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Published: 8 June 2024

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1

LI, LINGGUANG. "ON A CONJECTURE OF LAN–SHENG–ZUO ON SEMISTABLE HIGGS BUNDLES: RANK 3 CASE." International Journal of Mathematics 25, no. 02 (February 2014): 1450013. http://dx.doi.org/10.1142/s0129167x1450013x.

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Let X be a smooth projective curve of genus g over an algebraically closed field k of characteristic p > 2. We prove that any rank 3 nilpotent semistable Higgs bundle (E, θ) on X is a strongly semistable Higgs bundle. This gives a partially affirmative answer to a conjecture of Lan–Sheng–Zuo [Semistable Higgs bundles and representations of algebraic fundamental groups: positive characteristic case, preprint (2012), arXiv:1210.8280][(Very recently, A. Langer [Semistable modules over Lie algebroids in positive characteristic, preprint (2013), arXiv:1311.2794] and independently Lan–Sheng–Yang–Zuo [Semistable Higgs bundles of small ranks are strongly Higgs semistable, preprint (2013), arXiv:1311.2405] have proven the conjecture for ranks less than or equal to p case.)] In addition, we prove a tensor product theorem for strongly semistable Higgs bundles with p satisfying some bounds (Theorem 4.3). From this we reprove a tensor theorem for semistable Higgs bundles on the condition that the Lan–Sheng–Zuo conjecture holds (Corollary 4.4).

2

Andreatta, Fabrizio, and Adrian Iovita. "Semistable Sheaves and Comparison Isomorphisms in the Semistable Case." Rendiconti del Seminario Matematico della Università di Padova 128 (2012): 131–285. http://dx.doi.org/10.4171/rsmup/128-7.

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3

Mihalik, Michael L. "Bounded Depth Ascending HNN Extensions and -Semistability at infinity." Canadian Journal of Mathematics 72, no. 6 (July 22, 2019): 1529–50. http://dx.doi.org/10.4153/s0008414x19000385.

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AbstractA well-known conjecture is that all finitely presented groups have semistable fundamental groups at infinity. A class of groups whose members have not been shown to be semistable at infinity is the class ${\mathcal{A}}$ of finitely presented groups that are ascending HNN-extensions with finitely generated base. The class ${\mathcal{A}}$ naturally partitions into two non-empty subclasses, those that have “bounded” and “unbounded” depth. Using new methods introduced in a companion paper we show those of bounded depth have semistable fundamental group at infinity. Ascending HNN extensions produced by Ol’shanskii–Sapir and Grigorchuk (for other reasons), and once considered potential non-semistable examples are shown to have bounded depth. Finally, we devise a technique for producing explicit examples with unbounded depth. These examples are perhaps the best candidates to date in the search for a group with non-semistable fundamental group at infinity.

4

Pancheva, E. "Max-semistable laws." Journal of Mathematical Sciences 76, no. 1 (August 1995): 2177–80. http://dx.doi.org/10.1007/bf02363231.

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5

Hacking, Paul. "Semistable divisorial contractions." Journal of Algebra 278, no. 1 (August 2004): 173–86. http://dx.doi.org/10.1016/j.jalgebra.2004.03.008.

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6

Bayer, Arend, Martí Lahoz, Emanuele Macrì, Howard Nuer, Alexander Perry, and Paolo Stellari. "Stability conditions in families." Publications mathématiques de l'IHÉS 133, no. 1 (May 17, 2021): 157–325. http://dx.doi.org/10.1007/s10240-021-00124-6.

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AbstractWe develop a theory of Bridgeland stability conditions and moduli spaces of semistable objects for a family of varieties. Our approach is based on and generalizes previous work by Abramovich–Polishchuk, Kuznetsov, Lieblich, and Piyaratne–Toda. Our notion includes openness of stability, semistable reduction, a support property uniformly across the family, and boundedness of semistable objects. We show that such a structure exists whenever stability conditions are known to exist on the fibers.Our main application is the generalization of Mukai’s theory for moduli spaces of semistable sheaves on K3 surfaces to moduli spaces of Bridgeland semistable objects in the Kuznetsov component associated to a cubic fourfold. This leads to the extension of theorems by Addington–Thomas and Huybrechts on the derived category of special cubic fourfolds, to a new proof of the integral Hodge conjecture, and to the construction of an infinite series of unirational locally complete families of polarized hyperkähler manifolds of K3 type.Other applications include the deformation-invariance of Donaldson–Thomas invariants counting Bridgeland stable objects on Calabi–Yau threefolds, and a method for constructing stability conditions on threefolds via degeneration.

7

Meerschaert, Mark M., and Hans-Peter Scheffler. "Series representation for semistable laws and their domains of semistable attraction." Journal of Theoretical Probability 9, no. 4 (October 1996): 931–59. http://dx.doi.org/10.1007/bf02214258.

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8

Rajput, Balram S., and Kavi Rama-Murthy. "Spectral representation of semistable processes, and semistable laws on Banach spaces." Journal of Multivariate Analysis 21, no. 1 (February 1987): 139–57. http://dx.doi.org/10.1016/0047-259x(87)90103-5.

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9

Shimizu, Koji. "A -adic monodromy theorem for de Rham local systems." Compositio Mathematica 158, no. 12 (December 2022): 2157–205. http://dx.doi.org/10.1112/s0010437x2200776x.

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We study horizontal semistable and horizontal de Rham representations of the absolute Galois group of a certain smooth affinoid over a $p$ -adic field. In particular, we prove that a horizontal de Rham representation becomes horizontal semistable after a finite extension of the base field. As an application, we show that every de Rham local system on a smooth rigid analytic variety becomes horizontal semistable étale locally around every classical point. We also discuss potentially crystalline loci of de Rham local systems and cohomologically potentially good reduction loci of smooth proper morphisms.

10

Fujita, Kento. "On Berman–Gibbs stability and K-stability of -Fano varieties." Compositio Mathematica 152, no. 2 (November 26, 2015): 288–98. http://dx.doi.org/10.1112/s0010437x1500768x.

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The notion of Berman–Gibbs stability was originally introduced by Berman for $\mathbb{Q}$-Fano varieties $X$. We show that the pair $(X,-K_{X})$ is K-stable (respectively K-semistable) provided that $X$ is Berman–Gibbs stable (respectively semistable).

11

Melikyan, G. B., W. D. Niles, V. A. Ratinov, M. Karhanek, J. Zimmerberg, and F. S. Cohen. "Comparison of transient and successful fusion pores connecting influenza hemagglutinin expressing cells to planar membranes." Journal of General Physiology 106, no. 5 (November 1, 1995): 803–19. http://dx.doi.org/10.1085/jgp.106.5.803.

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Time-resolved admittance measurements were used to investigate the evolution of fusion pores formed between cells expressing influenza virus hemagglutinin (HA) and planar bilayer membranes. The majority of fusion pores opened in a stepwise fashion to semistable conductance levels of several nS. About 20% of the pores had measurable rise times to nS conductances; some of these opened to conductances of approximately 500 pS where they briefly lingered before opening further to semistable conductances. The fall times of closing were statistically similar to the rise times of opening. All fusion pores exhibited semistable values of conductance, varying from approximately 2-20 nS; they would then either close or fully open to conductances on the order of 1 microS. The majority of pores closed; approximately 10% fully opened. Once within the semistable stage, all fusion pores, even those that eventually closed, tended to grow. Statistically, however, before closing, transient fusion pores ceased to grow and reversed their conductance pattern: conductances decreased with a measurable time course until a final drop to closure. In contrast, pore enlargement to the fully open state tended to occur from the largest conductance values attained during a pore's semistable stage. This final enlargement was characterized by a stepwise increase in conductance. The density of HA on the cell surface did not strongly affect pore dynamics. But increased proteolytic treatment of cell surfaces did lead to faster growth within the semistable range. Transient pores and pores that fully opened had indistinguishable initial conductances and statistically identical time courses of early growth, suggesting they were the same upon formation. We suggest that transient and fully open pores evolved from common structures with stochastic factors determining their fate.

12

Molcho, Sam. "Universal stacky semistable reduction." Israel Journal of Mathematics 242, no. 1 (March 23, 2021): 55–82. http://dx.doi.org/10.1007/s11856-021-2118-0.

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13

Csorgő, Sandor, Sandor Csorgő, Zoltan Megyesi, Zoltan Megyesi, Zoltan Megyesi, and Zoltan Megyesi. "Merging to semistable laws." Teoriya Veroyatnostei i ee Primeneniya 47, no. 1 (2002): 90–109. http://dx.doi.org/10.4213/tvp2999.

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14

Фазекаш, Истван, and Istvan Fazekas. "Merging to semistable processes." Teoriya Veroyatnostei i ee Primeneniya 56, no. 4 (2011): 726–41. http://dx.doi.org/10.4213/tvp4420.

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15

Csorgo, S., and Z. Megyesi. "Merging to Semistable Laws." Theory of Probability & Its Applications 47, no. 1 (January 2003): 17–33. http://dx.doi.org/10.1137/s0040585x97979470.

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16

Fontaine, J. M. "Deforming semistable Galois representations." Proceedings of the National Academy of Sciences 94, no. 21 (October 14, 1997): 11138–41. http://dx.doi.org/10.1073/pnas.94.21.11138.

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17

Fazekas, I. "Merging to Semistable Processes." Theory of Probability & Its Applications 56, no. 4 (January 2012): 621–33. http://dx.doi.org/10.1137/s0040585x97985662.

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18

Shokurov, V. V. "SEMISTABLE 3-FOLD FLIPS." Russian Academy of Sciences. Izvestiya Mathematics 42, no. 2 (April 30, 1994): 371–425. http://dx.doi.org/10.1070/im1994v042n02abeh001541.

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19

Prokhorov, Yu G. "On semistable Mori contractions." Izvestiya: Mathematics 68, no. 2 (April 30, 2004): 365–74. http://dx.doi.org/10.1070/im2004v068n02abeh000478.

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20

Kern, Peter, and Lina Wedrich. "Dilatively semistable stochastic processes." Statistics & Probability Letters 99 (April 2015): 101–8. http://dx.doi.org/10.1016/j.spl.2015.01.008.

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21

Yurikusa, Toshiya. "Wide Subcategories are Semistable." Documenta Mathematica 23 (2018): 35–47. http://dx.doi.org/10.4171/dm/612.

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22

Krapavitskaite, D. "Discrete semistable probability distributions." Journal of Soviet Mathematics 38, no. 5 (September 1987): 2309–19. http://dx.doi.org/10.1007/bf01093832.

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23

Dani, S. G., and Riddhi Shah. "Contraction subgroups and semistable measures on p-adic Lie groups." Mathematical Proceedings of the Cambridge Philosophical Society 110, no. 2 (September 1991): 299–306. http://dx.doi.org/10.1017/s0305004100070377.

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Continuous one-parameter semigroups {μt}t≥0 of probability measures on a locally compact group which are semistable with respect to some automorphism τ of the group, namely such that τ(μt) = μct for all t ≥ 0, for a fixed c ∈ (0, 1), have attracted considerable attention of various researchers in recent years (cf. [3], [5] and other references cited therein). A detailed study of semistable measures on (real) Lie groups is carried out in [5]. In this context it is of interest to study semistable measures on the class of p-adic Lie groups, which is another significant class of locally compact groups.

24

Słowik, Oskar, Martin Hebenstreit, Barbara Kraus, and Adam Sawicki. "A link between symmetries of critical states and the structure of SLOCC classes in multipartite systems." Quantum 4 (July 20, 2020): 300. http://dx.doi.org/10.22331/q-2020-07-20-300.

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Central in entanglement theory is the characterization of local transformations among pure multipartite states. As a first step towards such a characterization, one needs to identify those states which can be transformed into each other via local operations with a non-vanishing probability. The classes obtained in this way are called SLOCC classes. They can be categorized into three disjoint types: the null-cone, the polystable states and strictly semistable states. Whereas the former two are well characterized, not much is known about strictly semistable states. We derive a criterion for the existence of the latter. In particular, we show that there exists a strictly semistable state if and only if there exist two polystable states whose orbits have different dimensions. We illustrate the usefulness of this criterion by applying it to tripartite states where one of the systems is a qubit. Moreover, we scrutinize all SLOCC classes of these systems and derive a complete characterization of the corresponding orbit types. We present representatives of strictly semistable classes and show to which polystable state they converge via local regular operators.

25

Česnavičius, Kęstutis, and Teruhisa Koshikawa. "The -cohomology in the semistable case." Compositio Mathematica 155, no. 11 (September 9, 2019): 2039–128. http://dx.doi.org/10.1112/s0010437x1800790x.

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For a proper, smooth scheme $X$ over a $p$-adic field $K$, we show that any proper, flat, semistable ${\mathcal{O}}_{K}$-model ${\mathcal{X}}$ of $X$ whose logarithmic de Rham cohomology is torsion free determines the same ${\mathcal{O}}_{K}$-lattice inside $H_{\text{dR}}^{i}(X/K)$ and, moreover, that this lattice is functorial in $X$. For this, we extend the results of Bhatt–Morrow–Scholze on the construction and the analysis of an $A_{\text{inf}}$-valued cohomology theory of $p$-adic formal, proper, smooth ${\mathcal{O}}_{\overline{K}}$-schemes $\mathfrak{X}$ to the semistable case. The relation of the $A_{\text{inf}}$-cohomology to the $p$-adic étale and the logarithmic crystalline cohomologies allows us to reprove the semistable conjecture of Fontaine–Jannsen.

26

Bertram, Aaron, and Cristian Martinez. "Change of Polarization for Moduli of Sheaves on Surfaces as Bridgeland Wall-crossing." International Mathematics Research Notices 2020, no. 7 (April 25, 2018): 2007–33. http://dx.doi.org/10.1093/imrn/rny065.

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Abstract We prove that the “Thaddeus flips” of L-twisted sheaves constructed by Matsuki and Wentworth explaining the change of polarization for Gieseker semistable sheaves on a surface can be obtained via Bridgeland wall-crossing. Similarly, we realize the change of polarization for moduli spaces of one-dimensional Gieseker semistable sheaves on a surface by varying a family of stability conditions.

27

Kedlaya, Kiran S. "Semistable reduction for overconvergent F-isocrystals, III: Local semistable reduction at monomial valuations." Compositio Mathematica 145, no. 1 (January 2009): 143–72. http://dx.doi.org/10.1112/s0010437x08003783.

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AbstractWe resolve the local semistable reduction problem for overconvergent F-isocrystals at monomial valuations (Abhyankar valuations of height 1 and residue transcendence degree zero). We first introduce a higher-dimensional analogue of the generic radius of convergence for a p-adic differential module, which obeys a convexity property. We then combine this convexity property with a form of the p-adic local monodromy theorem for so-called fake annuli.

28

Kedlaya, Kiran S. "Semistable reduction for overconvergent F-isocrystals, IV: local semistable reduction at nonmonomial valuations." Compositio Mathematica 147, no. 2 (February 21, 2011): 467–523. http://dx.doi.org/10.1112/s0010437x10005142.

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AbstractWe complete our proof that given an overconvergent F-isocrystal on a variety over a field of positive characteristic, one can pull back along a suitable generically finite cover to obtain an isocrystal which extends, with logarithmic singularities and nilpotent residues, to some complete variety. We also establish an analogue for F-isocrystals overconvergent inside a partial compactification. By previous results, this reduces to solving a local problem in a neighborhood of a valuation of height 1 and residual transcendence degree zero. We do this by studying the variation of some numerical invariants attached to p-adic differential modules, analogous to the irregularity of a complex meromorphic connection. This allows for an induction on the transcendence defect of the valuation, i.e., the discrepancy between the dimension of the variety and the rational rank of the valuation.

29

Choi, Gyeong Suck. "Criteria for recurrence and transience of semistable processes." Nagoya Mathematical Journal 134 (June 1994): 91–106. http://dx.doi.org/10.1017/s0027763000004876.

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The definition of semistable laws was originally given by Lévy in [11]. Two books, Kagan, Linnik and Rao [6] and Ramachandran and Lau [13], call a probability measure on R “semistable” when it is nondegenerate and its characteristic function (ch.f.) f(z) does not vanish on R and satisfies a functional equation of the formfor some real numbers b (0 < | b | <1) and c > 1.

30

Zhu, Ziwen. "Higher codimensional alpha invariants and characterization of projective spaces." International Journal of Mathematics 31, no. 02 (December 31, 2019): 2050012. http://dx.doi.org/10.1142/s0129167x20500123.

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We generalize the definition of alpha invariant to arbitrary codimension. We also give a lower bound of these alpha invariants for K-semistable [Formula: see text]-Fano varieties and show that we can characterize projective spaces among all K-semistable Fano manifolds in terms of higher codimensional alpha invariants. Our results demonstrate the relation between alpha invariants of any codimension and volumes of Fano manifolds in the characterization of projective spaces.

31

Choi, Jinwon, and Kiryong Chung. "Cohomology bounds for sheaves of dimension one." International Journal of Mathematics 25, no. 11 (October 2014): 1450103. http://dx.doi.org/10.1142/s0129167x14501031.

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We find sharp bounds on h0(F) for one-dimensional semistable sheaves F on a projective variety X. When X is the projective plane ℙ2, we study the stratification of the moduli space by the spectrum of sheaves. We show that the deepest stratum is isomorphic to a closed subset of a relative Hilbert scheme. This provides an example of a family of semistable sheaves having the biggest dimensional global section space.

32

Biswas, Indranil, Tomás L. Gómez, and Marina Logares. "Integrable Systems and Torelli Theorems for the Moduli Spaces of Parabolic Bundles and Parabolic Higgs Bundles." Canadian Journal of Mathematics 68, no. 3 (June 1, 2016): 504–20. http://dx.doi.org/10.4153/cjm-2015-039-5.

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AbstractWe prove a Torelli theorem for the moduli space of semistable parabolic Higgs bundles over a smooth complex projective algebraic curve under the assumption that the parabolic weight systemis generic. When the genus is at least two, using this result we also prove a Torelli theoremfor the moduli space of semistable parabolic bundles of rank at least two with generic parabolic weights. The key input in the proofs is a method of J.C. Hurtubise.

33

Langer, Adrian. "Semistable sheaves in positive characteristic." Annals of Mathematics 159, no. 1 (January 1, 2004): 251–76. http://dx.doi.org/10.4007/annals.2004.159.251.

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34

Ribet, Kenneth. "Images of semistable Galois representations." Pacific Journal of Mathematics 181, no. 3 (December 1, 1997): 277–97. http://dx.doi.org/10.2140/pjm.1997.181.277.

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35

Nemirovski, Stefan. "Levi problem and semistable quotients." Complex Variables and Elliptic Equations 58, no. 11 (November 2013): 1517–25. http://dx.doi.org/10.1080/17476933.2011.592579.

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36

Nitsure, Nitin. "Moduli of semistable logarithmic connections." Journal of the American Mathematical Society 6, no. 3 (September 1, 1993): 597. http://dx.doi.org/10.1090/s0894-0347-1993-1182671-2.

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37

Hazod, W., and R. Shah. "Semistable Selfdecomposable Laws on Groups." Journal of Applied Analysis 7, no. 1 (January 2001): 1–22. http://dx.doi.org/10.1515/jaa.2001.1.

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38

Keating, Kevin. "Galois scaffolds and semistable extensions." Journal of Number Theory 207 (February 2020): 110–21. http://dx.doi.org/10.1016/j.jnt.2019.07.002.

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39

Calegari, Frank. "Semistable abelian varieties over Q." manuscripta mathematica 113, no. 4 (April 1, 2004): 507–29. http://dx.doi.org/10.1007/s00229-004-0445-1.

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40

Garcia, Monica, and Alexander Garver. "Semistable subcategories for tiling algebras." Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 61, no. 1 (July 10, 2019): 47–71. http://dx.doi.org/10.1007/s13366-019-00461-y.

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41

Csörgő, Sándor. "Fourier Analysis of Semistable Distributions." Acta Applicandae Mathematicae 96, no. 1-3 (April 5, 2007): 159–74. http://dx.doi.org/10.1007/s10440-007-9111-4.

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42

Csörgő, Sándor. "Fourier Analysis of Semistable Distributions." Acta Applicandae Mathematicae 96, no. 1-3 (May 4, 2007): 175. http://dx.doi.org/10.1007/s10440-007-9143-9.

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43

Davis, Ryan, Charles Doran, Adam Gewiss, Andrey Novoseltsev, Dmitri Skjorshammer, Alexa Syryczuk, and Ursula Whitcher. "Short Tops and Semistable Degenerations." Experimental Mathematics 23, no. 4 (October 2, 2014): 351–62. http://dx.doi.org/10.1080/10586458.2014.910848.

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44

Nizioł, Wiesława. "Semistable conjecture via $K$ -theory." Duke Mathematical Journal 141, no. 1 (January 2008): 151–78. http://dx.doi.org/10.1215/s0012-7094-08-14114-6.

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45

Langer, Adrian. "On boundedness of semistable sheaves." Documenta Mathematica 27 (2022): 1–16. http://dx.doi.org/10.4171/dm/865.

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46

Prokhorov, Yu G. "On Semistable Mori Conic Bundles." Journal of Mathematical Sciences 131, no. 6 (December 2005): 6140–47. http://dx.doi.org/10.1007/s10958-005-0467-6.

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47

Pavel, Mihai. "Restriction theorems for semistable sheaves." Documenta Mathematica 29, no. 3 (May 8, 2024): 597–625. http://dx.doi.org/10.4171/dm/957.

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48

HUILLET, THIERRY, ANNA PORZIO, and MOHAMED BEN ALAYA. "ON LÉVY STABLE AND SEMISTABLE DISTRIBUTIONS." Fractals 09, no. 03 (September 2001): 347–64. http://dx.doi.org/10.1142/s0218348x01000786.

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This work emphasizes the special role played by semistable and log-semistable distributions as relevant statistical models of various observable and "internal" variables in physics. Besides of their representation, some of their remarkable properties (chiefly semi-self-similarity) are displayed in some detail. One of their characteristic features is a log-periodic variation of the scale parameter which appears in the standard Lévy α-stable distributions whose Fourier representations are re-derived in a self-contained way.

49

Chiarellotto, Bruno, and Christopher Lazda. "Around -independence." Compositio Mathematica 154, no. 1 (October 17, 2017): 223–48. http://dx.doi.org/10.1112/s0010437x17007527.

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In this article we study various forms of $\ell$-independence (including the case $\ell =p$) for the cohomology and fundamental groups of varieties over finite fields and equicharacteristic local fields. Our first result is a strong form of $\ell$-independence for the unipotent fundamental group of smooth and projective varieties over finite fields. By then proving a certain ‘spreading out’ result we are able to deduce a much weaker form of $\ell$-independence for unipotent fundamental groups over equicharacteristic local fields, at least in the semistable case. In a similar vein, we can also use this to deduce $\ell$-independence results for the cohomology of smooth and proper varieties over equicharacteristic local fields from the well-known results on $\ell$-independence for smooth and proper varieties over finite fields. As another consequence of this ‘spreading out’ result we are able to deduce the existence of a Clemens–Schmid exact sequence for formal semistable families. Finally, by deforming to characteristic $p$, we show a similar weak version of $\ell$-independence for the unipotent fundamental group of a semistable curve in mixed characteristic.

50

Diaconescu, Duiliu-Emanuel, Mauro Porta, and Francesco Sala. "McKay correspondence, cohomological Hall algebras and categorification." Representation Theory of the American Mathematical Society 27, no. 25 (October 5, 2023): 933–72. http://dx.doi.org/10.1090/ert/649.

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Let π : Y → X \pi \colon Y\to X denote the canonical resolution of the two dimensional Kleinian singularity X X of type ADE. In the present paper, we establish isomorphisms between the cohomological and K-theoretical Hall algebras of ω \omega -semistable properly supported sheaves on Y Y with fixed slope μ \mu and ζ \zeta -semistable finite-dimensional representations of the preprojective algebra of affine type ADE of slope zero respectively, under some conditions on ζ \zeta depending on the polarization ω \omega and μ \mu . These isomorphisms are induced by the derived McKay correspondence. In addition, they are interpreted as decategorified versions of a monoidal equivalence between the corresponding categorified Hall algebras. In the type A case, we provide a finer description of the cohomological, K-theoretical and categorified Hall algebra of ω \omega -semistable properly supported sheaves on Y Y with fixed slope μ \mu : for example, in the cohomological case, the algebra can be given in terms of Yangians of finite type ADE Dynkin diagrams.