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Journal articles on the topic 'Algebraic Geometry, Moduli spaces, Vector bundles'

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Published: 14 March 2023

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1

Bhosle, Usha N. "Moduli spaces of vector bundles on a real nodal curve." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 61, no. 4 (February 19, 2020): 615–26. http://dx.doi.org/10.1007/s13366-020-00489-5.

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2

BRADLOW, S. B., O. GARCÍA-PRADA, V. MERCAT, V. MUÑOZ, and P. E. NEWSTEAD. "ON THE GEOMETRY OF MODULI SPACES OF COHERENT SYSTEMS ON ALGEBRAIC CURVES." International Journal of Mathematics 18, no. 04 (April 2007): 411–53. http://dx.doi.org/10.1142/s0129167x07004151.

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Let C be an algebraic curve of genus g ≥ 2. A coherent system on C consists of a pair (E,V), where E is an algebraic vector bundle over C of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter α. We study the geometry of the moduli space of coherent systems for different values of α when k ≤ n and the variation of the moduli spaces when we vary α. As a consequence, for sufficiently large α, we compute the Picard groups and the first and second homotopy groups of the moduli spaces of coherent systems in almost all cases, describe the moduli space for the case k = n - 1 explicitly, and give the Poincaré polynomials for the case k = n - 2. In an appendix, we describe the geometry of the "flips" which take place at critical values of α in the simplest case, and include a proof of the existence of universal families of coherent systems when GCD (n,d,k) = 1.

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3

Lakshmibai, V., K. N. Raghavan, P. Sankaran, and P. Shukla. "Standard monomial bases, Moduli spaces of vector bundles, and Invariant theory." Transformation Groups 11, no. 4 (October 21, 2006): 673–704. http://dx.doi.org/10.1007/s00031-005-1123-4.

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4

Spies, Alexander. "Poisson-geometric Analogues of Kitaev Models." Communications in Mathematical Physics 383, no. 1 (March 9, 2021): 345–400. http://dx.doi.org/10.1007/s00220-021-03992-5.

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AbstractWe define Poisson-geometric analogues of Kitaev’s lattice models. They are obtained from a Kitaev model on an embedded graph $$\Gamma $$ Γ by replacing its Hopf algebraic data with Poisson data for a Poisson-Lie group G. Each edge is assigned a copy of the Heisenberg double $${\mathcal {H}}(G)$$ H ( G ) . Each vertex (face) of $$\Gamma $$ Γ defines a Poisson action of G (of $$G^*$$ G ∗ ) on the product of these Heisenberg doubles. The actions for a vertex and adjacent face form a Poisson action of the double Poisson-Lie group D(G). We define Poisson counterparts of vertex and face operators and relate them via the Poisson bracket to the vector fields generating the actions of D(G). We construct an isomorphism of Poisson D(G)-spaces between this Poisson-geometrical Kitaev model and Fock and Rosly’s Poisson structure for the graph $$\Gamma $$ Γ and the Poisson-Lie group D(G). This decouples the latter and represents it as a product of Heisenberg doubles. It also relates the Poisson-geometrical Kitaev model to the symplectic structure on the moduli space of flat D(G)-bundles on an oriented surface with boundary constructed from $$\Gamma $$ Γ .

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5

Aprodu, Marian, and Vasile Brînzănescu. "Moduli spaces of vector bundles over ruled surfaces." Nagoya Mathematical Journal 154 (1999): 111–22. http://dx.doi.org/10.1017/s0027763000025332.

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AbstractWe study moduli spaces M(c1, c2, d, r) of isomorphism classes of algebraic 2-vector bundles with fixed numerical invariants c1, c2, d, r over a ruled surface. These moduli spaces are independent of any ample line bundle on the surface. The main result gives necessary and sufficient conditions for the non-emptiness of the space M(c1, c2, d, r) and we apply this result to the moduli spaces ML(c1, c2) of stable bundles, where L is an ample line bundle on the ruled surface.

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6

Bochnak, J., W. Kucharz, and R. Silhol. "Morphisms, line bundles and moduli spaces in real algebraic geometry." Publications mathématiques de l'IHÉS 86, no. 1 (December 1997): 5–65. http://dx.doi.org/10.1007/bf02698900.

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7

Bochnak, Jacek, Wojciech Kucharz, and Robert Silhol. "Morphisms, line bundles and moduli spaces in real algebraic geometry." Publications mathématiques de l'IHÉS 92, no. 1 (December 2000): 195. http://dx.doi.org/10.1007/bf02698917.

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8

BOLOGNESI, MICHELE, and SONIA BRIVIO. "COHERENT SYSTEMS AND MODULAR SUBAVRIETIES OF $\mathcal{SU}_C(r)$." International Journal of Mathematics 23, no. 04 (April 2012): 1250037. http://dx.doi.org/10.1142/s0129167x12500371.

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Let C be an algebraic smooth complex curve of genus g > 1. The object of this paper is the study of the birational structure of certain moduli spaces of vector bundles and of coherent systems on C and the comparison of different type of notions of stability arising in moduli theory. Notably we show that in certain cases these moduli spaces are birationally equivalent to fibrations over simple projective varieties, whose fibers are GIT quotients (ℙr-1)rg// PGL (r), where r is the rank of the considered vector bundles. This allows us to compare different definitions of (semi-)stability (slope stability, α-stability, GIT stability) for vector bundles, coherent systems and point sets, and derive relations between them. In certain cases of vector bundles of low rank when C has small genus, our construction produces families of classical modular varieties contained in the Coble hypersurfaces.

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9

Schaffhauser, Florent. "Moduli spaces of vector bundles over a Klein surface." Geometriae Dedicata 151, no. 1 (August 1, 2010): 187–206. http://dx.doi.org/10.1007/s10711-010-9526-3.

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10

Serman, Olivier. "Moduli spaces of orthogonal and symplectic bundles over an algebraic curve." Compositio Mathematica 144, no. 3 (May 2008): 721–33. http://dx.doi.org/10.1112/s0010437x07003247.

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AbstractWe prove that, given a smooth projective curve C of genus g≥2, the forgetful morphism $\mathcal {M}_{\mathbf {O}_r} \longrightarrow \mathcal {M}_{\mathbf {GL}_r}$ (respectively $\mathcal M_{\mathbf {Sp}_{2r}}\longrightarrow \mathcal M_{\mathbf {GL}_{2r}}$) from the moduli space of orthogonal (respectively symplectic) bundles to the moduli space of all vector bundles over C is an embedding. Our proof relies on an explicit description of a set of generators for the polynomial invariants on the representation space of a quiver under the action of a product of classical groups.

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11

Bruzzo, U., and F. Pioli. "Complex Lagrangian embeddings of moduli spaces of vector bundles." Differential Geometry and its Applications 14, no. 2 (March 2001): 151–56. http://dx.doi.org/10.1016/s0926-2245(00)00040-1.

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12

RAN, ZIV. "JACOBI COHOMOLOGY, LOCAL GEOMETRY OF MODULI SPACES, AND HITCHIN CONNECTIONS." Proceedings of the London Mathematical Society 92, no. 3 (April 18, 2006): 545–80. http://dx.doi.org/10.1017/s0024611505015704.

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We develop some cohomological tools for the study of the local geometry of moduli and parameter spaces in complex Algebraic Geometry. Notably, we develop canonical formulae for the differential operators of arbitrary order and their natural action on suitable `natural' modules (for example, functions); in particular, we obtain a formula, in terms of the moduli problem, for the Lie bracket of vector fields on a moduli space. As an application, we obtain another construction and proof of flatness for the familiar KZW or Hitchin connection on moduli spaces of curves.

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13

MUÑOZ, VICENTE. "HODGE STRUCTURES OF THE MODULI SPACES OF PAIRS." International Journal of Mathematics 21, no. 11 (November 2010): 1505–29. http://dx.doi.org/10.1142/s0129167x10006604.

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Let X be a smooth projective curve of genus g ≥ 2 over ℂ. Fix n ≥ 2, d ∈ ℤ. A pair (E, ϕ) over X consists of an algebraic vector bundle E of rank n and degree d over X and a section ϕ ∈ H0(E). There is a concept of stability for pairs which depends on a real parameter τ. Let [Formula: see text] be the moduli space of τ-semistable pairs of rank n and degree d over X. Here we prove that the cohomology groups of [Formula: see text] are Hodge structures isomorphic to direct summands of tensor products of the Hodge structure H1(X). This implies a similar result for the moduli spaces of stable vector bundles over X.

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14

Gieseker, David, and Jun Li. "Irreducibility of moduli of rank-2 vector bundles on algebraic surfaces." Journal of Differential Geometry 40, no. 1 (1994): 23–104. http://dx.doi.org/10.4310/jdg/1214455287.

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15

Mata-Gutiérrez, O., and Frank Neumann. "Geometry of moduli stacks of (k,l)-stable vector bundles over algebraic curves." Journal of Geometry and Physics 111 (January 2017): 54–70. http://dx.doi.org/10.1016/j.geomphys.2016.10.003.

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16

CHOE, INSONG, and GEORGE H. HITCHING. "Lagrangian subbundles of symplectic bundles over a curve." Mathematical Proceedings of the Cambridge Philosophical Society 153, no. 2 (February 22, 2012): 193–214. http://dx.doi.org/10.1017/s0305004112000096.

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AbstractA symplectic bundle over an algebraic curve has a natural invariantsLagdetermined by the maximal degree of its Lagrangian subbundles. This can be viewed as a generalization of the classical Segre invariants of a vector bundle. We give a sharp upper bound onsLagwhich is analogous to the Hirschowitz bound on the classical Segre invariants. Furthermore, we study the stratifications induced bysLagon moduli spaces of symplectic bundles, and get a full picture for the case of rank four.

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17

Balaji, V., A. D. King, and P. E. Newstead. "Algebraic cohomology of the moduli space of rank 2 vector bundles on a curve." Topology 36, no. 2 (March 1997): 567–77. http://dx.doi.org/10.1016/0040-9383(96)00007-9.

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18

Schaffhauser, Florent. "Real points of coarse moduli schemes of vector bundles on a real algebraic curve." Journal of Symplectic Geometry 10, no. 4 (2012): 503–34. http://dx.doi.org/10.4310/jsg.2012.v10.n4.a2.

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19

Zograf, P. G., and L. A. Takhtadzhyan. "ON THE GEOMETRY OF MODULI SPACES OF VECTOR BUNDLES OVER A RIEMANN SURFACE." Mathematics of the USSR-Izvestiya 35, no. 1 (February 28, 1990): 83–100. http://dx.doi.org/10.1070/im1990v035n01abeh000687.

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20

NGUYEN, QUANG MINH. "VECTOR BUNDLES, DUALITIES AND CLASSICAL GEOMETRY ON A CURVE OF GENUS TWO." International Journal of Mathematics 18, no. 05 (May 2007): 535–58. http://dx.doi.org/10.1142/s0129167x07004230.

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Let C be a curve of genus two. We denote by [Formula: see text] the moduli space of semi-stable vector bundles of rank 3 and trivial determinant over C, and by Jd the variety of line bundles of degree d on C. In particular, J1 has a canonical theta divisor Θ. The space [Formula: see text] is a double cover of ℙ8 = |3Θ| branched along a sextic hypersurface, the Coble sextic. In the dual [Formula: see text], where J1 is embedded, there is a unique cubic hypersurface singular along J1, the Coble cubic. We prove that these two hypersurfaces are dual, inducing a non-abelian Torelli result. Moreover, by looking at some special linear sections of these hypersurfaces, we can observe and reinterpret some classical results of algebraic geometry in a context of vector bundles: the duality of the Segre–Igusa quartic with the Segre cubic, the symmetric configuration of 15 lines and 15 points, the Weddle quartic surface and the Kummer surface.

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21

Bradlow, S. B., O. García-Prada, V. Muñoz, and P. E. Newstead. "Coherent Systems and Brill–Noether Theory." International Journal of Mathematics 14, no. 07 (September 2003): 683–733. http://dx.doi.org/10.1142/s0129167x03002009.

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Let X be a curve of genus g. A coherent system on X consists of a pair (E,V), where E is an algebraic vector bundle over X of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter α. We study the variation of the moduli space of coherent systems when we move the parameter. As an application, we analyze the cases k=1,2,3 and n=2 explicitly. For small values of α, the moduli spaces of coherent systems are related to the Brill–Noether loci, the subschemes of the moduli spaces of stable bundles consisting of those bundles with at least a prescribed number of independent sections. The study of coherent systems is applied to find the dimension, prove the irreducibility, and in some cases calculate the Picard groups of the Brill–Noether loci with k≤3.

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22

Yu, Tony Yue. "Gromov compactness in non-archimedean analytic geometry." Journal für die reine und angewandte Mathematik (Crelles Journal) 2018, no. 741 (August 1, 2018): 179–210. http://dx.doi.org/10.1515/crelle-2015-0077.

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Abstract Gromov’s compactness theorem for pseudo-holomorphic curves is a foundational result in symplectic geometry. It controls the compactness of the moduli space of pseudo-holomorphic curves with bounded area in a symplectic manifold. In this paper, we prove the analog of Gromov’s compactness theorem in non-archimedean analytic geometry. We work in the framework of Berkovich spaces. First, we introduce a notion of Kähler structure in non-archimedean analytic geometry using metrizations of virtual line bundles. Second, we introduce formal stacks and non-archimedean analytic stacks. Then we construct the moduli stack of non-archimedean analytic stable maps using formal models, Artin’s representability criterion and the geometry of stable curves. Finally, we reduce the non-archimedean problem to the known compactness results in algebraic geometry. The motivation of this paper is to provide the foundations for non-archimedean enumerative geometry.

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23

Bochnak, J., and W. Kucharz. "K-theory of real algebraic surfaces and threefolds." Mathematical Proceedings of the Cambridge Philosophical Society 106, no. 3 (November 1989): 471–80. http://dx.doi.org/10.1017/s0305004100068213.

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LetXbe an affine real algebraic variety, i.e., up to biregular isomorphism an algebraic subset of ℝn. (For definitions and notions of real algebraic geometry we refer the reader to the book [6].) Letdenote the ring of regular functions onX([6], chapter 3). (IfXis an algebraic subset of ℝnthenis comprised of all functions of the formf/g, whereg, f: X→ ℝ are polynomial functions withg−1(O) = Ø.) In this paper, assuming thatXis compact, non-singular, and that dimX≤ 3, we compute the Grothendieck groupof projective modules over(cf. Section 1), and the Grothendieck groupand the Witt groupof symplectic spaces over(cf. Section 2), in terms of the algebraic cohomology groupsandgenerated by the cohomology classes associated with the algebraic subvarieties ofX. We also relate the groupto the Grothendieck groupKO(X) of continuous real vector bundles overX, and the groupsandto the Grothendieck groupK(X)of continuous complex vector bundles overX.

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24

Ballico, Edoardo. "On the Connectedness of the Real Part of Moduli Spaces of Vector Bundles on Real Algebraic Surfaces." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 68, no. 1 (February 2000): 41–54. http://dx.doi.org/10.1017/s1446788700001567.

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AbstractLet X be a smooth projective surface with q(X) = 0 defined over R and M(X;r;c1;c2;H) the moduli space of H-stable rank r vector bundles on X with Chern classes c1 and c2. Assume either r = 3 and X(R) connected or r = 3 and X(R) =ø or r=2 and X(R) = ø. We prove that quite often M is connected.

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25

Zuo, Kang. "Generic smoothness of the moduli spaces of rank two stable vector bundles over algebraic surfaces." Mathematische Zeitschrift 207, no. 1 (May 1991): 629–43. http://dx.doi.org/10.1007/bf02571412.

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26

Drézet, Jean-Marc. "Non-reduced moduli spaces of sheaves on multiple curves." Advances in Geometry 20, no. 2 (April 28, 2020): 285–96. http://dx.doi.org/10.1515/advgeom-2019-0033.

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AbstractSome coherent sheaves on projective varieties have a non-reduced versal deformation space; for example, this is the case for most unstable rank 2 vector bundles on ℙ2, see [18]. In particular, some moduli spaces of stable sheaves are non-reduced. We consider some sheaves on ribbons (double structures on smooth projective curves): let E be a quasi locally free sheaf of rigid type and let 𝓔 be a flat family of sheaves containing E. We find that 𝓔 is a reduced deformation of E when some canonical family associated to 𝓔 is also flat. We consider also a deformation of the ribbon to reduced projective curves with two components, and find that E can be deformed in two distinct ways to sheaves on the reduced curves. In particular some components M of the moduli spaces of stable sheaves deform to two components of the moduli spaces of sheaves on the reduced curves, and M appears as the “limit” of varieties with two components, whence the non-reduced structure of M.

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27

González–Martínez, Cristian. "The Hodge–Poincaré polynomial of the moduli spaces of stable vector bundles over an algebraic curve." Manuscripta Mathematica 137, no. 1-2 (May 8, 2011): 19–55. http://dx.doi.org/10.1007/s00229-011-0456-7.

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28

HU, YI, and WEI-PING LI. "VARIATION OF THE GIESEKER AND UHLENBECK COMPACTIFICATIONS." International Journal of Mathematics 06, no. 03 (June 1995): 397–418. http://dx.doi.org/10.1142/s0129167x95000134.

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In this article, we study the variation of the Gieseker and Uhlenbeck compactifications of the moduli spaces of Mumford-Takemoto stable vector bundles of rank 2 by changing polarizations. Some canonical rational morphisms among the Gieseker compactifications are shown to exist. In particular, we proved that when the second Chern class is sufficiently large, these morphisms are genuine rational maps. Moreover, as a consequence of studying the morphisms from the Gieseker compactifications to the Uhlenbeck compactifications, we show that there is an everywhere-defined canonical algebraic map between two adjacent Uhlenbeck compactifications which restricts to the identity on some Zariski open subset.

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29

Li, Jun. "The first two Betti numbers of the moduli spaces of vector bundles on surfaces." Communications in Analysis and Geometry 5, no. 4 (1997): 625–84. http://dx.doi.org/10.4310/cag.1997.v5.n4.a2.

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30

Moraru, Ruxandra. "Integrable Systems Associated to a Hopf Surface." Canadian Journal of Mathematics 55, no. 3 (June 1, 2003): 609–35. http://dx.doi.org/10.4153/cjm-2003-025-3.

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AbstractA Hopf surface is the quotient of the complex surface by an infinite cyclic group of dilations of . In this paper, we study the moduli spaces of stable -bundles on a Hopf surface , from the point of view of symplectic geometry. An important point is that the surface is an elliptic fibration, which implies that a vector bundle on can be considered as a family of vector bundles over an elliptic curve. We define a map that associates to every bundle on a divisor, called the graph of the bundle, which encodes the isomorphism class of the bundle over each elliptic curve. We then prove that the map G is an algebraically completely integrable Hamiltonian system, with respect to a given Poisson structure on . We also give an explicit description of the fibres of the integrable system. This example is interesting for several reasons; in particular, since the Hopf surface is not Kähler, it is an elliptic fibration that does not admit a section.

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31

Brugallé, Erwan, and Florent Schaffhauser. "Maximality of moduli spaces of vector bundles on curves." Épijournal de Géométrie Algébrique Volume 6 (January 6, 2023). http://dx.doi.org/10.46298/epiga.2023.8793.

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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.

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32

Schmitt, Alexander. "A Universal Construction for Moduli Spaces of Decorated Vector Bundles over Curves." Transformation Groups 9, no. 2 (April 2004). http://dx.doi.org/10.1007/s00031-004-7010-6.

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33

Belmans, Pieter, Sergey Galkin, and Swarnava Mukhopadhyay. "Decompositions of moduli spaces of vector bundles and graph potentials." Forum of Mathematics, Sigma 11 (2023). http://dx.doi.org/10.1017/fms.2023.14.

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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.

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Ellenberg, Jordan S., Matthew Satriano, and David Zureick-Brown. "Heights on stacks and a generalized Batyrev–Manin–Malle conjecture." Forum of Mathematics, Sigma 11 (2023). http://dx.doi.org/10.1017/fms.2023.5.

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Abstract We define a notion of height for rational points with respect to a vector bundle on a proper algebraic stack with finite diagonal over a global field, which generalizes the usual notion for rational points on projective varieties. We explain how to compute this height for various stacks of interest (for instance: classifying stacks of finite groups, symmetric products of varieties, moduli stacks of abelian varieties, weighted projective spaces). In many cases, our uniform definition reproduces ways already in use for measuring the complexity of rational points, while in others it is something new. Finally, we formulate a conjecture about the number of rational points of bounded height (in our sense) on a stack $\mathcal {X}$ , which specializes to the Batyrev–Manin conjecture when $\mathcal {X}$ is a scheme and to Malle’s conjecture when $\mathcal {X}$ is the classifying stack of a finite group.

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Deopurkar, Anand, and Anand Patel. "Vector bundles and finite covers." Forum of Mathematics, Sigma 10 (2022). http://dx.doi.org/10.1017/fms.2022.19.

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Abstract Motivated by the problem of finding algebraic constructions of finite coverings in commutative algebra, the Steinitz realization problem in number theory and the study of Hurwitz spaces in algebraic geometry, we investigate the vector bundles underlying the structure sheaf of a finite flat branched covering. We prove that, up to a twist, every vector bundle on a smooth projective curve arises from the direct image of the structure sheaf of a smooth, connected branched cover.

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CAVALIERI, RENZO, SIMON HAMPE, HANNAH MARKWIG, and DHRUV RANGANATHAN. "MODULI SPACES OF RATIONAL WEIGHTED STABLE CURVES AND TROPICAL GEOMETRY." Forum of Mathematics, Sigma 4 (2016). http://dx.doi.org/10.1017/fms.2016.7.

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We study moduli spaces of rational weighted stable tropical curves, and their connections with Hassett spaces. Given a vector $w$ of weights, the moduli space of tropical $w$-stable curves can be given the structure of a balanced fan if and only if $w$ has only heavy and light entries. In this case, the tropical moduli space can be expressed as the Bergman fan of an explicit graphic matroid. The tropical moduli space can be realized as a geometric tropicalization, and as a Berkovich skeleton, its algebraic counterpart. This builds on previous work of Tevelev, Gibney and Maclagan, and Abramovich, Caporaso and Payne. Finally, we construct the moduli spaces of heavy/light weighted tropical curves as fibre products of unweighted spaces, and explore parallels with the algebraic world.

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37

Favale, Filippo F., and Sonia Brivio. "On vector bundles over reducible curves with a node." Advances in Geometry, July 26, 2020. http://dx.doi.org/10.1515/advgeom-2020-0010.

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AbstractLet C be a curve with two smooth components and a single node, and let 𝓤C(w, r, χ) be the moduli space of w-semistable classes of depth one sheaves on C having rank r on both components and Euler characteristic χ. In this paper, under suitable assumptions, we produce a projective bundle over the product of the moduli spaces of semistable vector bundles of rank r on each component and we show that it is birational to an irreducible component of 𝓤C(w, r, χ). Then we prove the rationality of the closed subset containing vector bundles with given fixed determinant.

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38

Heinloth, Jochen. "Hilbert-Mumford stability on algebraic stacks and applications to $\mathcal{G}$-bundles on curves." Épijournal de Géométrie Algébrique Volume 1 (January 15, 2018). http://dx.doi.org/10.46298/epiga.2018.volume1.2062.

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In these notes we reformulate the classical Hilbert-Mumford criterion for GIT stability in terms of algebraic stacks, this was independently done by Halpern-Leinster. We also give a geometric condition that guarantees the existence of separated coarse moduli spaces for the substack of stable objects. This is then applied to construct coarse moduli spaces for torsors under parahoric group schemes over curves. Comment: 37 pages

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39

Elmanto, Elden, Marc Hoyois, Adeel A. Khan, Vladimir Sosnilo, and Maria Yakerson. "Modules over algebraic cobordism." Forum of Mathematics, Pi 8 (2020). http://dx.doi.org/10.1017/fmp.2020.13.

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Abstract We prove that the $\infty $ -category of $\mathrm{MGL} $ -modules over any scheme is equivalent to the $\infty $ -category of motivic spectra with finite syntomic transfers. Using the recognition principle for infinite $\mathbf{P} ^1$ -loop spaces, we deduce that very effective $\mathrm{MGL} $ -modules over a perfect field are equivalent to grouplike motivic spaces with finite syntomic transfers. Along the way, we describe any motivic Thom spectrum built from virtual vector bundles of nonnegative rank in terms of the moduli stack of finite quasi-smooth derived schemes with the corresponding tangential structure. In particular, over a regular equicharacteristic base, we show that $\Omega ^\infty _{\mathbf{P} ^1}\mathrm{MGL} $ is the $\mathbf{A} ^1$ -homotopy type of the moduli stack of virtual finite flat local complete intersections, and that for $n>0$ , $\Omega ^\infty _{\mathbf{P} ^1} \Sigma ^n_{\mathbf{P} ^1} \mathrm{MGL} $ is the $\mathbf{A} ^1$ -homotopy type of the moduli stack of finite quasi-smooth derived schemes of virtual dimension $-n$ .

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Franco, Emilio, and Marcos Jardim. "Mirror symmetry for Nahm branes." Épijournal de Géométrie Algébrique Volume 6 (March 1, 2022). http://dx.doi.org/10.46298/epiga.2022.6604.

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The Dirac--Higgs bundle is a hyperholomorphic bundle over the moduli space of stable Higgs bundles of coprime rank and degree. We provide an algebraic generalization to the case of trivial degree and the rank higher than $1$. This allow us to generalize to this case the Nahm transform defined by Frejlich and the second named author, which, out of a stable Higgs bundle, produces a vector bundle with connection over the moduli space of rank 1 Higgs bundles. By performing the higher rank Nahm transform we obtain a hyperholomorphic bundle with connection over the moduli space of stable Higgs bundles of rank $n$ and degree 0, twisted by the gerbe of liftings of the projective universal bundle. Such hyperholomorphic vector bundles over the moduli space of stable Higgs bundles can be seen, in the physicist's language, as BBB-branes twisted by the above mentioned gerbe. We refer to these objects as Nahm branes. Finally, we study the behaviour of Nahm branes under Fourier--Mukai transform over the smooth locus of the Hitchin fibration, checking that the resulting objects are supported on a Lagrangian multisection of the Hitchin fibration, so they describe partial data of BAA-branes.

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Birkbeck, Christopher, Tony Feng, David Hansen, Serin Hong, Qirui Li, Anthony Wang, and Lynnelle Ye. "EXTENSIONS OF VECTOR BUNDLES ON THE FARGUES-FONTAINE CURVE." Journal of the Institute of Mathematics of Jussieu, May 14, 2020, 1–46. http://dx.doi.org/10.1017/s1474748020000183.

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We completely classify the possible extensions between semistable vector bundles on the Fargues–Fontaine curve (over an algebraically closed perfectoid field), in terms of a simple condition on Harder–Narasimhan (HN) polygons. Our arguments rely on a careful study of various moduli spaces of bundle maps, which we define and analyze using Scholze’s language of diamonds. This analysis reduces our main results to a somewhat involved combinatorial problem, which we then solve via a reinterpretation in terms of the Euclidean geometry of HN polygons.

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Bies, Martin, Mirjam Cvetič, Ron Donagi, Ling Lin, Muyang Liu, and Fabian Ruehle. "Machine learning and algebraic approaches towards complete matter spectra in 4d F-theory." Journal of High Energy Physics 2021, no. 1 (January 2021). http://dx.doi.org/10.1007/jhep01(2021)196.

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Abstract Motivated by engineering vector-like (Higgs) pairs in the spectrum of 4d F-theory compactifications, we combine machine learning and algebraic geometry techniques to analyze line bundle cohomologies on families of holomorphic curves. To quantify jumps of these cohomologies, we first generate 1.8 million pairs of line bundles and curves embedded in dP3, for which we compute the cohomologies. A white-box machine learning approach trained on this data provides intuition for jumps due to curve splittings, which we use to construct additional vector-like Higgs-pairs in an F-Theory toy model. We also find that, in order to explain quantitatively the full dataset, further tools from algebraic geometry, in particular Brill-Noether theory, are required. Using these ingredients, we introduce a diagrammatic way to express cohomology jumps across the parameter space of each family of matter curves, which reflects a stratification of the F-theory complex structure moduli space in terms of the vector-like spectrum. Furthermore, these insights provide an algorithmically efficient way to estimate the possible cohomology dimensions across the entire parameter space.

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Suhas, B. N., Praveen Kumar Roy, and Amit Kumar Singh. "On the rationality of moduli spaces of vector bundles over chain-like curves." Journal of Geometry and Physics, June 2022, 104590. http://dx.doi.org/10.1016/j.geomphys.2022.104590.

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