Moduli Space Of Vector Bundles Research Articles

Rational Curves on Moduli Spaces of Vector Bundles

Rational Curves on Moduli Spaces of Vector Bundles

Interpolation and moduli spaces of vector bundles on very general blowups of the projective plane

In this paper, we study certain moduli spaces of vector bundles on the blowup of the projective plane in at least 10 very general points. Moduli spaces of sheaves on general type surfaces may be nonreduced, reducible and even disconnected. In contrast, moduli spaces of sheaves on minimal rational surfaces and certain del Pezzo surfaces are irreducible and smooth along the locus of stable bundles. We find examples of moduli spaces of vector bundles on more general blowups of the projective plane that are disconnected and have components of different dimensions. In fact, assuming the SHGH Conjecture, we can find moduli spaces with arbitrarily many components of arbitrarily large dimension.

Higgs fields, non-abelian Cauchy kernels and the Goldman symplectic structure

We consider the moduli space of vector bundles of rank n and degree ng over a fixed Riemann surface of genus g⩾2 with the explicit parametrization in terms of the Tyurin data. The ‘non-abelian’ theta divisor consists of bundles such that h1⩾1 . On the complement of this divisor we construct a non-abelian (i.e. matrix) Cauchy kernel explicitly in terms of the Tyurin data. With the additional datum of a non-special divisor, we can construct a reference flat holomorphic connection which also depends holomorphically on the moduli of the bundle. This allows us to identify the bundle of Higgs fields, i.e. the cotangent bundle of the moduli space, with the affine bundle of holomorphic connections and provide a monodromy map into the GLn character variety. We show that the Goldman symplectic structure on the character variety pulls back along this map to the complex canonical symplectic structure on the cotangent bundle and hence also on the space of connections. The pull-back of the Liouville one-form to the affine bundle of connections is then shown to be a logarithmic form with poles along the non-abelian theta divisor and residue given by h 1.

Moduli spaces of vector bundles on a curve and opers

Let X be a compact connected Riemann surface of genus g, with $$g\, \ge \,2$$ , and let $$\xi $$ be a holomorphic line bundle on X with $$\xi ^{\otimes 2}\,=\, {\mathcal O}_X$$ . Fix a theta characteristic $${\mathbb {L}}$$ on X. Let $${\mathcal M}_X(r,\xi )$$ be the moduli space of stable vector bundles E on X of rank r such that $$\bigwedge ^r E\,=\, \xi $$ and $$H^0(X,\, E\otimes {\mathbb L})\,=\, 0$$ . Consider the quotient of $${\mathcal M}_X(r,\xi )$$ by the involution given by $$E\, \longmapsto \, E^*$$ . We construct an algebraic morphism from this quotient to the moduli space of $$\textrm{SL}(r,{\mathbb C})$$ opers on X. Since $$\dim {\mathcal M}_X(r,\xi )$$ coincides with the dimension of the moduli space of $$\textrm{SL}(r,{\mathbb C})$$ opers, it is natural to ask about the injectivity and surjectivity of this map.

Moduli of vector bundles on primitive multiple schemes

A primitive multiple scheme is a Cohen–Macaulay scheme Y such that the associated reduced scheme X = Yred is smooth, irreducible, and that Y can be locally embedded in a smooth variety of dimension [Formula: see text]. If n is the multiplicity of Y, there is a canonical filtration [Formula: see text], such that [Formula: see text] is a primitive multiple scheme of multiplicity i. The simplest example is the trivial primitive multiple scheme of multiplicity n associated to a line bundle L on X: it is the nth infinitesimal neighborhood of X, embedded in the line bundle [Formula: see text] by the zero section. The main subject of this paper is the construction and properties of fine moduli spaces of vector bundles on primitive multiple schemes. Suppose that [Formula: see text] is of multiplicity n, and can be extended to [Formula: see text] of multiplicity [Formula: see text], and let [Formula: see text] be a fine moduli space of vector bundles on [Formula: see text]. With suitable hypotheses, we construct a fine moduli space [Formula: see text] for the vector bundles on [Formula: see text] whose restriction to [Formula: see text] belongs to [Formula: see text]. It is an affine bundle over the subvariety [Formula: see text] of bundles that can be extended to [Formula: see text]. In general this affine bundle is not banal. This applies in particular to Picard groups. We give also many new examples of primitive multiple schemes Y such that the dualizing sheaf [Formula: see text] is trivial.

Motives of moduli spaces of bundles on curves via variation of stability and flips

AbstractWe study the rational Chow motives of certain moduli spaces of vector bundles on a smooth projective curve with additional structure (such as a parabolic structure or Higgs field). In the parabolic case, these moduli spaces depend on a choice of stability condition given by weights; our approach is to use explicit descriptions of variation of this stability condition in terms of simple birational transformations (standard flips/flops and Mukai flops) for which we understand the variation of the Chow motives. For moduli spaces of parabolic vector bundles, we describe the change in motive under wall‐crossings, and for moduli spaces of parabolic Higgs bundles, we show the motive does not change under wall‐crossings. Furthermore, we prove a motivic analogue of a classical theorem of Harder and Narasimhan relating the rational cohomology of moduli spaces of vector bundles with and without fixed determinant. For rank 2 vector bundles of odd degree, we obtain formulae for the rational Chow motives of moduli spaces of semistable vector bundles, moduli spaces of Higgs bundles and moduli spaces of parabolic (Higgs) bundles that are semistable with respect to a generic weight (all with and without fixed determinant).

Maximality of moduli spaces of vector bundles on curves

We prove that moduli spaces of semistable vector bundles of coprime rank and degree over a non-singular real projective curve are maximal real algebraic varieties if and only if the base curve itself is maximal. This provides a new family of maximal varieties, with members of arbitrarily large dimension. We prove the result by comparing the Betti numbers of the real locus to the Hodge numbers of the complex locus and showing that moduli spaces of vector bundles over a maximal curve actually satisfy a property which is stronger than maximality and that we call Hodge-expressivity. We also give a brief account on other varieties for which this property was already known.

Derived Category and ACM Bundles of Moduli Space of Vector Bundles on a Curve

Abstract We show that the derived category of a curve is embedded into the derived category of the moduli space of vector bundles on the curve of coprime rank and degree. We also generalize the semiorthogonal decomposition constructed by Narasimhan and Belmans-Mukhopadhyay. Finally, we produce a one-dimensional family of ACM bundles over the moduli space.

Decompositions of moduli spaces of vector bundles and graph potentials

Abstract We propose a conjectural semiorthogonal decomposition for the derived category of the moduli space of stable rank 2 bundles with fixed determinant of odd degree, independently formulated by Narasimhan. We discuss some evidence for and furthermore propose semiorthogonal decompositions with additional structure. We also discuss two other decompositions. One is a decomposition of this moduli space in the Grothendieck ring of varieties, which relates to various known motivic decompositions. The other is the critical value decomposition of a candidate mirror Landau–Ginzburg model given by graph potentials, which in turn is related under mirror symmetry to Muñoz’s decomposition of quantum cohomology. This corresponds to an orthogonal decomposition of the Fukaya category. We discuss how decompositions on different levels (derived category of coherent sheaves, Grothendieck ring of varieties, Fukaya category, quantum cohomology, critical sets of graph potentials) are related and support each other.

On the rationality of moduli spaces of vector bundles over chain-like curves

Let C be a chain-like curve over C. In this paper, we investigate the rationality of moduli spaces of w-semistable vector bundles on C of arbitrary rank and fixed determinant by putting some restrictions on the Euler characteristics.

25 open questions about vector bundles and their moduli

We present 25 open questions about moduli spaces of vector bundles and related topics, and discuss some longstanding conjectures. We hope to inspire young researchers to engage in this area of re- search.

Minimal rational curves on the moduli spaces of symplectic and orthogonal bundles

Let $C$ be an algebraic curve of genus $g$ and $L$ a line bundle over $C$. Let $\mathcal{MS}_C(n,L)$ and $\mathcal{MO}_C(n,L)$ be the moduli spaces of $L$-valued symplectic and orthogonal bundles respectively, over $C$ of rank $n$. We construct rational curves on these moduli spaces which generalize Hecke curves on the moduli space of vector bundles. As a main result, we show that these Hecke type curves have the minimal degree among the rational curves passing through a general point of the moduli spaces. As its byproducts, we show the non-abelian Torelli theorem and compute the automorphism group of moduli spaces.

Quantizations of local surfaces and rebel instantons

We construct explicit deformation quantizations of the noncompact complex surfaces Z_k:=\operatorname{Tot}(\mathcal{O}_{\mathbb{P}^1}(-k)) and describe their effect on moduli spaces of vector bundles and instanton moduli spaces. We introduce the concept of rebel instantons, as being those which react badly to some quantizations, misbehaving by shooting off extra families of noncommutative instantons. We then show that the quantum instanton moduli space can be viewed as the étale space of a constructible sheaf over the classical instanton moduli space with support on rebel instantons.

On the splitting type of holomorphic vector bundles induced from regular systems of differential equation

Abstract We calculate the splitting type of holomorphic vector bundles on the Riemann sphere induced by a Fuchsian system of differential equations. Using this technique, we indicate the relationship between Hölder continuous matrix functions and a moduli space of vector bundles on the Riemann sphere. For second order systems with three singular points we give a complete characterization of the corresponding vector bundles by the invariants of Fuchsian system.

Meromorphic connections, determinant line bundles and the Tyurin parametrization

We develop a holomorphic equivalence between on one hand the space of pairs (stable bundle, flat connection on the bundle) and the of holomorphic (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 of holomorphic bundles and holomorphic connections, in particular those arising from families of stable bundles over the surface. These generalize the Tyurin parametrization of stable vector bundles $E$ over a compact connected Riemann surface, and one can build above them spaces of (equivalence classes of) connections, which are again symplectic. These spaces are also symplectically biholomorphically equivalent to the sheaf of connections for the determinant bundle over the Tyurin family. The last portion of the paper shows how this extends to moduli of framed bundles.

Infinite dimensional families of Calabi–Yau threefolds and moduli of vector bundles

We study noncompact Calabi–Yau threefolds, their moduli spaces of vector bundles and deformation theory. We present Calabi–Yau threefolds that have infinitely many distinct deformations, constructing them explicitly, and describe the effect that such deformations produce on moduli spaces of vector bundles.

Moduli spaces of vector bundles on a real nodal curve

Let Y be a geometrically irreducible nodal projective algebraic curve of arithmetic genus $$g \ge 2$$ defined over $${{\mathbb {R}}}$$ . Let $$Y_{{\mathbb {C}}} = Y \times _{{\mathbb {R}}} {\mathbb {C}}$$ . Fix an $${\mathbb {R}}$$ -valued point $$\xi $$ of $$Pic^d(Y)$$ . For integers $$r \ge 2$$ and d, let $$M(r,\xi )$$ [respectively $$U(r,\xi )$$ ] be the moduli stack of vector bundles (respectively the moduli space of stable vector bundles) of rank r and determinant $$\xi $$ on Y. We determine the Picard group of $$M(r,\xi )$$ . We compute the Picard group and Brauer group of $$U(r,\xi )$$ .

Moduli spaces of vector bundles with fixed determinant over a real curve

Let $$(\Sigma ,\tau )$$ denote a Riemann surface of genus $$g \ge 2$$ equipped with an anti-holomorphic involution $$\tau $$. In this paper we study the topology of the moduli space $$M(r,\xi )^\tau $$ of stable Real vector bundles over $$(\Sigma ,\tau )$$ of rank r and fixed determinant $$\xi $$ of degree coprime to r. We prove that $$M(r,\xi )^{\tau }$$ is an orientable and monotone Lagrangian submanifold of the complex moduli space $$M(r,\xi )$$ so it determines an object in the appropriate Fukaya category. We derive recursive formulas for the $${\mathbb {Z}}_2$$-Betti numbers of $$M(r,\xi )^\tau $$ and compute $${\mathbb {Z}}_p$$-Betti numbers for odd p through a range of degrees. We deduce that if r is even and $$ g>>0$$, then $$M(r,\xi )^{\tau }$$ and $$M(r,\xi ')^{\tau }$$ have non-isomorphic cohomology groups unless $$\xi $$ and $$\xi '$$ have equivalent Stieffel–Whitney classes modulo automorphisms of $$(\Sigma ,\tau )$$. If r is even, and $$g>>0$$ is even, we prove that the Betti numbers of $$M(r,\xi )^{\tau }$$ distinguish topological types of $$(\Sigma , \tau ; \xi )$$. If $$r=2$$ and g is odd, we compute all $${\mathbb {Z}}_p$$-Betti numbers of $$M(2,\xi )^\tau $$.

Finite generation of the algebra of type A conformal blocks via birational geometry II: higher genus

We prove finite generation of the algebra of type A conformal blocks over arbitrary stable curves of any genus. As an application we construct a flat family of irreducible normal projective varieties over the moduli stack of stable pointed curves, whose fiber over a smooth curve is a moduli space of semistable parabolic bundles. This generalizes a construction of a degeneration of the moduli space of vector bundles presented in a recent work of Belkale and Gibney.

Theta divisors and Ulrich bundles on geometrically ruled surfaces

We consider the following question: for which invariants $g$ and $e$ is there a geometrically ruled surface $S \rightarrow C$ over a curve $C$ of genus $g$ with invariant $e$ such that $S$ is the support of an Ulrich line bundle with respect to a very ample line bundle? A surprising relation between the existence of certain proper Theta divisors on some moduli spaces of vector bundles on $C$ with the existence of Ulrich line bundles on $S$ will be the key to completely solve the above question. The relation is realized by translating the vanishing conditions characterizing Ulrich line bundles to specific geometric conditions on the symmetric powers of the defining vector bundle of a given ruled surface. This general principle leads to some finer existence results of Ulrich line bundles in particular cases. Another focus is on the rank two case where, with very few exceptions, we show the existence of large families of special Ulrich bundles on arbitrary polarized ruled surfaces.