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Let X be a compact and connected Riemann surface of genus g ≥ 2. In this paper, moduli spaces of Higgs bundles over X with structure group SL(n, ℂ), for n ≥ 3, and Spin(2n, ℂ), for n ≥ 4, are considered. In the case of structure group SL(n, ℂ), two involutions of the Higgs bundle moduli space are defined and an alternative proof is given to show, using specific properties of the structure group, that the stable fixed points can be described as certain Higgs pairs with structure group SO(n, ℂ) or Sp(n, ℂ). Moreover, the notions of stability, semistability, and polystability of the obtained Higgs pairs are established. Also, two involutions of the moduli space of Higgs bundles with structure group Spin(2n, ℂ) are defined for n ≥ 4 analogously and it is shown, also using specific properties of the group, that their stable fixed points can be described as certain Higgs pairs whose structure group is of the form Spin(2r + 1, ℂ) × Spin(2n − 2r − 1, ℂ) for some 0 ≤ r ≤ n − 1. [ABSTRACT FROM AUTHOR]

We generalize to vector bundles the techniques introduced for line bundles in Liu (TAMS 374:367–405, 2021). We then use this method to prove the injectivity of the Petri map for vector bundles and the surjectivity of a map related to deformation theory of Poincaré sheaves. [ABSTRACT FROM AUTHOR]

Let X be a compact connected Riemann surface, and let Q (r , d) denote the quot scheme parametrizing the torsion quotients of O X ⊕ r of degree d. Given a projective structure P on X, we show that the cotangent bundle T ∗ U of a certain nonempty Zariski open subset U ⊂ Q (r , d) , equipped with the natural Liouville symplectic form, admits a canonical deformation quantization. When r = 1 = d , then Q (r , d) = X ; this case was addressed earlier in Ben-Zvi and Biswas (Lett. Math. Phys.54 (2000) 73–82). [ABSTRACT FROM AUTHOR]

Let X be a compact connected Riemann surface and (V , ϕ) a holomorphic Lie algebroid on X such that the holomorphic vector bundle V is stable. We give a necessary and sufficient condition on holomorphic vector bundles E on X to admit a Lie algebroid connection. [ABSTRACT FROM AUTHOR]

Let X be a compact Riemann surface of genus g ≥ 2 and let G be the group Spin(8, ℂ) or PSO(8, ℂ). Let τ be a choice of a non-trivial outer automorphism of G of order 3, called triality. Suppose that X is equipped with a holomorphic order 3 automorphism α. Then α acts on the moduli space M(G) of principal G-bundles over X by taking the pull-back, and τ also acts on M(G), by modifying the action of G on the principal bundles through any automorphism of G of order 3 representing τ. The combination of these two actions gives an automorphism of order 3 of the moduli space M(G). In this work, the notion of α-trialitarian G-bundle is introduced to describe the fixed points of the automorphism of M(G) mentioned above. In particular, an equivalent notion of stability and polystability for such bundles is proved, and an explicit construction of them is provided. [ABSTRACT FROM AUTHOR]

Let X be an irreducible smooth projective curve defined over F ¯ p and E a vector bundle on X of rank at least two. For any 1 ≤ r < rank (E) , let Gr r (E) be the Grassmann bundle over X parametrizing all the r dimensional quotients of the fibers of E. We prove that the effective cone in NS (Gr r (E)) ⊗ Z R coincides with the pseudo-effective cone in NS (Gr r (E)) ⊗ Z R . When r = 1 or rank (E) - 1 , this was proved in [4]. [ABSTRACT FROM AUTHOR]

Let f : X ⟶ Y be a generically smooth nonconstant morphism between irreducible projective curves, defined over an algebraically closed field, which is étale on an open subset of Y that contains both the singular locus of Y and the image, in Y, of the singular locus of X. We prove that the following statements are equivalent: The homomorphism of étale fundamental groups f ∗ : π 1 et (X) ⟶ π 1 et (Y) induced by f is surjective. There is no nontrivial étale covering ϕ : Y ′ ⟶ Y admitting a morphism q : X ⟶ Y ′ such that ϕ ∘ q = f . The fiber product X × Y X is connected. dim H 0 (X , f ∗ f ∗ O X) = 1 . O Y ⊂ f ∗ O X is the maximal semistable subsheaf. The pullback f ∗ E of every stable sheaf E on Y is also stable. [ABSTRACT FROM AUTHOR]

In enumerative geometry, Virasoro constraints were first conjectured in Gromov-Witten theory with many new recent developments in the sheaf theoretic context. In this paper, we rephrase the sheaf theoretic Virasoro constraints in terms of primary states coming from a natural conformal vector in Joyce's vertex algebra. This shows that Virasoro constraints are preserved under wall-crossing. As an application, we prove the conjectural Virasoro constraints for moduli spaces of torsion-free sheaves on any curve and on surfaces with only (p , p) cohomology classes by reducing the statements to the rank 1 case. [ABSTRACT FROM AUTHOR]

Let X be a K3 surface, let C be a smooth curve of genus g on X, and let A be a line bundle of degree d on C. Then a line bundle M on X with M ⊗ O C = A is called a lift of A. In this paper, we prove that if the dimension of the linear system |A| is r ≥ 2 , g > 2 d - 3 + (r - 1) 2 , d ≥ 2 r + 4 , and A computes the Clifford index of C, then there exists a base point free lift M of A such that the general member of |M| is a smooth curve of genus r. In particular, if |A| is a base point free net which defines a double covering π : C ⟶ C 0 of a smooth curve C 0 ⊂ P 2 of degree k ≥ 4 branched at distinct 6k points on C 0 , then, by using the aforementioned result, we can also show that there exists a 2:1 morphism π ~ : X ⟶ P 2 such that π ~ | C = π . [ABSTRACT FROM AUTHOR]

Let D be a reduced effective strict normal crossing divisor on a smooth complex variety X, and let X D be the associated root stack over C . Suppose that X admits an anti-holomorphic involution (real structure) that keeps D invariant. We show that the root stack X D naturally admits a real structure compatible with X. We also establish an equivalence of categories between the category of real logarithmic connections on this root stack and the category of real parabolic connections on X. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic 2, X a smooth projective curve of genus g = 2 over k, M X s (r , d) the moduli space of stable vector bundles of rank r and degree d on X. We classify the Frobenius destabilized stable vector bundles of rank 4 and degree d on X in terms of the Harder-Narasimhan Polygons and study the geometry of the Frobenius strata in M X s (4 , d) when 2 ∣ d . [ABSTRACT FROM AUTHOR]

We introduce a categorical construction of Harder–Narasimhan filtration via a slope method which does not need a degree function. With a theorem of existence and uniqueness of Harder–Narasimhan filtration in our categorical setting, we give a categorical interpretation of Stuhler–Grayson filtration in the case of not necessarily Hermitian normed lattices. [ABSTRACT FROM AUTHOR]

This present paper is devoted to the study of a class of Nakayama algebras N n (r) given by the path algebra of the equioriented quiver A n subject to the nilpotency degree r for each sequence of r consecutive arrows. We show that the Nakayama algebras N n (r) for certain pairs (n, r) can be realized as endomorphism algebras of tilting objects in the bounded derived category of coherent sheaves over a weighted projective line, or in its stable category of vector bundles. Moreover, we classify all the Nakayama algebras N n (r) of Fuchsian type, that is, derived equivalent to the bounded derived categories of extended canonical algebras. We also provide a new way to prove the classification result on Nakayama algebras of piecewise hereditary type, which have been done by Happel–Seidel before. [ABSTRACT FROM AUTHOR]

Let B g (r) be the moduli space of triples of the form (X , K X 1 / 2 , F) , where X is a compact connected Riemann surface of genus g, with g ≥ 2 , K X 1 / 2 is a theta characteristic on X, and F is a stable vector bundle on X of rank r and degree zero. We construct a T ∗ B g (r) -torsor H g (r) over B g (r) . This generalizes on the one hand the torsor over the moduli space of stable vector bundles of rank r, on a fixed Riemann surface Y, given by the moduli space of algebraic connections on the stable vector bundles of rank r on Y, and on the other hand the torsor over the moduli space of Riemann surfaces given by the moduli space of Riemann surfaces with a projective structure. It is shown that H g (r) has a holomorphic symplectic structure compatible with the T ∗ B g (r) -torsor structure. We also describe H g (r) in terms of the second order matrix valued differential operators. It is shown that H g (r) is identified with the T ∗ B g (r) -torsor given by the sheaf of holomorphic connections on the theta line bundle over B g (r) . [ABSTRACT FROM AUTHOR]

Given a singular connection D on a vector bundle E over an irreducible smooth projective curve X, defined over an algebraically closed field, we show that there is a unique maximal subsheaf of E on which D induces a nonsingular connection. Given a generically smooth map ϕ : Y → X between irreducible smooth projective curves, and a singular connection (V , D) on Y, the direct image ϕ * ⁢ V has a singular connection. Let 퐑 ⁢ (ϕ * ⁢ 풪 Y) be the unique maximal subsheaf on which the singular connection on ϕ * ⁢ 풪 Y – corresponding to the trivial connection on 풪 Y – induces a nonsingular connection. We prove that the homomorphism of étale fundamental groups ϕ * : π 1 et ⁢ (Y , y 0) → π 1 et ⁢ (X , ϕ ⁢ (y 0)) induced by ϕ is surjective if and only if 풪 X ⊂ 퐑 ⁢ (ϕ * ⁢ 풪 Y) is the unique maximal semistable subsheaf. When the characteristic of the base field is zero, this homomorphism ϕ * is surjective if and only if 풪 X = 퐑 ⁢ (ϕ * ⁢ 풪 Y) . For any nonsingular connection D on a vector bundle V over X, there is a natural map V ↪ 퐑 ⁢ (ϕ * ⁢ ϕ * ⁢ V) . When the characteristic of the base field is zero, we prove that the map ϕ is genuinely ramified if and only if V = 퐑 ⁢ (ϕ * ⁢ ϕ * ⁢ V) . [ABSTRACT FROM AUTHOR]

Let X be a compact Riemann surface of genus g ≥ 2 and let n ≥ 3 be an integer number. The group of automorphisms of the moduli space of vector bundles over X with rank n and trivial determinant is isomorphic to H 1 (X , Z / (n)) ⋊ (Out (SL (n , C)) × Aut (X)) . Several papers have studied the subvarieties of fixed points for the action of the unique outer involution of SL (n , C) on this moduli space. In this paper, explicit descriptions of the fixed points for the actions of the elements of H 1 (X , Z / (n)) , H 1 (X , Z / (n)) ⋊ Out (SL (n , C)) , and Out (SL (n , C)) × Aut (X) on the moduli space of rank n and trivial determinant vector bundles over X are provided. For the description of the fixed points for the action of the elements of Out (SL (n , C)) × Aut (X) , the notion of Galois bundle is introduced. Specifically, Galois bundles over X admitting a nontrivial automorphism which commutes with the Galois structure are constructed associated with an involution σ X of X. Finally, it is discussed how the description of fixed points for the action of the elements of is covered by the descriptions above. [ABSTRACT FROM AUTHOR]

Let C be a smooth irreducible complex projective curve of genus g ≥ 2 and M the moduli space of stable vector bundles on C of rank n and degree d with gcd (n , d) = 1 . A generalised Picard sheaf is the direct image on M of the tensor product of a universal bundle on M × C by the pullback of a vector bundle E 0 on C. In this paper, we investigate the stability of generalised Picard sheaves and, in the case where these are locally free, their deformations. When g ≥ 3 , n ≥ 2 (with some additional restrictions for g = 3 , 4 ) and the rank and degree of E 0 are coprime, this leads to the construction of a fine moduli space for deformations of Picard bundles. [ABSTRACT FROM AUTHOR]

In this article, we will construct a universal moduli space of stable parabolic vector bundles over the moduli space of marked Deligne–Mumford stable curves M ¯ g , n . The objects that appear over the boundary of M ¯ g , n , i.e., over singular curves will remain vector bundles. The total space and the fibers over M ¯ g , n will have good singularities. [ABSTRACT FROM AUTHOR]

We briefly review an open conjecture about Higgs bundles that are semistable after pulling back to any curve, and prove it in the rank 2 case. We also prove some results in higher rank under suitable additional assumptions. Moreover, we establish a set of inequalities holding for H-nef Higgs bundles that generalize some of the Fulton–Lazarsfeld inequalities for numerically effective vector bundles. [ABSTRACT FROM AUTHOR]

The Lawson surface ξ 1 , g of genus g is constructed by rotating and reflecting the Plateau solution f t with respect to a particular geodesic 4-gon Γ t across its boundary, where (t = 1 2 g + 2 , π 2 , t , π 2) are the angles of Γ t . Recent progress in integrable surface theory allows for a more explicit construction of these surfaces and for better understanding of their geometric properties using so-called Fuchsian DPW potentials for t ∼ 0 . In this paper we combine the existence and regularity of the Plateau solution f t in t ∈ (0 , 1 4) with a detailed investigation of the moduli space of Fuchsian systems on the 4-punctured sphere to obtain the existence of a Fuchsian DPW potential η t for every f t with t ∈ (0 , 1 4 ] . Moreover, the coefficients of η t are shown to depend real analytically on t. This implies that the Taylor expansions of the DPW potential η t and of the area in Heller et al. (Complete families of embedded high genus CMC surfaces in the 3-sphere. ) computed at t = 0 already determine these quantities for all ξ 1 , g . In particular, this leads to an algorithm to conformally parametrize all Lawson surfaces ξ 1 , g . [ABSTRACT FROM AUTHOR]

Let XX be a compact Riemann surface of genus g≥2g\ge 2, GG be a semisimple complex Lie group and ρ:G→GL(V)\rho :G\to {\rm{GL}}\left(V) be a complex representation of GG. Given a principal GG-bundle EE over XX, a vector bundle E(V)E\left(V) whose typical fiber is a copy of VV is induced. A (G,ρ)\left(G,\rho )-Higgs pair is a pair (E,φ)\left(E,\varphi ), where EE is a principal GG-bundle over XX and φ\varphi is a holomorphic global section of E(V)⊗LE\left(V)\otimes L, LL being a fixed line bundle over XX. In this work, Higgs pairs of this type are considered for G=Spin(8,C)G={\rm{Spin}}\left(8,{\mathbb{C}}) and the three irreducible eight-dimensional complex representations which Spin(8,C){\rm{Spin}}\left(8,{\mathbb{C}}) admits. In particular, the reduced notions of stability, semistability, and polystability for these specific Higgs pairs are given, and it is proved that the corresponding moduli spaces are isomorphic, and a precise expression for the stable and not simple Higgs pairs associated with one of the three announced representations of Spin(8,C){\rm{Spin}}\left(8,{\mathbb{C}}) is described.

We prove that the moduli spaces of parabolic symplectic/orthogonal bundles on a smooth curve are globally F-regular type. As a consequence, all higher cohomologies of the theta line bundle vanish. During the proof, we develop a method to estimate codimension.

Given a smooth genus three curve C, the moduli space of rank two stable vector bundles on C with trivial determinant embeds in ${\mathbb {P}}^8$ as a hypersurface whose singular locus is the Kummer threefold of C; this hypersurface is the Coble quartic. Gruson, Sam and Weyman realized that this quartic could be constructed from a general skew-symmetric four-form in eight variables. Using the lines contained in the quartic, we prove that a similar construction allows to recover $\operatorname {\mathrm {SU}}_C(2,L)$ , the moduli space of rank two stable vector bundles on C with fixed determinant of odd degree L, as a subvariety of $G(2,8)$ . In fact, each point $p\in C$ defines a natural embedding of $\operatorname {\mathrm {SU}}_C(2,{\mathcal {O}}(p))$ in $G(2,8)$ . We show that, for the generic such embedding, there exists a unique quadratic section of the Grassmannian which is singular exactly along the image of $\operatorname {\mathrm {SU}}_C(2,{\mathcal {O}}(p))$ and thus deserves to be coined the Coble quadric of the pointed curve $(C,p)$ .

Let $\alpha \colon X \to Y$ be a finite cover of smooth curves. Beauville conjectured that the pushforward of a general vector bundle under $\alpha $ is semistable if the genus of Y is at least $1$ and stable if the genus of Y is at least $2$ . We prove this conjecture if the map $\alpha $ is general in any component of the Hurwitz space of covers of an arbitrary smooth curve Y.

Let X be a hyperelliptic algebraic curve and let M (E 6) be the moduli space of polystable principal E 6 -bundles over X. Suppose, in addition, that the outer involution σ of E 6 acts as the hyperelliptic involution of X. Then, an automorphism of M (E 6) is defined which acts by E ↦ σ ∗ (E ∗) , where E is a principal E 6 -bundle over X seen as a vector bundle through the fundamental irreducible 27-dimensional representation of E 6 . In this paper, Galois E 6 -bundles over X are defined and related to the fixed points of the above automorphism of M (E 6) . If P E 6 is the centerless group with Lie algebra e 6 , then Galois P E 6 -bundles over X are also defined and related to Galois E 6 -bundles. Finally, a specific expression for a certain family of Galois E 6 -bundles over X is given and some implications of the study in terms of representations of the fundamental group π 1 (X) of the base curve are drawn. [ABSTRACT FROM AUTHOR]

We consider Higgs bundles satisfying a notion of numerical flatness (H-nflatness) that was introduced in [5; 4], and show that they have Jordan-Hölder filtrations whose quotients are stable, locally free and H-nflat. This is applied to show that curve semistable Higgs bundles on simply connected Calabi–Yau varieties have vanishing discriminant. [ABSTRACT FROM AUTHOR]

Let X be a compact connected Riemann surface of genus g, with g ≥ 2 , and let ξ be a holomorphic line bundle on X with ξ ⊗ 2 = O X . Fix a theta characteristic L on X. Let M X (r , ξ) be the moduli space of stable vector bundles E on X of rank r such that ⋀ r E = ξ and H 0 (X , E ⊗ L) = 0 . Consider the quotient of M X (r , ξ) by the involution given by E ⟼ E ∗ . We construct an algebraic morphism from this quotient to the moduli space of SL (r , C) opers on X. Since dim M X (r , ξ) coincides with the dimension of the moduli space of SL (r , C) opers, it is natural to ask about the injectivity and surjectivity of this map. [ABSTRACT FROM AUTHOR]

Given a compact Riemann surface X of genus at least 2 with automorphism group G, we provide formulae that enable us to compute traces of automorphisms of X on the space of global sections of G-linearized line bundles defined on certain blow-ups of projective spaces along the curve X. The method is an adaptation of one used by Thaddeus to compute the dimensions of those spaces. In particular, we can compute the traces of automorphisms of X on the Verlinde spaces corresponding to the moduli space S U X (2 , ξ) when ξ is a line bundle G-linearized of suitable degree. [ABSTRACT FROM AUTHOR]

We provide a spectral sequence computing the extension groups of tautological bundles on symmetric products of curves. One main consequence is that, if E ≠ O X is simple, then the natural map Ext 1 (E , E) → Ext 1 (E [ n ] , E [ n ]) is injective for every n. Along with previous results, this implies that E ↦ E [ n ] defines an embedding of the moduli space of stable bundles of slope μ ∉ [ - 1 , n - 1 ] on the curve X into the moduli space of stable bundles on the symmetric product X (n) . The image of this embedding is, in most cases, contained in the singular locus. For line bundles on a non-hyperelliptic curve, the embedding identifies the Brill–Noether loci of X with the loci in the moduli space of stable bundles on X (n) where the dimension of the tangent space jumps. We also prove that E [ n ] is simple if E is simple. [ABSTRACT FROM AUTHOR]

Let Y be a possibly non-connected smooth projective curve, y 1 1 , y 2 1 , ... , y 1 ν , y 2 ν 2 ν different points of Y, r ∈ N , d ∈ Z , δ ∈ Q > 0 , κ ̲ = (κ 1 , ... , κ ν) ∈ Q ≥ 0 ν and e ̲ = (e 1 , ... , e ν) ∈ Z ≥ 0 ν with e i ≤ r . We construct a projective moduli space of (κ ̲ , δ) -(semi)stable singular principal G-bundles of rank r, degree d, with generalized parabolic structure of type e ̲ supported on the divisors D 1 = y 1 1 + y 2 1 , ... , D ν = y 1 ν + y 2 ν . In case Y is the normalization of a connected and reducible projective nodal curve X, there exists a closed subscheme coarsely representing the subfunctor corresponding to those bundles that descend to X. We prove that the descent operation gives a bijection between the set of isomorphism classes of singular principal G-bundles of type e ̲ on X and the set of isomorphism classes of descending singular principal G-bundles with generalized parabolic structures of type e ̲ satisfying certain condition on Y. If the stable locus is dense inside the moduli space of descending singular principal G-bundles, the descent operation induces a birational, surjective and proper morphism onto the schematic closure of the space of δ -stable singular principal G-bundles of type e ̲ . This generalizes the known results over irreducible curves. [ABSTRACT FROM AUTHOR]

Let XX be a compact Riemann surface of genus g≥2g\ge 2 and ℳ(E6){\mathcal{ {\mathcal M} }}\left({E}_{6}) be the moduli space of E6{E}_{6}-Higgs bundles over XX. We consider the automorphisms σ+{\sigma }_{+} of ℳ(E6){\mathcal{ {\mathcal M} }}\left({E}_{6}) defined by σ+(E,φ)=(E∗,−φt){\sigma }_{+}\left(E,\varphi )=\left({E}^{\ast },-{\varphi }^{t}), induced by the action of the outer involution of E6{E}_{6} in ℳ(E6){\mathcal{ {\mathcal M} }}\left({E}_{6}), and σ−{\sigma }_{-} defined by σ−(E,φ)=(E∗,φt){\sigma }_{-}\left(E,\varphi )=\left({E}^{\ast },{\varphi }^{t}), which results from the combination of σ+{\sigma }_{+} with the involution of ℳ(E6){\mathcal{ {\mathcal M} }}\left({E}_{6}), which consists on a change of sign in the Higgs field. In this work, we describe the fixed points of σ+{\sigma }_{+} and σ−{\sigma }_{-}, as F4{F}_{4}-Higgs bundles, F4{F}_{4}-Higgs pairs associated with the fundamental irreducible representation of F4{F}_{4}, and PSp(8,C){\rm{PSp}}\left(8,{\mathbb{C}})-Higgs pairs associated with the second symmetric power or the second wedge power of the fundamental representation of Sp(8,C){\rm{Sp}}\left(8,{\mathbb{C}}). Finally, we describe the reduced notions of semistability and polystability for these objects.

The twistor space of the moduli space of solutions of Hitchin's self-duality equations can be identified with the Deligne-Hitchin moduli space of λ -connections. We use real projective structures on Riemann surfaces to prove the existence of new components of real holomorphic sections of the Deligne-Hitchin moduli space. Applying the twistorial construction we show the existence of new hyper-Kähler manifolds associated to any compact Riemann surface of genus g ≥ 2 . These hyper-Kähler manifolds can be considered as moduli spaces of (certain) singular solutions of the self-duality equations. [ABSTRACT FROM AUTHOR]

We study torsion-free, rank 2 Higgs sheaves on genus one fibered surfaces, (semi)stable with respect to suitable polarizations in the sense of Friedman and O'Grady. We prove that slope-semistability of a Higgs sheaf on the surface implies semistability on the generic fiber. In the case of Higgs sheaves of odd fiber degree on elliptic surfaces in characteristic ≠ 2 , we prove that any moduli space of Higgs sheaves with fixed numerical invariants splits canonically as the product of the moduli space of ordinary sheaves (with the same invariants), and the space of global regular 1-forms on the surface. For elliptic surfaces with section in characteristic zero, and in the case arbitrary fiber degree, we prove that if a Higgs sheaf has reduced Friedman spectral curve, or is regular on a general fiber with non-reduced spectral cover, then its Higgs field takes values in the saturation of the pull-back of the canonical bundle of the base curve in the cotangent bundle of the surface. [ABSTRACT FROM AUTHOR]

Let X be a smooth quartic hypersurface in P 3 . By the Brill-Noether theory of curves on K3 surfaces, if a rank 2 aCM bundle on X is globally generated, then it is the Lazarsfeld-Mukai bundle E C , Z associated with a smooth curve C on X and a base point free pencil Z on C. In this paper, we will focus on the classification of such bundles on X to investigate aCM bundles of rank 2 on X. Concretely, we will give a necessary condition for a rank 2 vector bundle of type E C , Z to be indecomposable initialized and aCM, in the case where the class of C in Pic (X) is contained in the sublattice of rank 2 generated by the hyperplane class of X and a non-trivial initialized aCM line bundle on X. [ABSTRACT FROM AUTHOR]