motivated by a conjecture of Fine and Panov, we prove that any compact monotone tall complexity one $T$-space is equivariantly symplectomorphic to a Fano manifold endowed with a suitable symplectic form and a complexity one $T$-action.

math.SG↗

arXiv2022

Generalizing the Mukai Conjecture to the symplectic category and the Kostant game

In this paper we pose the question of whether the (generalized) Mukai inequalities hold for compact, positive monotone symplectic manifolds. We first provide a method that enables one to check whether the (generalized) Mukai inequalities hold true. This only makes use of the almost complex structure of the manifold and the analysis of the zeros of the so-called generalized Hilbert polynomial, which takes into account the Atiyah-Singer indices of all possible line bundles. We apply this method to generalized flag varieties. In order to find the zeros of the corresponding generalized Hilbert polynomial we introduce a modified version of the Kostant game and study its combinatorial properties.

math.SG↗

arXiv2020

Fano quiver moduli

We exhibit a large class of quiver moduli spaces which are Fano varieties, by studying line bundles on quiver moduli and their global sections in general, and work out several classes of examples, comprising moduli spaces of point configurations, Kronecker moduli, and toric quiver moduli.

math.AG↗

arXiv2020

Rigidity of elliptic genera: from number theory to geometry and back

In this paper we derive topological and number theoretical consequences of the rigidity of elliptic genera, which are special modular forms associated to each compact almost complex manifold. In particular, on the geometry side, we prove that rigidity implies relations between the Betti numbers and the index of a compact symplectic manifold of dimension $2n$ admitting a Hamiltonian action of a circle with isolated fixed points. We investigate the case of maximal index and toric actions. On the number theoretical side we prove that from each compact almost complex manifold of index greater than one, that can be endowed with the action of a circle with isolated fixed points, one can derive non-trivial relations among Eisenstein series. We give explicit formulas coming from the standard action on $\mathbb{C} P^n$.

math.AT↗

arXiv2019

On topological properties of positive complexity one spaces

Motivated by work of Fine and Panov, and of Lindsay and Panov, we prove that every closed symplectic complexity one space that is positive (e.g. positive monotone) enjoys topological properties that Fano varieties with a complexity one holomorphic torus action possess. In particular, such spaces are simply connected, have Todd genus equal to one and vanishing odd Betti numbers.

math.SG↗

arXiv2018

Faithful Semitoric Systems

This paper consists of two parts. The first provides a review of the basic properties of integrable and almost-toric systems, with a particular emphasis on the integral affine structure associated to an integrable system. The second part introduces faithful semitoric systems, a generalization of semitoric systems (introduced by Vu Ngoc and classified by Pelayo and Vu Ngoc) that provides the language to develop surgeries on almost-toric systems in dimension 4. We prove that faithful semitoric systems are natural building blocks of almost-toric systems. Moreover, we show that they enjoy many of the properties that their (proper) semitoric counterparts do.

math.SG↗

arXiv2016

What do central counterparties default funds really cover? A network-based stress test answer

In the last years, increasing efforts have been put into the development of effective stress tests to quantify the resilience of financial institutions. Here we propose a stress test methodology for central counterparties based on a network characterization of clearing members, whose links correspond to direct credits and debits. This network constitutes the ground for the propagation of financial distress: equity losses caused by an initial shock with both exogenous and endogenous components reverberate within the network and are amplified through credit and liquidity contagion channels. At the end of the dynamics, we determine the vulnerability of each clearing member, which represents its potential equity loss. We apply the proposed framework to the Fixed Income asset class of CC&G, the central counterparty operating in Italy whose main cleared securities are Italian Government Bonds. We consider two different scenarios: a distributed, plausible initial shock, as well as a shock corresponding to the cover 2 regulatory requirement (the simultaneous default of the two most exposed clearing members). Although the two situations lead to similar results after an unlimited reverberation of shocks on the network, the distress propagation is much more hasty in the latter case, with a large number of additional defaults triggered at early stages of the dynamics. Our results thus show that setting a default fund to cover insolvencies only on a cover 2 basis may not be adequate for taming systemic events, and only very conservative default funds, such as CC&G's one, can face total losses due to the shock propagation. Overall, our network-based stress test represents a refined tool for calibrating default fund amounts.

q-fin.RM↗

arXiv2016

12, 24 and Beyond

We generalize the well-known "12" and "24" Theorems for reflexive polytopes of dimension 2 and 3 to any smooth reflexive polytope. Our methods apply to a wider category of objects, here called reflexive GKM graphs, that are associated with certain monotone symplectic manifolds which do not necessarily admit a toric action. As an application, we provide bounds on the Betti numbers for certain monotone Hamiltonian spaces which depend on the minimal Chern number of the manifold.

math.CO↗

arXiv2016

On the Chern numbers and the Hilbert polynomial of an almost complex manifold with a circle action

Let $(M,J)$ be a compact, connected, almost complex manifold of dimension $2n$ endowed with a $J$-preserving circle action with isolated fixed points. In this note we analyse the `geography problem' for such manifolds, deriving equations relating the Chern numbers to the index $k_0$ of $(M,J)$. We study the symmetries and zeros of the Hilbert polynomial associated to $(M,J)$, which imply many rigidity results for the Chern numbers when $k_0\neq 1$. We apply these results to the category of compact, connected symplectic manifolds. A long-standing question posed by McDuff and Salamon, also known as the `McDuff conjecture', asked about the existence of non-Hamiltonian actions with isolated fixed points. This question was answered recently by Tolman, with an explicit construction of a six-dimensional manifold with such an action. One issue that this raises is whether one can find topological criteria that ensure the manifold can only support a Hamiltonian or only a non-Hamiltonian action. In this vein, we are able to deduce such criteria from our rigidity theorems in terms of relatively few Chern numbers, depending on the index. This improves upon results of Feldman for which one needs to know the entire Todd genus. Another consequence of our work is that, in the situation above with a Hamiltonian action, the minimal Chern number coincides with the index and is at most $n+1$, mirroring results of Michelsohn in the complex category and Hattori in the almost complex category.

math.AT↗

arXiv2015

Canonical bases for the equivariant cohomology and K-theory rings of symplectic toric manifolds

Let $M$ be a symplectic toric manifold acted on by a torus $\mathbb{T}$. In this work we exhibit an explicit basis for the equivariant K-theory ring $\mathcal{K}_{\mathbb{T}}(M)$ which is canonically associated to a generic component of the moment map. We provide a combinatorial algorithm for computing the restrictions of the elements of this basis to the fixed point set; these, in turn, determine the ring structure of $\mathcal{K}_{\mathbb{T}}(M)$. The construction is based on the notion of local index at a fixed point, similar to that introduced by Guillemin and Kogan in [GK]. We apply the same techniques to exhibit an explicit basis for the equivariant cohomology ring $H_{\mathbb{T}}(M; \mathbb{Z})$ which is canonically associated to a generic component of the moment map. Moreover we prove that the elements of this basis coincide with some well-known sets of classes: the equivariant Poincaré duals to the closures of unstable manifolds, and also the canonical classes introduced by Goldin and Tolman in [GT], which exist whenever the moment map is index increasing.

math.SG↗

arXiv2014

Fermat and the number of fixed points of periodic flows

We obtain a general lower bound for the number of fixed points of a circle action on a compact almost complex manifold $M$ of dimension $2n$ with nonempty fixed point set, provided the Chern number $c_1c_{n-1}[M]$ vanishes. The proof combines techniques originating in equivariant K-theory with celebrated number theory results on polygonal numbers, introduced by Pierre de Fermat. This lower bound confirms in many cases a conjecture of Kosniowski from 1979, and is better than existing bounds for some symplectic actions. Moreover, if the fixed point set is discrete, we prove divisibility properties for the number of fixed points, improving similar statements obtained by Hirzebruch in 1999. Our results apply, for example, to a class of manifolds which do not support any Hamiltonian circle action, namely those for which the first Chern class is torsion. This includes, for instance, all symplectic Calabi Yau manifolds.

math.AT↗

arXiv2013

An integer optimization problem for non-Hamiltonian periodic flows

Let C be the class of compact 2n-dimensional symplectic manifolds M for which the first or (n-1) Chern class vanish. We point out an integer optimization problem to find a lower bound B(n) on the number of equilibrium points of non-Hamiltonian symplectic periodic flows on manifolds M in C. As a consequence, we confirm in dimensions 2n in {8,10,12,14,18,20, 22} a conjecture for unitary manifolds made by Kosniowski in 1979 for the subclass C.

math.SG↗

arXiv2013

From semi-toric systems to Hamiltonian S^1-spaces

This paper studies the local and global aspects of semi-toric integrable systems, introduced by Vu Ngoc, using ideas stemming from the theory of Hamiltonian S^1-spaces developed by Karshon. First, we show how any labeled convex polygon associated to a semi-toric system (as defined by Vu Ngoc) determines Karshon's labeled directed graph which classifies the underlying Hamiltonian S^1-space up to isomorphism. Then we characterize adaptable semi-toric systems, i.e. those whose underlying Hamiltonian S^1-action can be extended to an effective Hamiltonian T^2-action, as those which have at least one associated convex polygon which satisfies the Delzant condition.

math.SG↗

arXiv2013

Moduli spaces of toric manifolds

We construct a distance on the moduli space of symplectic toric manifolds of dimension four. Then we study some basic topological properties of this space, in particular, path-connectedness, compactness, and completeness. The construction of the distance is related to the Duistermaat-Heckman measure and the Hausdorff metric. While the moduli space, its topology and metric, may be constructed in any dimension, the tools we use in the proofs are four-dimensional, and hence so is our main result.

math.SG↗

arXiv2013

Polynomial Assignments

The concept of assignments was introduced in [GGK99] as a method for extracting geometric information about group actions on manifolds from combinatorial data encoded in the infinitesimal orbit-type stratification. In this paper we will answer in the affirmative a question posed in [GGK99] by showing that the equivariant cohomology ring of $M$ is to a large extent determined by this data.

math.AT↗

arXiv2012

New Techniques for obtaining Schubert-type formulas for Hamiltonian manifolds

In [GT], Goldin and the second author extend some ideas from Schubert calculus to the more general setting of Hamiltonian torus actions on compact symplectic manifolds with isolated fixed points. (See also [Kn99] and [Kn08].) The main goal of this paper is to build on this work by finding more effective formulas. More explicitly, given a generic component of the moment map, they define a canonical class $α_p$ in the equivariant cohomology of the manifold $M$ for each fixed point $p \in M$. When they exist, canonical classes form a natural basis of the equivariant cohomology of $M$. In particular, when $M$ is a flag variety, these classes are the equivariant Schubert classes. It is a long standing problem in combinatorics to find positive integral formulas for the equivariant structure constants associated to this basis. Since computing the restriction of the canonical classes to the fixed points determines these structure constants, it is important to find effective formulas for these restrictions. In this paper, we introduce new techniques for calculating the restrictions of a canonical class $α_p$ to a fixed point $q$. Our formulas are nearly always simpler, in the sense that they count the contributions over fewer paths. Moreover, our formula is manifestly positive and integral in certain important special cases.

math.SG↗