Let $X$ be a projective variety with an action of a reductive group $G$. Each ample $G$-line bundle $L$ on $X$ defines an open subset $X^{\rm ss}(L)$ of semi-stable points. Following Dolgachev and Hu, define a GIT-class as the set of algebraic equivalence classes of $L'$s with fixed $X^{\rm ss}(L)$. We show that the GIT-classes are the relative interiors of rational polyhedral convex cones, which form a fan in the $G$-ample cone. We also study the corresponding variations of quotients $X^{\rm ss}(L)//G$. This sharpens results of Thaddeus and Dolgachev-Hu.
We give a description of the image of tensor products of tautological bundles on Hilbert schemes of points on surfaces under the Bridgeland-King-Reid-Haiman equivalence. Using this, some new formulas for cohomological invariants of these bundles are obtained. In particular, we give formulas for the Euler characteristic of arbitrary tensor products on the Hilbert scheme of two points and of triple tensor products in general.
In this paper, we study triples of the form (E, theta, phi) over a compact Riemann Surface, where (E, theta) is a Higgs bundle and phi is a global holomorphic section of the Higgs bundle. Our main result is a description of a birational equivalence which relates geometrically the moduli space of Higgs bundles of rank r and degree d to the moduli space of Higgs bundles of rank (r-1) and degree d.
Hartshorne in "Ample vector bundles" proved that $E$ is ample if and only if $\OOO_{P(E)}(1)$ is ample. Here we generalize this result to flag manifolds associated to a vector bundle $E$ on a complex manifold $X$: For a partition $a$ we show that the line bundle $\it Q_a^s$ on the corresponding flag manifold $\mathcal{F}l_s(E)$ is ample if and only if $ \SSS_aE $ is ample. In particular $\det Q$ on $\it{G}_r(E)$ is ample if and only if $\wedge ^rE$ is ample.\\ We give also a proof of the Ampleness Dominance theorem that does not depend on the saturation property of the Littlewood-Richardson se…
In this paper, we prove that the notions of Hilbert stability and Mumford stability agree for vector bundles of arbitrary rank over smooth curves. The notion of Hilbert stability was introduced by Gieseker and Morrison in 1984, and they showed that for smooth curves and vector bundles of rank two it agrees with Mumford stability. A different proof for the rank two case was given by M. Teixidor i Bigas. Our proof uses a new approach and avoids complicated computations. Our results might serve as a first step in the construction of the Hilbert stable compactification of the universal moduli spac…
We develop a holomorphic equivalence between on one hand the space of pairs (stable bundle, flat connection on the bundle) and the "sheaf of holomorphic connections" (the sheaf of splittings of the one-jet sequence) for the determinant (Quillen) line bundle over the moduli space of vector bundles on a compact connected Riemann surface. This equivalence is shown to be holomorphically symplectic. The equivalences, both holomorphic and symplectic, seem to be quite general, in that they extend to other general families of holomorphic bundles and holomorphic connections, in particular those arising…
We investigate a construction providing pairs of Calabi-Yau varieties described as zero loci of pushforwards of a hyperplane section on a roof as described by Kanemitsu. We discuss the implications of such construction at the level of Hodge equivalence, derived equivalence and $\mathbb L$-equivalence. For the case of $K3$ surfaces, we provide alternative interpretations for the Fourier-Mukai duality in the family of $K3$ surfaces of degree 12 of Mukai. In all these constructions the derived equivalence lifts to an equivalence of matrix factorizations categories.
We construct an equivalence of categories from a strong categorical sl(2) action, following the work of Chuang-Rouquier. As an application, we give an explicit, natural equivalence between the derived categories of coherent sheaves on cotangent bundles to complementary Grassmannians.
Mehta and Seshadri have proved that the set of equivalence classes of irreducible unitary representations of the fundamental group of a punctured compact Riemann surface, can be identified with equivalence classes of stable parabolic bundles of parabolic degree zero on the compact Riemann surface. In this note, we discuss the Mehta-Seshadri correspondence over an irreducible projective curve with at most nodes as singularities.
Brown proved that the I-function of a toric fibration lies on the overruled Lagrangian cone of its genus zero Gromov-Witten theory, introduced by Coates and Givental. In this paper, we prove the theorem for partial flag variety fibrations. To do so, we will construct new moduli spaces generalising the idea of Ciocan-Fontanine, Kim and Maulik.
We describe the GIT-equivalence classes of linearized ample line bundles for the diagonal actions of the linear algebraic groups $SL(V)$ and $SO(V)$ on ${\mathbb{P}(V)^{m_1}\times \mathbb{P}(V^*)^{m_2}}$ and $\mathbb{P}(V)^m$ respectively.
In this paper, we investigate the structure of the convergent quantization of the 1-shifted cotangent bundle $S$ of a smooth scheme $X$ over a perfect field of positive characteristic. The quantization is an $E_2$-algebra over the Frobenius twist $S'$ of the 1-shifted cotangent bundle which restricted to the zero section $X'\rightarrow S'$ is weakly $E_2$-Morita equivalent to the structure sheaf $\mathscr O_{X'}$ of the Frobenius twist $X'$ of $X$. Explicitly, we show that the $(\infty,2)$-category of coherent (left-)modules over $\mathscr O_{X'}$ is equivalent to the full subcategory of the $…
The aim of this paper is to establish an equivalence of certain categories of Higgs bundles on a non-isotrivial elliptic surface $π: X \rightarrow C$ with $χ(X) > 0$ and certain categories of Parabolic Higgs bundles on $C$
Let $C$ be a hyperelliptic curve of genus $g \geq 3$. We give a new description of the theta map for moduli spaces of rank 2 semistable vector bundles with trivial determinant. In orther to do this, we describe a fibration of (a birational model of) the moduli space, whose fibers are GIT quotients $(\mathbb{P}^1)^{2g}//\operatorname{PGL(2)}$. Then, we use recent results of Kumar to identify the restriction of the theta map to these GIT quotients with some explicit osculating projection. As a corollary of this construction, we obtain a birational equivalence between the ramification locus of th…
Let $M$ be a quasi-regular compact connected Sasakian manifold, and let $N = M/S^1$ be the base projective variety. We establish an equivalence between the class of Sasakian $G$-Higgs bundles over $M$ and the class of parabolic (or equivalently, ramified) $G$-Higgs bundles over the base $N$.
By considering the equivalence between the category of locally free sheaves and the category of algebraic vector bundles, we show how elementary transformations of vector bundles can be used to prove a case of the maximal rank hypothesis. We in turn show how this can be applied in the study of minimal free resolutions.
A homogeneous roof is a rational homogeneous variety of Picard rank 2 and index $r$ equipped with two different $\mathbb P^{r-1}$-bundle structures. We consider bundles of homogeneous roofs over a smooth projective variety, formulating a relative version of the duality of Calabi--Yau pairs associated to roofs of projective bundles. We discuss how derived equivalence of such pairs can lift to Calabi--Yau fibrations, extending a result of Bridgeland and Maciocia to higher-dimensional cases. We formulate an approach to prove that the $DK$-conjecture holds for a class of simple $K$-equivalent maps…
It is known that a tilting generator on an algebraic variety $X$ gives a derived equivalence between $X$ and a certain non-commutative algebra. In this paper, we explain a method to construct a tilting generator from an ample line bundle, and construct it in several examples.
Given an algebraic torus action on a normal projective variety with finitely generated total coordinate ring, we study the GIT-equivalence for not necessarily ample linearized divisors, and we provide a combinatorial description of the partially ordered set of GIT-equivalence classes. As an application, we extend in the $\QQ$-factorial case a basic feature of the collection of ample GIT-classes to the partially ordered collection of maximal subsets with a quasiprojective quotient: for any two members there is at most one minimal member comprising both of them. Moreover, we demonstrate in an ex…
We consider the problem of comparing t-structures under the derived McKay correspondence and for tilting equivalences. We relate the t-structures using certain natural torsion theories. As an application, we give a criterion for rationality for surfaces with a tilting bundle. In particular we show that every smooth projective surface which admits a full, strong, exceptional collection of line bundles is rational.
We construct vector bundles $R^r_μ$ on a smooth projective curve $X$ having the property that for all sheaves $E$ of slope $μ$ and rank $r$ on $X$ we have an equivalence: $E$ is a semistable vector bundle $\iff$ $Hom(R^r_μ,E)=0$. As a byproduct of our construction we obtain effective bounds on $r$ such that the linear system $|R \cdot Θ|$ has base points on the moduli space $U_X(r,r(g-1))$.
Let $k$ be an algebraically closed field. Consider a reductive group $G$ over $k$. Let $X$ be a projective variety over $k$ with a $G$-action and let $L$ be a very ample $G$-linearized line bundle on $X$. Suppose that $L$ descends to the GIT quotient of $X$ by $G$. If $L$ satisfies the property $N_p$ one can ask if its descent also has $N_p$ property. In this article, we show this is the case under certain conditions. We then apply our results to some cases of interest. As a consequence of our results, we show that if $G$ is a finite group and $L$ satisfies $N_p$ property and its descent satis…
We study the problem of classifying projectivizations of rank-two vector bundles over ${\mathbb P}^2$ up to various notions of equivalence that arise naturally in ${\mathbb A}^1$-homotopy theory, namely ${\mathbb A}^1$-weak equivalence and ${\mathbb A}^1$-$h$-cobordism. First, we classify such varieties up to ${\mathbb A}^1$-weak equivalence: over algebraically closed fields having characteristic unequal to two the classification can be given in terms of characteristic classes of the underlying vector bundle. When the base field is ${\mathbb C}$, this classification result can be compared to a…
Let L be a homogeneous ample line bundle on any flag variety G/P and let T be a maximal torus of G. We prove a general necessary and sufficient condition for L to descend as a line bundle on the GIT quotient of G/P by T. We use this result to explicitly determine exactly which L descend to the GIT quotient for any simple complex algebraic group G and any parabolic subgroup P.
We show by an example that the (equivalence class of) singularity of a plurisubharmonic function cannot be determined by the data of its Lelong numbers, in a nontrivial sense. Such an example is provided by Siu-type singular hermitian metrics associated to an effective line bundle. We also show that a Siu-type metric has analytic singularities if and only if the section ring of the line bundle is finitely generated.
We say that an exact equivalence between the derived categories of two algebraic varieties is tilting-type if it is constructed by using tilting bundles. The aim of this article is to understand the behavior of tilting-type equivalences for crepant resolutions under deformations. As an application of the method that we establish in this article, we study the derived equivalence for stratified Mukai flops and stratified Atiyah flops in terms of tilting bundles.
Special kinds of rank 2 vector bundles with (possibly irregular) connections on P^1 are considered. We construct an equivalence between the derived category of quasi-coherent sheaves on the moduli stack of such bundles and the derived category of modules over a TDO ring on certain non-separated curve. We identify this curve with the coarse moduli space of some parabolic bundles on P^1. Then our equivalence becomes an example of the categorical Langlands correspondence.
Let $X$ be a compact Kähler manifold. The set $\cha(X)$ of one-dimensional complex valued characters of the fundamental group of $X$ forms an algebraic group. Consider the subset of $\cha(X)$ consisting of those characters for which the corresponding local system has nontrivial cohomology in a given degree $d$. This set is shown to be a union of finitely many components that are translates of algebraic subgroups of $\cha(X)$. When the degree $d$ equals 1, it is shown that some of these components are pullbacks of the character varieties of curves under holomorphic maps. As a corollary, it is s…
We present equivalences between certain categories of vector bundles on projective varieties, namely cokernel bundles, Steiner bundles, syzygy bundles, and monads, and full subcategories of representations of certain quivers. As an application, we provide decomposability criteria for such bundles.
For line bundles on arithmetic varieties we construct height functions using arithmetic intersection theory. In the case of an arithmetic surface, generically of genus g, for line bundles of degree g equivalence is shown to the height on the Jacobian defined by the Theta divisor.
In the setting of a variety $X$ admitting a tilting bundle $T$ we consider the problem of constructing $X$ as a quiver GIT quotient of the endomorphism algebra $A=\textrm{End}_X(T)^{\textrm{op}}$ corresponding to the tilting bundle. We prove that if the tilting equivalence restricts to a bijection between the skyscraper sheaves of $X$ and the closed points of a quiver GIT moduli functor for $A$ then $X$ is indeed a fine moduli space for this quiver GIT moduli functor, and we prove this result without any assumptions on the singularities of $X$. As an application we consider varieties which are…
A conjecture of Hirschowitz's predicts that a globally generated vector bundle $W$ on a compact complex manifold $A$ satisfies the formal principle, i.e., the formal neighborhood of its zero section determines the germ of neighborhoods in the underlying complex manifold of the vector bundle $W$. By applying Cartan's equivalence method to a suitable differential system on the universal family of the Douady space of the complex manifold, we prove that this conjecture is true if $A$ is a Fano manifold, or if the global sections of $W$ separate points of $A$. Our method shows more generally that f…
These notes give an introduction to Geometric Invariant Theory and symplectic reduction, with lots of pictures and simple examples. We describe their applications to moduli of bundles and varieties, and their infinite dimensional analogues in gauge theory and the theory of special metrics on algebraic varieties. Donaldson's "quantisation" link between the infinite and finite dimensional situations is described, as are surprisingly strong connections between the bundle and variety cases.
We define the category of B-branes in a (not necessarily affine) Landau-Ginzburg B-model, incorporating the notion of R-charge. Our definition is a direct generalization of the category of perfect complexes. We then consider pairs of Landau-Ginzburg B-models that arise as different GIT quotients of a vector space by a one-dimensional torus, and show that for each such pair the two categories of B-branes are quasi-equivalent. In fact we produce a whole set of quasi-equivalences indexed by the integers, and show that the resulting auto-equivalences are all spherical twists.
We consider the descent of line bundles to GIT quotients of products of flag varieties. Let $G$ be a simple, connected, algebraic group over $\mathbb{C}$. We fix a Borel subgroup $B$ and consider the diagonal action of $G$ on the projective variety $X = G/B \times G/B \times G/B$. For any triple $(λ, μ, ν)$ of dominant regular characters there is a $G$-equivariant line bundle $\mathcal{L}$ on $X$. Then, $\mathcal{L}$ is said to descend to the GIT quotient $π:[X(\mathcal{L})]^{ss} \rightarrow X(\mathcal{L})//G$ if there exists a line bundle $\hat{\mathcal{L}}$ on $X(\mathcal{L})//G$ such that $…
Kirillov has described a McKay correspondence for finite subgroups of PSL_{2}(C) that associates to each `height' function an affine Dynkin quiver together with a derived equivalence between equivariant sheaves on the projective line P^1 and representations of this quiver. The equivalences for different height functions are then related by reflection functors for quiver representations. The main goal of this paper is to develop an analogous story for the cotangent bundle of P^1. We show that each height function gives rise to a derived equivalence between equivariant sheaves on the cotangent b…
For a finite subgroup G in SL(3,C), Bridgeland, King and Reid proved that the moduli space of G-clusters is a crepant resolution of the quotient C^3/G. This paper considers the moduli spaces M_θ, introduced by Kronheimer and further studied by Sardo Infirri, which coincide with G-Hilb for a particular choice of the GIT parameter θ. For G Abelian, we prove that every projective crepant resolution of C^3/G is isomorphic to M_θfor some parameter θ. The key step is the description of GIT chambers in terms of the K-theory of the moduli space via the appropriate Fourier--Mukai transform. We also unc…
Geometric Invariant Theory gives a method for constructing quotients for group actions on algebraic varieties which in many cases appear as moduli spaces parametrizing isomorphism classes of geometric objects (vector bundles, polarized varieties, etc.). The quotient depends on a choice of an ample linearized line bundle. Two choices are equivalent if they give rise to identical quotients. A priori, there are infinitely many choices since there are infinitely many isomorphism classes of linearized ample line bundles. Hence several fundamental questions naturally arise. Is the set of equivalence…
It is well known that degree two Deligne cohomology groups can be identified with groups of isomorphism classes of holomorphic line bundles with connections. There is also a geometric description of degree three Deligne cohomology, due to J-L. Brylinski and P. Deligne, in terms of gerbes with connective structures and curvings. This paper gives a geometric interpretation of Deligne cohomology of all degrees, in terms of equivalence classes of higher line bundles with $k$-connections. It is also shown that the classical Abel-Jacobi isomorphism $Pic^0(X) \cong J(X)$ generalizes to the isomorphis…
In this paper, using the Atiyah-Ward equivalence and a theorem of Hitchin, one makes to correspond to certain bundles on the projective space, which are extensions of instanton bundles (in particular, these new bundles may have the first Chern class not congruent to 0 modulo the rank), bundles on the grassmanian Gr(2,4) or on the sphere S4, equiped with certain connections constructed from data associated to the bundle. Monads for these extensions are described.
This survey intends to present the basic notions of Geometric Invariant Theory (GIT) through its paradigmatic application in the construction of the moduli space of holomorphic vector bundles. Special attention is paid to the notion of stability from different points of view and to the concept of maximal unstability, represented by the Harder-Narasimhan filtration and, from which, correspondences with the GIT picture and results derived from stratifications on the moduli space are discussed.
In this paper we consider the total space of the canonical bundle of P^2 and we use a proposal by Hosono, together with results in Seidel and Auroux-Katzarkov-Orlov, to deduce the right physical mirror equivalence between D^b(K_{P^2}) and the derived Fukaya category of its mirror. By construction, our equivalence is compatible with the mirror map between moduli spaces and with the computation of Gromov--Witten invariants.
