Zeitschriftenartikel: „Semistable points“

1

Luks, Tomasz, und Yimin Xiao. „Multiple Points of Operator Semistable Lévy Processes“. Journal of Theoretical Probability 33, Nr. 1 (14.09.2018): 153–79. http://dx.doi.org/10.1007/s10959-018-0859-4.

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2

Heinzner, Peter, und Henrik Stötzel. „Semistable points with respect to real forms“. Mathematische Annalen 338, Nr. 1 (23.12.2006): 1–9. http://dx.doi.org/10.1007/s00208-006-0063-1.

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3

Pattanayak, S. K. „Minimal Schubert Varieties Admitting Semistable Points for Exceptional Cases“. Communications in Algebra 42, Nr. 9 (23.04.2014): 3811–22. http://dx.doi.org/10.1080/00927872.2013.795578.

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4

Lai, K. F. „C2 building and projective space“. Journal of the Australian Mathematical Society 76, Nr. 3 (Juni 2004): 383–402. http://dx.doi.org/10.1017/s1446788700009939.

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AbstractWe study the stability map from the rigid analytic space of semistable points in P3 to convex sets in the building of Sp2 over a local field and construct a pure affinoid covering of the space of stable points.

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Vyugin, Il'ya Vladimirovich, und Lada Andreevna Dudnikova. „Stable vector bundles and the Riemann-Hilbert problem on a Riemann surface“. Sbornik: Mathematics 215, Nr. 2 (2024): 141–56. http://dx.doi.org/10.4213/sm9781e.

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The paper is devoted to holomorphic vector bundles with logarithmic connections on a compact Riemann surface and the applications of the results obtained to the question of solvability of the Riemann-Hilbert problem on a Riemann surface. We give an example of a representation of the fundamental group of a Riemann surface with four punctured points which cannot be realized as the monodromy representation of a logarithmic connection with four singular points on a semistable bundle. For an arbitrary pair of a bundle and a logarithmic connection on it we prove an estimate for the slopes of the associated Harder-Narasimhan filtration quotients. In addition, we present results on the realizability of a representation as a direct summand in the monodromy representation of a logarithmic connection on a semistable bundle of degree zero. Bibliography: 9 titles.

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Castella, Francesc. „ON THE EXCEPTIONAL SPECIALIZATIONS OF BIG HEEGNER POINTS“. Journal of the Institute of Mathematics of Jussieu 17, Nr. 1 (04.02.2016): 207–40. http://dx.doi.org/10.1017/s1474748015000444.

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We extend the $p$-adic Gross–Zagier formula of Bertolini et al. [Generalized Heegner cycles and $p$-adic Rankin $L$-series, Duke Math. J.162(6) (2013), 1033–1148] to the semistable non-crystalline setting, and combine it with our previous work [Castella, On the $p$-adic variation of Heegner points, Preprint, 2014, arXiv:1410.6591] to obtain a derivative formula for the specializations of Howard’s big Heegner points [Howard, Variation of Heegner points in Hida families, Invent. Math.167(1) (2007), 91–128] at exceptional primes in the Hida family.

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Abramovich, Dan, und Anthony Várilly-Alvarado. „Campana points, Vojta’s conjecture, and level structures on semistable abelian varieties“. Journal de Théorie des Nombres de Bordeaux 30, Nr. 2 (2018): 525–32. http://dx.doi.org/10.5802/jtnb.1037.

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8

Tamagawa, Akio. „Ramification of torsion points on curves with ordinary semistable Jacobian varieties“. Duke Mathematical Journal 106, Nr. 2 (Februar 2001): 281–319. http://dx.doi.org/10.1215/s0012-7094-01-10623-6.

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9

Kalabušić, S., M. R. S. Kulenović und E. Pilav. „Multiple Attractors for a Competitive System of Rational Difference Equations in the Plane“. Abstract and Applied Analysis 2011 (2011): 1–35. http://dx.doi.org/10.1155/2011/295308.

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We investigate global dynamics of the following systems of difference equationsxn+1=β1xn/(B1xn+yn),yn+1=(α2+γ2yn)/(A2+xn),n=0,1,2,…, where the parametersβ1,B1,β2,α2,γ2,A2are positive numbers, and initial conditionsx0andy0are arbitrary nonnegative numbers such thatx0+y0>0. We show that this system has up to three equilibrium points with various dynamics which depends on the part of parametric space. We show that the basins of attractions of different locally asymptotically stable equilibrium points or nonhyperbolic equilibrium points are separated by the global stable manifolds of either saddle points or of nonhyperbolic equilibrium points. We give an example of globally attractive nonhyperbolic equilibrium point and semistable non-hyperbolic equilibrium point.

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Chen, Huachen. „O’Grady’s birational maps and strange duality via wall-hitting“. International Journal of Mathematics 30, Nr. 09 (August 2019): 1950044. http://dx.doi.org/10.1142/s0129167x19500447.

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We prove that O’Grady’s birational maps [K. G O’Grady, The weight-two Hodge structure of moduli spaces of sheaves on a K3 surface, J. Algebr. Geom. 6(4) (1997) 599–644] between moduli of sheaves on an elliptic K3 surface can be interpreted as intermediate wall-crossing (wall-hitting) transformations at so-called totally semistable walls, studied by Bayer and Macrì [A. Bayer and E. Macrì, MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations, Inventiones Mathematicae 198(3) (2014) 505–590]. As a key ingredient, we describe the first totally semistable wall for ideal sheaves of [Formula: see text] points on the elliptic [Formula: see text]. As an application, we give new examples of strange duality isomorphisms, based on a result of Marian and Oprea [A. Marian and D. Oprea, Generic strange duality for K3 surfaces, with an appendix by Kota Yoshioka, Duke Math. J. 162(8) (2013) 1463–1501].

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Nayek, Arpita, und S. K. Pattanayak. „Torus quotient of Richardson varieties in orthogonal and symplectic grassmannians“. Journal of Algebra and Its Applications 19, Nr. 10 (11.10.2019): 2050186. http://dx.doi.org/10.1142/s0219498820501868.

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For any simple, simply connected algebraic group [Formula: see text] of type [Formula: see text] and [Formula: see text] and for any maximal parabolic subgroup [Formula: see text] of [Formula: see text], we provide a criterion for a Richardson variety in [Formula: see text] to admit semistable points for the action of a maximal torus [Formula: see text] with respect to an ample line bundle on [Formula: see text].

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Voskuil, Harm. „On the action of the unitary group on the projective plane over a local field“. Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 62, Nr. 3 (Juni 1997): 371–97. http://dx.doi.org/10.1017/s1446788700001075.

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AbstractLet G be a unitary group of rank one over a non-archimedean local field K (whose residue field has a characteristic ≠ 2). We consider the action of G on the projective plane. A G(K) equivariant map from the set of points in the projective plane that are semistable for every maximal K split torus in G to the set of convex subsets of the building of G(K) is constructed. This map gives rise to an equivariant map from the set of points that are stable for every maximal K split torus to the building. Using these maps one describes a G(K) invariant pure affinoid covering of the set of stable points. The reduction of the affinoid covering is given.

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HITCHING, GEORGE H. „RANK FOUR SYMPLECTIC BUNDLES WITHOUT THETA DIVISORS OVER A CURVE OF GENUS TWO“. International Journal of Mathematics 19, Nr. 04 (April 2008): 387–420. http://dx.doi.org/10.1142/s0129167x08004716.

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The moduli space [Formula: see text] of rank four semistable symplectic vector bundles over a curve X of genus two is an irreducible projective variety of dimension ten. Its Picard group is generated by the determinantal line bundle Ξ. The base locus of the linear system |Ξ| consists of precisely those bundles without theta divisors, that is, admitting nonzero maps from every line bundle of degree -1 over X. We show that this base locus consists of six distinct points, which are in canonical bijection with the Weierstrass points of the curve. We relate our construction of these bundles to another of Raynaud and Beauville using Fourier–Mukai transforms. As an application, we prove that the map sending a symplectic vector bundle to its theta divisor is a surjective map from [Formula: see text] to the space of even 4Θ divisors on the Jacobian variety of the curve.

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Trautwein, Samuel. „Convergence of the Yang–Mills–Higgs flow on Gauged Holomorphic maps and applications“. International Journal of Mathematics 29, Nr. 04 (April 2018): 1850024. http://dx.doi.org/10.1142/s0129167x18500246.

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The symplectic vortex equations admit a variational description as global minimum of the Yang–Mills–Higgs functional. We study its negative gradient flow on holomorphic pairs [Formula: see text] where [Formula: see text] is a connection on a principal [Formula: see text]-bundle [Formula: see text] over a closed Riemann surface [Formula: see text] and [Formula: see text] is an equivariant map into a Kähler Hamiltonian [Formula: see text]-manifold. The connection [Formula: see text] induces a holomorphic structure on the Kähler fibration [Formula: see text] and we require that [Formula: see text] descends to a holomorphic section of this fibration. We prove a Łojasiewicz type gradient inequality and show uniform convergence of the negative gradient flow in the [Formula: see text]-topology when [Formula: see text] is equivariantly convex at infinity with proper moment map, [Formula: see text] is holomorphically aspherical and its Kähler metric is analytic. As applications we establish several results inspired by finite dimensional GIT: First, we prove a certain uniqueness property for the critical points of the Yang–Mills–Higgs functional which is the analogue of the Ness uniqueness theorem. Second, we extend Mundet’s Kobayashi–Hitchin correspondence to the polystable and semistable case. The arguments for the polystable case lead to a new proof in the stable case. Third, in proving the semistable correspondence, we establish the moment–weight inequality for the vortex equation and prove the analogue of the Kempf existence and uniqueness theorem.

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Disegni, Daniel. „The -adic Gross–Zagier formula on Shimura curves“. Compositio Mathematica 153, Nr. 10 (11.07.2017): 1987–2074. http://dx.doi.org/10.1112/s0010437x17007308.

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We prove a general formula for the $p$-adic heights of Heegner points on modular abelian varieties with potentially ordinary (good or semistable) reduction at the primes above $p$. The formula is in terms of the cyclotomic derivative of a Rankin–Selberg $p$-adic $L$-function, which we construct. It generalises previous work of Perrin-Riou, Howard, and the author to the context of the work of Yuan–Zhang–Zhang on the archimedean Gross–Zagier formula and of Waldspurger on toric periods. We further construct analytic functions interpolating Heegner points in the anticyclotomic variables, and obtain a version of our formula for them. It is complemented, when the relevant root number is $+1$ rather than $-1$, by an anticyclotomic version of the Waldspurger formula. When combined with work of Fouquet, the anticyclotomic Gross–Zagier formula implies one divisibility in a $p$-adic Birch and Swinnerton-Dyer conjecture in anticyclotomic families. Other applications described in the text will appear separately.

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RYAN, TIM. „THE EFFECTIVE CONE OF MODULI SPACES OF SHEAVES ON A SMOOTH QUADRIC SURFACE“. Nagoya Mathematical Journal 232 (04.09.2017): 151–215. http://dx.doi.org/10.1017/nmj.2017.24.

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Let $\unicode[STIX]{x1D709}$ be a stable Chern character on $\mathbb{P}^{1}\times \mathbb{P}^{1}$, and let $M(\unicode[STIX]{x1D709})$ be the moduli space of Gieseker semistable sheaves on $\mathbb{P}^{1}\times \mathbb{P}^{1}$ with Chern character $\unicode[STIX]{x1D709}$. In this paper, we provide an approach to computing the effective cone of $M(\unicode[STIX]{x1D709})$. We find Brill–Noether divisors spanning extremal rays of the effective cone using resolutions of the general elements of $M(\unicode[STIX]{x1D709})$ which are found using the machinery of exceptional bundles. We use this approach to provide many examples of extremal rays in these effective cones. In particular, we completely compute the effective cone of the first fifteen Hilbert schemes of points on $\mathbb{P}^{1}\times \mathbb{P}^{1}$.

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Heinzner, Peter, Gerald W. Schwarz und Henrik Stötzel. „Stratifications with respect to actions of real reductive groups“. Compositio Mathematica 144, Nr. 1 (Januar 2008): 163–85. http://dx.doi.org/10.1112/s0010437x07003259.

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AbstractWe study the action of a real reductive group G on a real submanifold X of a Kähler manifold Z. We suppose that the action of G extends holomorphically to an action of the complexified group $G^{\mathbb {C}}$ and that with respect to a compatible maximal compact subgroup U of $G^{\mathbb {C}}$ the action on Z is Hamiltonian. There is a corresponding gradient map $\mu _{\mathfrak {p}}\colon X\to \mathfrak {p}^*$ where $\mathfrak {g}=\mathfrak {k}\oplus \mathfrak {p}$ is a Cartan decomposition of $\mathfrak {g}$. We obtain a Morse-like function $\eta _{\mathfrak {p}}:=\Vert \mu _{\mathfrak {p}}\Vert ^2$ on X. Associated with critical points of $\eta _{\mathfrak {p}}$ are various sets of semistable points which we study in great detail. In particular, we have G-stable submanifolds Sβ of X which are called pre-strata. In cases where $\mu _{\mathfrak {p}}$ is proper, the pre-strata form a decomposition of X and in cases where X is compact they are the strata of a Morse-type stratification of X. Our results are generalizations of results of Kirwan obtained in the case where $G=U^{\mathbb {C}}$ and X=Z is compact.

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Corbera, Montserrat, Jaume Llibre und Marco Antonio Teixeira. „Symmetric periodic orbits near a heteroclinic loop in formed by two singular points, a semistable periodic orbit and their invariant manifolds“. Physica D: Nonlinear Phenomena 238, Nr. 6 (April 2009): 699–705. http://dx.doi.org/10.1016/j.physd.2009.01.002.

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19

Herrero, Andres Fernandez. „On automorphisms of semistable G-bundles with decorations“. Advances in Geometry, 18.07.2023. http://dx.doi.org/10.1515/advgeom-2023-0016.

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Abstract We prove a rigidity result for automorphisms of points of certain stacks admitting adequate moduli spaces. It encompasses as special cases variations of the moduli of G-bundles on a smooth projective curve for a reductive algebraic group G. For example, our result applies to the stack of semistable G-bundles, to stacks of semistable Hitchin pairs, and to stacks of semistable parabolic G-bundles. Similar arguments apply to Gieseker semistable G-bundles in higher dimensions. We present two applications of the main result. First, we show that in characteristic 0 every stack of semistable decorated G-bundles admitting a quasiprojective good moduli space can be written naturally as a G-linearized global quotient Y/G, so the moduli problem can be interpreted as a GIT problem. Secondly, we give a proof that the stack of semistable meromorphic G-Higgs bundles on a family of curves is smooth over any base in characteristic 0.

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Haulmark, M., und M. Mihalik. „Relatively hyperbolic groups with semistable peripheral subgroups“. International Journal of Algebra and Computation, 08.03.2022, 1–31. http://dx.doi.org/10.1142/s0218196722500321.

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Suppose [Formula: see text] is a finitely presented group that is hyperbolic relative to [Formula: see text] a finite collection of finitely generated proper subgroups of [Formula: see text]. Our main theorem states that if each [Formula: see text] has semistable fundamental group at [Formula: see text], then [Formula: see text] has semistable fundamental group at [Formula: see text]. The problem reduces to the case when [Formula: see text] and the members of [Formula: see text] are all one ended and finitely presented. In that case, if the boundary [Formula: see text] has no cut point, then [Formula: see text] was already known to have semistable fundamental group at [Formula: see text]. We consider the more general situation when [Formula: see text] contains cut points.

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Asai, Sota, und Osamu Iyama. „Semistable torsion classes and canonical decompositions in Grothendieck groups“. Proceedings of the London Mathematical Society 129, Nr. 5 (16.10.2024). http://dx.doi.org/10.1112/plms.12639.

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AbstractWe study two classes of torsion classes that generalize functorially finite torsion classes, that is, semistable torsion classes and morphism torsion classes. Semistable torsion classes are parametrized by the elements in the real Grothendieck group up to TF equivalence. We give a close connection between TF equivalence classes and the cones given by canonical decompositions of the spaces of projective presentations due to Derksen–Fei. More strongly, for ‐tame algebras and hereditary algebras, we prove that TF equivalence classes containing lattice points are exactly the cones given by canonical decompositions. One of the key steps in our proof is a general description of semistable torsion classes in terms of morphism torsion classes. We also answer a question by Derksen–Fei negatively by giving examples of algebras that do not satisfy the ray condition. As an application of our results, we give an explicit description of TF equivalence classes of preprojective algebras of type .

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Biswas, Indranil, Swarnava Mukhopadhyay und Richard Wentworth. „A Hitchin connection on nonabelian theta functions for parabolic 𝐺-bundles“. Journal für die reine und angewandte Mathematik (Crelles Journal), 14.09.2023. http://dx.doi.org/10.1515/crelle-2023-0049.

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Abstract For a simple, simply connected complex affine algebraic group 𝐺, we prove the existence of a flat projective connection on the bundle of nonabelian theta functions on the moduli spaces of semistable parabolic 𝐺-bundles for families of smooth projective curves with marked points.

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BÉRCZI, GERGELY, und FRANCES KIRWAN. „GRADED UNIPOTENT GROUPS AND GROSSHANS THEORY“. Forum of Mathematics, Sigma 5 (2017). http://dx.doi.org/10.1017/fms.2017.19.

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Let $U$ be a unipotent group which is graded in the sense that it has an extension $H$ by the multiplicative group of the complex numbers such that all the weights of the adjoint action on the Lie algebra of $U$ are strictly positive. We study embeddings of $H$ in a general linear group $G$ which possess Grosshans-like properties. More precisely, suppose $H$ acts on a projective variety $X$ and its action extends to an action of $G$ which is linear with respect to an ample line bundle on $X$. Then, provided that we are willing to twist the linearization of the action of $H$ by a suitable (rational) character of $H$, we find that the $H$-invariants form a finitely generated algebra and hence define a projective variety $X/\!/H$; moreover, the natural morphism from the semistable locus in $X$ to $X/\!/H$ is surjective, and semistable points in $X$ are identified in $X/\!/H$ if and only if the closures of their $H$-orbits meet in the semistable locus. A similar result applies when we replace $X$ by its product with the projective line; this gives us a projective completion of a geometric quotient of a $U$-invariant open subset of $X$ by the action of the unipotent group $U$.

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Li, Muxi, und Mao Sheng. „Characterization of Beauville’s Numbers via Hodge Theory“. International Mathematics Research Notices, 09.08.2021. http://dx.doi.org/10.1093/imrn/rnab211.

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Abstract We provide a new Hodge theoretical characterization of the set of complex numbers that arises from the complete list, due to A. Beauville, of semistable families of elliptic curves over ${\mathbb {P}}^1$ with four singular fibers. The characterization is approached via a detailed analysis of the periodicity of the uniformizing Higgs bundle attached to ${\mathbb {P}}^1$ minus four points over the field of complex numbers.

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Doi, Mamoru, und Naoto Yotsutani. „Differential geometric global smoothings of simple normal crossing complex surfaces with trivial canonical bundle“. Complex Manifolds 10, Nr. 1 (01.01.2023). http://dx.doi.org/10.1515/coma-2022-0143.

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Abstract Let X X be a simple normal crossing (SNC) compact complex surface with trivial canonical bundle which includes triple intersections. We prove that if X X is d d -semistable, then there exists a family of smoothings in a differential geometric sense. This can be interpreted as a differential geometric analogue of the smoothability results due to Friedman, Kawamata-Namikawa, Felten-Filip-Ruddat, Chan-Leung-Ma, and others in algebraic geometry. The proof is based on an explicit construction of local smoothings around the singular locus of X X , and the first author’s existence result of holomorphic volume forms on global smoothings of X X . In particular, these volume forms are given as solutions of a nonlinear elliptic partial differential equation. As an application, we provide several examples of d d -semistable SNC complex surfaces with trivial canonical bundle including double curves, which are smoothable to complex tori, primary Kodaira surfaces, and K 3 K3 surfaces. We also provide several examples of such complex surfaces including triple points, which are smoothable to K 3 K3 surfaces.

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Greb, Daniel, und Christian Miebach. „Hamiltonian actions of unipotent groups on compact K\"ahler manifolds“. Épijournal de Géométrie Algébrique Volume 2 (09.11.2018). http://dx.doi.org/10.46298/epiga.2018.volume2.4486.

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We study meromorphic actions of unipotent complex Lie groups on compact K\"ahler manifolds using moment map techniques. We introduce natural stability conditions and show that sets of semistable points are Zariski-open and admit geometric quotients that carry compactifiable K\"ahler structures obtained by symplectic reduction. The relation of our complex-analytic theory to the work of Doran--Kirwan regarding the Geometric Invariant Theory of unipotent group actions on projective varieties is discussed in detail. Comment: v2: 30 pages, final version as accepted by EPIGA

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Halpern-Leistner, Daniel, Andres Fernandez Herrero und Trevor Jones. „Moduli spaces of sheaves via affine Grassmannians“. Journal für die reine und angewandte Mathematik (Crelles Journal), 20.02.2024. http://dx.doi.org/10.1515/crelle-2023-0099.

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Abstract We develop a new method for analyzing moduli problems related to the stack of pure coherent sheaves on a polarized family of projective schemes. It is an infinite-dimensional analogue of Geometric Invariant Theory. We apply this to two familiar moduli problems: the stack of Λ-modules and the stack of pairs. In both examples, we construct a Θ-stratification of the stack, defined in terms of a polynomial numerical invariant, and we construct good moduli spaces for the open substacks of semistable points. One of the essential ingredients is the construction of higher-dimensional analogues of the affine Grassmannian for the moduli problems considered.

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Hahn, Marvin Anas. „Mustafin Models of Projective Varieties and Vector Bundles“. International Mathematics Research Notices, 05.07.2021. http://dx.doi.org/10.1093/imrn/rnab148.

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Abstract Mustafin varieties are well-studied degenerations of projective spaces induced by a choice of integral points in a Bruhat–Tits building. In recent work, Annette Werner and the author initiated the study of degenerations of plane curves obtained by Mustafin varieties by means of arithmetic geometry. Moreover, we applied these techniques to construct models of vector bundles on plane curves with strongly semistable reduction. In this work, we take a Groebner basis approach to the more general problem of studying degenerations of projective varieties. Our methods include determining the behaviour of Groebner bases under substitution over unique factorisation rings. Finally, we outline applications to the $p-$adic Simpson correspondence, when the respective projective variety is a curve.

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Haiden, F., L. Katzarkov und C. Simpson. „Spectral Networks and Stability Conditions for Fukaya Categories with Coefficients“. Communications in Mathematical Physics 405, Nr. 11 (12.10.2024). http://dx.doi.org/10.1007/s00220-024-05138-9.

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AbstractGiven a holomorphic family of Bridgeland stability conditions over a surface, we define a notion of spectral network which is an object in a Fukaya category of the surface with coefficients in a triangulated DG-category. These spectral networks are analogs of special Lagrangian submanifolds, combining a graph with additional algebraic data, and conjecturally correspond to semistable objects of a suitable stability condition on the Fukaya category with coefficients. They are closely related to the spectral networks of Gaiotto–Moore–Neitzke. One novelty of our approach is that we establish a general uniqueness results for spectral network representatives. We also verify the conjecture in the case when the surface is disk with six marked points on the boundary and the coefficients category is the derived category of representations of an $$A_2$$ A 2 quiver. This example is related, via homological mirror symmetry, to the stacky quotient of an elliptic curve by the cyclic group of order six.

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Araujo, Carolina, Thiago Fassarella, Inder Kaur und Alex Massarenti. „On Automorphisms of Moduli Spaces of Parabolic Vector Bundles“. International Mathematics Research Notices, 08.07.2019. http://dx.doi.org/10.1093/imrn/rnz132.

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AbstractFix $n\geq 5$ general points $p_1, \dots , p_n\in{\mathbb{P}}^1$ and a weight vector ${\mathcal{A}} = (a_{1}, \dots , a_{n})$ of real numbers $0 \leq a_{i} \leq 1$. Consider the moduli space $\mathcal{M}_{{\mathcal{A}}}$ parametrizing rank two parabolic vector bundles with trivial determinant on $\big ({\mathbb{P}}^1, p_1,\dots , p_n\big )$ that are semistable with respect to ${\mathcal{A}}$. Under some conditions on the weights, we determine and give a modular interpretation for the automorphism group of the moduli space $\mathcal{M}_{{\mathcal{A}}}$. It is isomorphic to $\left (\frac{\mathbb{Z}}{2\mathbb{Z}}\right )^{k}$ for some $k\in \{0,\dots , n-1\}$ and is generated by admissible elementary transformations of parabolic vector bundles. The largest of these automorphism groups, with $k=n-1$, occurs for the central weight ${\mathcal{A}}_{F}= \left (\frac{1}{2},\dots ,\frac{1}{2}\right )$. The corresponding moduli space ${\mathcal M}_{{\mathcal{A}}_F}$ is a Fano variety of dimension $n-3$, which is smooth if $n$ is odd, and has isolated singularities if $n$ is even.

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Kučikienė, Domantė, Ravichandran Rajkumar, Katharina Timpte, Jan Heckelmann, Irene Neuner, Yvonne Weber und Stefan Wolking. „EEG microstates show different features in focal epilepsy and psychogenic nonepileptic seizures“. Epilepsia, 30.01.2024. http://dx.doi.org/10.1111/epi.17897.

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AbstractObjectiveElectroencephalography (EEG) microstate analysis seeks to cluster the scalp's electric field into semistable topographical EEG activity maps at different time points. Our study aimed to investigate the features of EEG microstates in subjects with focal epilepsy and psychogenic nonepileptic seizures (PNES).MethodsWe included 62 adult subjects with focal epilepsy or PNES who received video‐EEG monitoring at the epilepsy monitoring unit. The subjects (mean age = 42.8 ± 21.2 years) were distributed equally between epilepsy and PNES groups. We extracted microstates from a 4.4 ± 1.0‐min, 21‐channel resting‐state EEG. We excluded subjects with interictal epileptiform discharges during resting‐state EEGs. After preprocessing, we derived five main EEG microstates—MS1 to MS5—for the full frequency band (1–30 Hz) and frequency subbands (delta, 1–4 Hz; theta, 4–8 Hz; alpha, 8–12 Hz; beta, 12–30 Hz), using the MATLAB‐based EEGLAB toolkit. Statistical features of microstates (duration, occurrence, contribution, global field power [GFP]) were compared between the groups, using logistic regression corrected for age and sex.ResultsWe detected no differences in microstate parameters in the full frequency band. We found a longer duration (delta: B = −7.680, p = .046; theta: B = −16.200, p = .043) and a higher contribution (delta: B = −7.414, p = .035; theta: B = −7.509, p = .031) of MS4 in lower frequency bands in the epilepsy group. The PNES group showed a higher occurrence of MS5 in the delta subband (B = 3.283, p = .032). In the theta subband, a higher GFP of MS1 was associated with the PNES group (B = 5.674, p = .025), whereas a higher GFP of MS2 was associated with the epilepsy group (B = −6.579, p = .026).SignificanceMicrostate features show differences between patients with focal epilepsy and PNES. EEG microstates could be a promising parameter, helping to understand changes in brain dynamics in subjects with epilepsy, and should be explored as a potential biomarker.

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