We define vector bundles for tropical schemes, and explore their properties. The paper largely consists of three parts; (1) we study free modules over zero-sum free semirings, which provide the necessary algebraic background for the theory (2) we relate vector bundles on tropical schemes to topological vector bundles and vector bundles on monoid schemes, and finally (3) we show that all line bundles on a tropical scheme can be lifted to line bundles on a usual scheme in the affine case.
We introduce tropical vector bundles, morphisms and rational sections of these bundles and define the pull-back of a tropical vector bundle and of a rational section along a morphism. Afterwards we use the bounded rational sections of a tropical vector bundle to define the Chern classes of this bundle and prove some basic properties of Chern classes. Finally we give a complete classification of all vector bundles on an elliptic curve up to isomorphisms.
The moduli space $M_g^{trop}$ of tropical curves of genus $g$ is a generalized cone complex that parametrizes metric vertex-weighted graphs of genus $g$. For each such graph $Γ$, the associated canonical linear system $\vert K_Γ\vert$ has the structure of a polyhedral complex. In this article we propose a tropical analogue of the Hodge bundle on $M_g^{trop}$ and study its basic combinatorial properties. Our construction is illustrated with explicit computations and examples.
Using the notion of a root datum of a reductive group $G$ we propose a tropical analogue of a principal $G$-bundle on a metric graph. We focus on the case $G=\mathrm{GL}_n$, i.e. the case of vector bundles. Here we give a characterization of vector bundles in terms of multidivisors and use this description to prove analogues of the Weil--Riemann--Roch theorem and the Narasimhan--Seshadri correspondence. We proceed by studying the process of tropicalization. In particular, we show that the non-Archimedean skeleton of the moduli space of semistable vector bundles on a Tate curve is isomorphic to…
We use the tropical geometry approach to compute absolute and relative Gromov-Witten invariants of complex surfaces which are $\CC P^1$-bundles over an elliptic curve. We also show that the tropical multiplicity used to count curves can be refined by the standard Block-Göttsche refined multiplicity to give tropical refined invariants. We then give a concrete algorithm using floor diagrams to compute these invariants along with the associated interpretation as operators acting on some Fock space. The floor diagram algorithm allows one to prove the piecewise polynomiality of the relative invaria…
Let $R$ be a complete discrete valuation ring of equi-characteristic zero with fractional field $K$. Let $X$ be a connected, smooth projective variety of dimension $d$ over $K$, and let $L$ be an ample line bundle over $X$. We assume that there exist a regular strictly semistable model $\mathscr{X}$ of $X$ over $R$ and a relatively ample line bundle $\mathscr{L}$ over $\mathscr{X}$ with $\mathscr{L}|_{X} \cong L$. Let $S(\mathscr{X})$ be the skeleton associated to $\mathscr{X}$ in the Berkovich analytification $X^{\mathrm{an}}$ of $X$. In this article, we study when $S(\mathscr{X})$ is faithfu…
Tropical recurrent sequences are introduced satisfying a given vector (being a tropical counterpart of classical linear recurrent sequences). We consider the case when Newton polygon of the vector has a single (bounded) edge. In this case there are periodic tropical recurrent sequences which are similar to classical linear recurrent sequences. A question is studied when there exists a non-periodic tropical recurrent sequence satisfying a given vector, and partial answers are provided to this question. Also an algorithm is designed which tests existence of non-periodic tropical recurrent sequen…
In this short writing we prove that the set of m-dissimilarity vectors of n-trees is contained in the (m,n)-tropical Grassmannian, answering a question of Pachter and Speyer. We do this by proving a generalization of an equivalent conjecture proposed by Cools.
In 2004 Pachter and Speyer introduced the higher dissimilarity maps for phylogenetic trees and asked two important questions about their relation to the tropical Grassmannian. Multiple authors, using independent methods, answered affirmatively the first of these questions, showing that dissimilarity vectors lie on the tropical Grassmannian, but the second question, whether the set of dissimilarity vectors forms a tropical subvariety, remained opened. We resolve this question by showing that the tropical balancing condition fails. However, by replacing the definition of the dissimilarity map wi…
In this paper we investigate the idea of a tropical critical point of the superpotential for the full flag variety of type A. Recall that associated to an irreducible representation of G=SLn(C) are various polytopes whose integral points parameterize a basis for the representation, e.g. the Gelfand-Zetlin polytope. Such polytopes can be constructed via the theory of geometric crystals by tropicalising a certain function, and in fact, the function involved coincides with the superpotential from the Landau-Ginzburg model for G/B coming from mirror symmetry. In mirror symmetry a special role is p…
We define tropical analogues of the notions of linear space and Plucker coordinate and study their combinatorics. We introduce tropical analogues of intersection and dualization and define a tropical linear space built by repeated dualization and transverse intersection to be constructible. Our main result that all constructible tropical linear spaces have the same f-vector and are ``series-parallel''. We conjecture that this f-vector is maximal for all tropical linear spaces with equality precisely for the series-parallel tropical linear spaces. We present many partial results towards this co…
In this paper we study general tropical linear spaces locally: For any basis B of the matroid underlying a tropical linear space L, we define the local tropical linear space L_B to be the subcomplex of L consisting of all vectors v that make B a basis of maximal v-weight. The tropical linear space L can then be expressed as the union of all its local tropical linear spaces, which we prove are homeomorphic to Euclidean space. Local tropical linear spaces have a simple description in terms of polyhedral matroid subdivisions, and we prove that they are dual to mixed subdivisions of Minkowski sums…
The spinor variety is cut out by the quadratic Wick relations among the principal Pfaffians of an n x n skew-symmetric matrix. Its points correspond to n-dimensional isotropic subspaces of a 2n-dimensional vector space. In this paper we tropicalize this picture, and we develop a combinatorial theory of tropical Wick vectors and tropical linear spaces that are tropically isotropic. We characterize tropical Wick vectors in terms of subdivisions of Delta-matroid polytopes, and we examine to what extent the Wick relations form a tropical basis. Our theory generalizes several results for tropical l…
In this article, we will prove that the set of 4-dissimilarity vectors of n-trees is contained in the tropical Grassmannian G_{4,n}. We will also propose three equivalent conjectures related to the set of m-dissimilarity vectors of n-trees for the case m > 4. Using a computer algebra system, we can prove these conjectures for m = 5.
Using the notion of a valuation into the semifield of piecewise linear functions, we give a classification of torus equivariant flat families of finite type over a toric variety base, by certain piecewise linear maps between fans. As a consequence we derive a classification of toric vector bundles phrased in terms of tropicalized linear spaces. We use these tools to give a characterization of the Mori dream space property for a projectivized toric vector bundle.
We show that the weights on a tropical variety can be recovered from the tropical scheme structure proposed by the Giansiracusas in arXiv:1308.0042, so there is a well-defined Hilbert-Chow morphism from a tropical scheme to the underlying tropical cycle. For a subscheme of projective space given by a homogeneous ideal I we show that this tropical scheme structure contains the same information as the set of valuated matroids of the vector spaces I_d for d \geq 0. We also give a combinatorial criterion to determine whether a given relation is in the congruence defining the tropical scheme struct…
The Cayley-Menger variety is the Zariski closure of the set of vectors specifying the pairwise squared distances between $n$ points in $\mathbb{R}^d$. This variety is fundamental to algebraic approaches in rigidity theory. We study the tropicalization of the Cayley-Menger variety. In particular, when $d = 2$, we show that it is the Minkowski sum of the set of ultrametrics on $n$ leaves with itself, and we describe its polyhedral structure. We then give a new, tropical, proof of Laman's theorem.
We investigate stratified-algebraic vector bundles on a real algebraic variety X. A stratification of X is a finite collection of pairwise disjoint, Zariski locally closed subvarieties whose union is X. A topological vector bundle on X is called a stratified-algebraic vector bundle if, roughly speaking, there exists a stratification of X such that the restriction of the bundle to each stratum is an algebraic vector bundle. In particular, every algebraic vector bundle on X is stratified-algebraic. It turns out that stratified-algebraic vector bundles have many surprising properties, which disti…
In this paper we show that semistable vector bundles on a Castelnuovo curve of genus g >= 2 have theta divisors. As a corollary, we deduce that semistable vector bundles on a smooth, general curve of genus g >= 2 which extend to semistable vector bundles on any Castelnuovo degeneration of the general curve admit a theta divisor.
Given a vector bundle on a $\mathbb{P}^1$ bundle, the base is stratified by degeneracy loci measuring the spitting type of the vector bundle restricted to each fiber. The classes of these degeneracy loci in the Chow ring or cohomology ring of the base are natural invariants characterizing the degenerations of the vector bundle. When these degeneracy loci occur in the expected codimension, we find their classes. This yields universal formulas for degeneracy classes in terms of naturally arising vector bundles on the base. Our results hold over arbitrary fields of any characteristic.
In this paper, it is proved that certain stable rank-3 vector bundles can be written as extensions of line bundles and stable rank-2 bundles. As an application, we show the rationality of certain moduli spaces of stable rank-3 bundles over the projective plane P^2.
We study vector bundles on the moduli stack of elliptic curves over a local ring R. If R is a field or a discrete valuation ring of (residue) characteristic not 2 or 3, all these vector bundles are sums of line bundles. For R the 3-local integers, we construct higher rank indecomposable vector bundles and give a classification of vector bundles that are iterated extensions of line bundles. For R the 2-local integers, we show that there are even indecomposable vector bundles of arbitrary high rank.
We give a criterion for a projectivized toric vector bundle to be a Mori dream space and describe its Cox ring using generators and relations. Both of these results are in terms of the matroids of all symmetric powers of the bundle. We also give a criterion for a toric vector bundle to be big and describe several interesting examples of toric vector bundles which highlights how positivity properties for toric vector bundles are more complicated than for toric line bundles.
We show that on a generic curve and under some conditions on the degree and genus, there exists a component B of the locus of stable vector bundles of rank r and degree d with at least k sections of the expected dimension such that for a generic vector bundle E of B, the Petri map is injective.
In this paper we study Brill-Noether loci for rank-two vector bundles and describe the general member of some components as suitable extensions of line bundles.
We give a cohomological criterion for a parabolic vector bundle on a curve to be semistable. It says that a parabolic vector bundle $E$ with rational parabolic weights is semistable if and only if there is another parabolic vector bundle $F$ with rational parabolic weights such that the cohomologies of the vector bundle underlying the para- bolic tensor product $E \otimes F$ vanish. This criterion generalizes the known semistability criterion of Faltings for vector bundles on curves and significantly improves the result in [Bis07].
We prove that vector bundles give equations of cactus varieties. We derive from it that equations coming from vector bundles are not enough to define secant varieties of Veronese varieties in general.
Among recently introduced new notions in real algebraic geometry is that of regulous functions. Such functions form a foundation for the development of regulous geometry. Several interesting results on regulous varieties and regulous sheaves are already available. In this paper, we define and investigate regulous vector bundles. We establish algebraic and geometric properties of such vector bundles, and identify them with stratified-algebraic vector bundles. Furthermore, using new results on curve-rational functions, we characterize regulous vector bundles among families of vector spaces param…
Given a parabolic vector bundle, we construct for it a projectivization and tautological line bundle. These are analogs of the projectivization and tautological line bundle for an usual vector bundle. Using these we give a construction of the parabolic Chern classes.
Let $(X, H)$ be a polarized smooth projective algebraic surface and $E$ is globally generated, stable vector bundle on $X$. Then the Syzygy bundle $M_E$ associated to it is defined as the kernel bundle corresponding to the evaluation map. In this article we will study the stability property of $M_E$ with respect to $H$.
We systematically study the splitting of vector bundles on a smooth, projective variety, whose restriction to the zero locus of a regular section of an ample vector bundle splits. First, we find ampleness and genericity conditions which ensure that the splitting of the vector bundle along the subvariety implies its global splitting. Second, we obtain a simple splitting criterion for vector bundles on the Grassmannian and on partial flag varieties.
We characterize the vector bundles on G(1,4) that have no intermediate cohomology. We obtain them from extensions of the universal bundles and others related with them. In particular, we get a characterization of the universal vector bundles from their cohomology.
We study the vector bundles without intermediate cohomology on Fano threefolds of index two, degree d=3,4,5 and Betti number one. We obtain a complete characterization in the case of rank-two vector bundles. For arbitrary rank, we give all possible Chern classes of such vector bundles, under some general condition.
Here we classify the weakly uniform rank two vector bundles on multiprojective spaces. Moreover we show that every rank $r>2$ weakly uniform vector bundle with splitting type $a_{1,1}=...=a_{r,s}=0$ is trivial and every rank $r>2$ uniform vector bundle with splitting type $a_1>...>a_r$, splits.
Shiv Nadar University（希夫纳达尔大学）；Indian Institute of Technology (Indian School of Mines) Dhanbad（印度技术学院（印度矿业学院）丹巴德）；Indraprastha Institute of Information Technology, Delhi（德里因德拉普拉斯塔信息技术学院）
In this paper, we give a simple proof of a triviality criterion due to I.Biswas and J.Pedro and P.Dos Santos. We also prove a vector bundle on a homogenous space is trivial if and only if the restrictions of the vector bundle to Schubert lines are trivial. Using this result and Chern classes of vector bundles, we give a general criterion of a uniform vector bundle on a homogenous space to be splitting. As an application, we prove a uniform vector bundle on classical Grassmannians and quadrics of low rank is splitting.
Given a complex manifold $X$, a normal crossing divisor $D\subset X$ whose irreducible components $D_1,...,D_s$ are smooth, and a choice of natural numbers $r=(r_1,...,r_s)$, we construct a manifold $X(D,\ur)$ with an action of a torus $Γ$ and we prove that some full subcategory of the category of $Γ$-equivariant vector bundles on $X(D,r)$ is equivalent to the category of parabolic vector bundles on $(X,D)$ in which the lengths of the filtrations over each irreducible component of $X$ are given by $r$. When $X$ is Kaehler, we study the Kaehler cone of $X(D,r)$ and the relation between the corr…
We prove an existence result for stable vector bundles with arbitrary rank on an algebraic surface, and determine the birational structure of certain moduli space of stable bundles on a rational ruled surface.
