Готові списки джерел за темами / Irreducible holomorphic symplectic manifolds

Добірка наукової літератури з теми "Irreducible holomorphic symplectic manifolds"

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Опубліковано: 8 червня 2024

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Статті в журналах з теми "Irreducible holomorphic symplectic manifolds":

1

Camere, Chiara. "Lattice polarized irreducible holomorphic symplectic manifolds." Annales de l’institut Fourier 66, no. 2 (2016): 687–709. http://dx.doi.org/10.5802/aif.3022.

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2

Boissière, Samuel, Marc Nieper-Wißkirchen, and Alessandra Sarti. "Smith theory and irreducible holomorphic symplectic manifolds." Journal of Topology 6, no. 2 (February 13, 2013): 361–90. http://dx.doi.org/10.1112/jtopol/jtt002.

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3

Franco, Emilio, Marcos Jardim, and Grégoire Menet. "Brane involutions on irreducible holomorphic symplectic manifolds." Kyoto Journal of Mathematics 59, no. 1 (April 2019): 195–235. http://dx.doi.org/10.1215/21562261-2018-0009.

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4

Braverman, Maxim. "Symplectic cutting of Kähler manifolds." Journal für die reine und angewandte Mathematik (Crelles Journal) 1999, no. 508 (March 12, 1999): 85–98. http://dx.doi.org/10.1515/crll.1999.508.85.

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Abstract We obtain estimates on the character of the cohomology of an S1-equivariant holomorphic vector bundle over a Kähler manifold M in terms of the cohomology of the Lerman symplectic cuts and the symplectic reduction of M. In particular, we prove and extend inequalities conjectured by Wu and Zhang. The proof is based on constructing a flat family of complex spaces Mt (t ∈ ℂ) such that Mt is isomorphic to M for t ≠ 0, while M0 is a singular reducible complex space, whose irreducible components are the Lerman symplectic cuts.
5

Camere, Chiara. "Some remarks on moduli spaces of lattice polarized holomorphic symplectic manifolds." Communications in Contemporary Mathematics 20, no. 04 (May 20, 2018): 1750044. http://dx.doi.org/10.1142/s0219199717500444.

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We construct quasi-projective moduli spaces of [Formula: see text]-general lattice polarized irreducible holomorphic symplectic manifolds. Moreover, we study their Baily–Borel compactification and investigate a relation between one-dimensional boundary components and equivalence classes of rational Lagrangian fibrations defined on mirror manifolds.
6

Amerik, Ekaterina, and Misha Verbitsky. "Construction of automorphisms of hyperkähler manifolds." Compositio Mathematica 153, no. 8 (May 31, 2017): 1610–21. http://dx.doi.org/10.1112/s0010437x17007138.

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Let $M$ be an irreducible holomorphic symplectic (hyperkähler) manifold. If $b_{2}(M)\geqslant 5$, we construct a deformation $M^{\prime }$ of $M$ which admits a symplectic automorphism of infinite order. This automorphism is hyperbolic, that is, its action on the space of real $(1,1)$-classes is hyperbolic. If $b_{2}(M)\geqslant 14$, similarly, we construct a deformation which admits a parabolic automorphism (and many other automorphisms as well).
7

Brecan, Ana-Maria, Tim Kirschner, and Martin Schwald. "Unobstructedness of hyperkähler twistor spaces." Mathematische Zeitschrift 300, no. 3 (October 6, 2021): 2485–517. http://dx.doi.org/10.1007/s00209-021-02841-4.

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AbstractA family of irreducible holomorphic symplectic (ihs) manifolds over the complex projective line has unobstructed deformations if its period map is an embedding. This applies in particular to twistor spaces of ihs manifolds. Moreover, a family of ihs manifolds over a subspace of the period domain extends to a universal family over an open neighborhood in the period domain.
8

Camere, Chiara, Grzegorz Kapustka, Michał Kapustka, and Giovanni Mongardi. "Verra Four-Folds, Twisted Sheaves, and the Last Involution." International Mathematics Research Notices 2019, no. 21 (February 1, 2018): 6661–710. http://dx.doi.org/10.1093/imrn/rnx327.

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Abstract We study the geometry of some moduli spaces of twisted sheaves on K3 surfaces. In particular we introduce induced automorphisms from a K3 surface on moduli spaces of twisted sheaves on this K3 surface. As an application we prove the unirationality of moduli spaces of irreducible holomorphic symplectic manifolds of K3[2]-type admitting non-symplectic involutions with invariant lattices U(2) ⊕ D4(−1) or U(2) ⊕ E8(−2). This complements the results obtained in [43], [13], and the results from [29] about the geometry of irreducible holomorphic symplectic (IHS) four-folds constructed using the Hilbert scheme of (1, 1) conics on Verra four-folds. As a byproduct we find that IHS four-folds of K3[2]-type with Picard lattice U(2) ⊕ E8(−2) naturally contain non-nodal Enriques surfaces.
9

Golla, Marco, and Laura Starkston. "The symplectic isotopy problem for rational cuspidal curves." Compositio Mathematica 158, no. 7 (July 2022): 1595–682. http://dx.doi.org/10.1112/s0010437x2200762x.

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We define a suitably tame class of singular symplectic curves in 4-manifolds, namely those whose singularities are modeled on complex curve singularities. We study the corresponding symplectic isotopy problem, with a focus on rational curves with irreducible singularities (rational cuspidal curves) in the complex projective plane. We prove that every such curve is isotopic to a complex curve in degrees up to five, and for curves with one singularity whose link is a torus knot. Classification results of symplectic isotopy classes rely on pseudo-holomorphic curves together with a symplectic version of birational geometry of log pairs and techniques from four-dimensional topology.
10

Knutsen, Andreas Leopold, Margherita Lelli-Chiesa, and Giovanni Mongardi. "Wall divisors and algebraically coisotropic subvarieties of irreducible holomorphic symplectic manifolds." Transactions of the American Mathematical Society 371, no. 2 (September 20, 2018): 1403–38. http://dx.doi.org/10.1090/tran/7340.

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Статті в журналах

Дисертації з теми "Irreducible holomorphic symplectic manifolds":

1

Cattaneo, Alberto. "Non-symplectic automorphisms of irreducible holomorphic symplectic manifolds." Thesis, Poitiers, 2018. http://www.theses.fr/2018POIT2322/document.

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Nous allons étudier les automorphismes des variétés symplectiques holomorphes irréductibles de type K3^[n], c'est-à-dire des variétés équivalentes par déformation au schéma de Hilbert de n points sur une surface K3, pour n > 1.Dans la première partie de la thèse, nous classifions les automorphismes du schéma de Hilbert de n points sur une surface K3 projective générique, dont le réseau de Picard est engendré par un fibré ample. Nous montrons que le groupe des automorphismes est soit trivial soit engendré par une involution non-symplectique et nous déterminons des conditions numériques et géométriques pour l’existence de l’involution.Dans la deuxième partie, nous étudions les automorphismes non-symplectiques d’ordre premier des variétés de type K3^[n]. Nous déterminons les propriétés du réseau invariant de l'automorphisme et de son complément orthogonal dans le deuxième réseau de cohomologie de la variété et nous classifions leurs classes d’isométrie. Dans le cas des involutions, e des automorphismes d’ordre premier impair pour n = 3, 4, nous montrons que toutes les actions en cohomologie dans notre classification sont réalisées par un automorphism non-symplectique sur une variété de type K3^[n]. Nous construisons explicitement l’immense majorité de ces automorphismes et, en particulier, nous présentons la construction d’un nouvel automorphisme d’ordre trois sur une famille de dimension dix de variétés de Lehn-Lehn-Sorger-van Straten de type K3^[4]. Pour n < 6, nous étudions aussi les espaces de modules de dimension maximal des variétés de type K3^[n] munies d’une involution non-symplectique
We study automorphisms of irreducible holomorphic symplectic manifolds of type K3^[n], i.e. manifolds which are deformation equivalent to the Hilbert scheme of n points on a K3 surface, for some n > 1. In the first part of the thesis we describe the automorphism group of the Hilbert scheme of n points on a generic projective K3 surface, i.e. a K3 surface whose Picard lattice is generated by a single ample line bundle. We show that, if it is not trivial, the automorphism group is generated by a non-symplectic involution, whose existence depends on some arithmetic conditions involving the number of points n and the polarization of the surface. We also determine necessary and sufficient conditions on the Picard lattice of the Hilbert scheme for the existence of the involution.In the second part of the thesis we study non-symplectic automorphisms of prime order on manifolds of type K3^[n]. We investigate the properties of the invariant lattice and its orthogonal complement inside the second cohomology lattice of the manifold, providing a classification of their isometry classes. We then approach the problem of constructing examples (or at least proving the existence) of manifolds of type K3^[n] with a non-symplectic automorphism inducing on cohomology each specific action in our classification. In the case of involutions, and of automorphisms of odd prime order for n=3,4, we are able to realize all possible cases. In order to do so, we present a new non-symplectic automorphism of order three on a ten-dimensional family of Lehn-Lehn-Sorger-van Straten eightfolds of type K3^[4]. Finally, for n < 6 we describe deformation families of large dimension of manifolds of type K3^[n] equipped with a non-symplectic involution
2

CATTANEO, ALBERTO. "NON-SYMPLECTIC AUTOMORPHISMS OF IRREDUCIBLE HOLOMORPHIC SYMPLECTIC MANIFOLDS." Doctoral thesis, Università degli Studi di Milano, 2018. http://hdl.handle.net/2434/606455.

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La tesi si concentra sullo studio degli automorfismi di varietà olomorfe simplettiche irriducibili di tipo K3^[n], ovvero varietà equivalenti per deformazione allo schema di Hilbert di n punti su una superficie K3, per n > 1. Negli ultimi anni, molti teoremi classici riguardanti la classificazione degli automorfismi non-simplettici di superfici K3 sono stati estesi alle varietà di tipo K3^[2]. Siamo quindi interessati a comprendere se tali risultati possono essere ulteriormente generalizzati anche al caso di varietà di tipo K3^[n], per n > 2. Nella prima parte della tesi descriviamo il gruppo degli automorfismi dello schema di Hilbert di n punti su una superficie K3 proiettiva generica, il cui reticolo di Picard è generato da un singolo fibrato ampio. Mostriamo che, se il gruppo non è triviale, esso è generato da una involuzione non-simplettica, la cui esistenza è determinata da condizioni aritmetiche che coinvolgono il numero n di punti e la polarizzazione della superficie. In aggiunta a tale caratterizzazione numerica, individuiamo anche delle condizioni necessarie e sufficienti per l'esistenza dell'involuzione riguardanti la struttura del reticolo di Picard dello schema di Hilbert. La seconda parte della tesi è dedicata allo studio degli automorfismi non-simplettici di ordine primo su varietà di tipo K3^[n]. Dopo aver investigato le proprietà del reticolo invariante dell'automorfismo e del suo complemento ortogonale all'interno del secondo reticolo di coomologia della varietà, forniamo una classificazione per le loro classi di isometria. Affrontiamo quindi il problema di individuare varietà di tipo K3^[n] dotate di automorfismi non-simplettici che inducano ognuna delle possibili azioni in coomologia presenti nella nostra classificazione. Nel caso delle involuzioni, e degli automorfismi di ordine primo dispari per n=3, 4, siamo in grado di realizzare tutti i casi ammissibili, presentando una costruzione esplicita della varietà o almeno dimostrandone l'esistenza. Tra i numerosi esempi esibiti, è di particolare rilievo un nuovo automorfismo di ordine tre su una famiglia di dimensione dieci di varietà di Lehn-Lehn-Sorger-van Straten di tipo K3^[4]. Infine, per n < 6 descriviamo le famiglie di deformazione massimali di varietà di tipo K3^[n] dotate di una involuzione non-simplettica.
We study automorphisms of irreducible holomorphic symplectic manifolds of type K3^[n], i.e. manifolds which are deformation equivalent to the Hilbert scheme of n points on a K3 surface, for some n > 1. In the first part of the thesis we describe the automorphism group of the Hilbert scheme of n points on a generic projective K3 surface, i.e. a K3 surface whose Picard lattice is generated by a single ample line bundle. We show that, if it is not trivial, the automorphism group is generated by a non-symplectic involution, whose existence depends on some arithmetic conditions involving the number of points n and the polarization of the surface. We also determine necessary and sufficient conditions on the Picard lattice of the Hilbert scheme for the existence of the involution. In the second part of the thesis we study non-symplectic automorphisms of prime order on manifolds of type K3^[n]. We investigate the properties of the invariant lattice and its orthogonal complement inside the second cohomology lattice of the manifold, providing a classification of their isometry classes. We then approach the problem of constructing examples (or at least proving the existence) of manifolds of type K3^[n] with a non-symplectic automorphism inducing on cohomology each specific action in our classification. In the case of involutions, and of automorphisms of odd prime order for n=3,4, we are able to realize all possible cases. In order to do so, we present a new non-symplectic automorphism of order three on a ten-dimensional family of Lehn-Lehn-Sorger-van Straten eightfolds of type K3^[4]. Finally, for n < 6 we describe deformation families of large dimension of manifolds of type K3^[n] equipped with a non-symplectic involution.
Nous allons étudier les automorphismes des variétés symplectiques holomorphes irréductibles de type K3^[n], c'est-à-dire des variétés équivalentes par déformation au schéma de Hilbert de n points sur une surface K3, pour n > 1. Dans la première partie de la thèse, nous classifions les automorphismes du schéma de Hilbert de n points sur une surface K3 projective générique, dont le réseau de Picard est engendré par un fibré ample. Nous montrons que le groupe des automorphismes est soit trivial soit engendré par une involution non-symplectique et nous déterminons des conditions numériques et géométriques pour l’existence de l’involution. Dans la deuxième partie, nous étudions les automorphismes non-symplectiques d’ordre premier des variétés de type K3^[n]. Nous déterminons les propriétés du réseau invariant de l'automorphisme et de son complément orthogonal dans le deuxième réseau de cohomologie de la variété et nous classifions leurs classes d’isométrie. Dans le cas des involutions, e des automorphismes d’ordre premier impair pour n = 3, 4, nous montrons que toutes les actions en cohomologie dans notre classification sont réalisées par un automorphism non-symplectique sur une variété de type K3^[n]. Nous construisons explicitement l’immense majorité de ces automorphismes et, en particulier, nous présentons la construction d’un nouvel automorphisme d’ordre trois sur une famille de dimension dix de variétés de Lehn-Lehn-Sorger-van Straten de type K3^[4]. Pour n < 6, nous étudions aussi les espaces de modules de dimension maximal des variétés de type K3^[n] munies d’une involution non-symplectique.
3

Onorati, Claudio. "Irreducible holomorphic symplectic manifolds and monodromy operators." Thesis, University of Bath, 2018. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.767583.

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One of the most important tools to study the geometry of irreducible holomorphic symplectic manifolds is the monodromy group. The first part of this dissertation concerns the construction and studyof monodromy operators on irreducible holomorphic symplectic manifolds which are deformation equivalent to the 10-dimensional example constructed by O'Grady. The second part uses the knowledge of the monodromy group to compute the number of connected components of moduli spaces of bothmarked and polarised irreducible holomorphic symplectic manifolds which are deformationequivalent to generalised Kummer varieties.
4

NOVARIO, SIMONE. "LINEAR SYSTEMS ON IRREDUCIBLE HOLOMORPHIC SYMPLECTIC MANIFOLDS." Doctoral thesis, Università degli Studi di Milano, 2021. http://hdl.handle.net/2434/886303.

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In questa tesi studiamo alcuni sistemi lineari completi associati a divisori di schemi di Hilbert di 2 punti su una superficie K3 proiettiva complessa con gruppo di Picard di rango 1, e le mappe razionali indotte. Queste varietà sono chiamate quadrati di Hilbert su superfici K3 generiche, e sono esempi di varietà irriducibili olomorfe simplettiche (varietà IHS). Nella prima parte della tesi, usando la teoria dei reticoli, gli operatori di Nakajima e il modello di Lehn–Sorger, diamo una base per il sottospazio vettoriale dell’anello di coomologia singolare a coefficienti razionali generato dalle classi di Hodge razionali di tipo (2, 2) sul quadrato di Hilbert di una qualsiasi superficie K3 proiettiva. In seguito sfruttiamo un teorema di Qin e Wang insieme a un risultato di Ellingsrud, Göttsche e Lehn per ottenere una base del reticolo delle classi di Hodge integrali di tipo (2, 2) sul quadrato di Hilbert di una qualsiasi superficie K3 proiettiva. Nella seconda parte della tesi studiamo il problema seguente: se X è il quadrato di Hilbert di una superficie K3 generica che ammette un divisore ampio D con q(D) = 2, dove q è la forma quadratica di Beauville-Bogomolov-Fujiki, descrivere geometricamente la mappa razionale indotta dal sistema lineare completo |D|. Il risultato principale della tesi mostra che tale X, tranne nel caso del quadrato di Hilbert di una superficie quartica generica di P^3, è una doppia EPW sestica, cioè il ricoprimento doppio di una EPW sestica, una ipersuperficie normale di P^5, ramificato nel suo luogo singolare. Inoltre la mappa razionale indotta da |D| coincide proprio con tale ricoprimento doppio. Gli strumenti principali per ottenere questo risultato sono la descrizione del reticolo delle classi integrali di Hodge di tipo (2, 2) della prima parte della tesi e l’esistenza di un’involuzione anti-simplettica su tali varietà per un teorema di Boissière, Cattaneo, Nieper-Wißkirchen e Sarti.
In this thesis we study some complete linear systems associated to divisors of Hilbert schemes of 2 points on complex projective K3 surfaces with Picard group of rank 1, together with the rational maps induced. We call these varieties Hilbert squares of generic K3 surfaces, and they are examples of irreducible holomorphic symplectic (IHS) manifold. In the first part of the thesis, using lattice theory, Nakajima operators and the model of Lehn–Sorger, we give a basis for the subvector space of the singular cohomology ring with rational coefficients generated by rational Hodge classes of type (2, 2) on the Hilbert square of any projective K3 surface. We then exploit a theorem by Qin and Wang together with a result by Ellingsrud, Göttsche and Lehn to obtain a basis of the lattice of integral Hodge classes of type (2, 2) on the Hilbert square of any projective K3 surface. In the second part of the thesis we study the following problem: if X is the Hilbert square of a generic K3 surface admitting an ample divisor D with q(D)=2, where q is the Beauville–Bogomolov–Fujiki form, describe geometrically the rational map induced by the complete linear system |D|. The main result of the thesis shows that such an X, except on the case of the Hilbert square of a generic quartic surface of P^3, is a double EPW sextic, i.e., the double cover of an EPW sextic, a normal hypersurface of P^5, ramified over its singular locus. Moreover, the rational map induced by |D| is a morphism and coincides exactly with this double covering. The main tools to obtain this result are the description of integral Hodge classes of type (2, 2) of the first part of the thesis and the existence of an anti-symplectic involution on such varieties due to a theorem by Boissière, Cattaneo, Nieper-Wißkirchen and Sarti.
Dans cette thèse, nous étudions certains systèmes linéaires complets associés aux diviseurs des schémas de Hilbert de 2 points sur des surfaces K3 projectives complexes avec groupe de Picard de rang 1, et les fonctions rationnelles induites. Ces variétés sont appelées carrés de Hilbert sur des surfaces K3 génériques, et sont un exemple de variété symplectique holomorphe irréductible (variété IHS). Dans la première partie de la thèse, en utilisant la théorie des réseaux, les opérateurs de Nakajima et le modèle de Lehn–Sorger, nous donnons une base pour le sous-espace vectoriel de l’anneau de cohomologie singulière à coefficients rationnels engendré par les classes de Hodge rationnels de type (2, 2) sur le carré de Hilbert de toute surface K3 projective. Nous exploitons ensuite un théorème de Qin et Wang ainsi qu’un résultat de Ellingsrud, Göttsche et Lehn pour obtenir une base du réseau des classes de Hodge intégraux de type (2, 2) sur le carré de Hilbert d’une surface K3 projective quelconque. Dans la deuxième partie de la thèse, nous étudions le problème suivant : si X est le carré de Hilbert d’une surface K3 générique tel que X admet un diviseur ample D avec q(D) = 2, où q est la forme quadratique de Beauville–Bogomolov–Fujiki, on veut décrire géométriquement la fonction rationnelle induite par le système linéaire complet |D|. Le résultat principal de la thèse montre qu’une telle X, sauf dans le cas du carré de Hilbert d’une surface quartique générique de P^3, est une double sextique EPW, c’est-à-dire le revêtement double d’une sextique EPW, une hypersurface normale de P^5, ramifié sur son lieu singulier. En plus la fonction rationnelle induite par |D| est exactement ce revêtement double. Les outils principaux pour obtenir ce résultat sont la description des classes de Hodge intégraux de type (2, 2) de la première partie de la thèse et l’existence d’une involution anti-symplectique sur de telles variétés par un théorème de Boissière, Cattaneo, Nieper-Wißkirchen et Sarti.
5

Denisi, Francesco Antonio. "Positivité sur les variétés irréductibles holomorphes symplectiques." Electronic Thesis or Diss., Université de Lorraine, 2023. http://www.theses.fr/2023LORR0162.

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Dans cette thèse, nous étudions certains aspects de la positivité des diviseurs sur les variétés irréductibles holomorphes symplectiques (IHS). Fixons une variété IHS projective X de dimension complexe 2n. Inspirés par le travail de Bauer, Küronya et Szemberg, nous montrons que le cône big de X a une décomposition localement finie en sous-cônes localement rationnelles polyhédraux, qu'on appelle chambres de Boucksom-Zariski. Ces sous-cônes ont une signification géométrique : sur chacun d'eux, la fonction volume est exprimée par un polynôme homogène de degré 2n. De plus, à l'intérieur de toute chambre de Boucksom-Zariski, la partie divisorielle du lieu base augmenté des diviseurs big reste la même. Après avoir analysé le cône big, nous déterminons la structure du cône pseudo-effectif de X, généralisant ainsi un résultat bien connu de Kovács pour les surfaces K3. En particulier, nous montrons que si le nombre de Picard de X est au moins 3, le cône pseudo-effectif de X est soit circulaire, soit ne contient pas de parties circulaires et est égal à la clôture du cône engendré par les classes des diviseurs premiers exceptionnels. De ce résultat en géométrie convexe, nous déduisons quelques propriétés géométriques de X et nous montrons l'existence de diviseurs rigides uniréglés sur certaines variétés symplectiques singulières. Nous étudions le comportement des lieux de base asymptotiques des diviseurs big sur X et nous en donnons une caractérisation numérique. En conséquence de cette caractérisation numérique, nous obtenons une description des duaux des cônes mathrm{Amp}_k(X), pour tout 1leq k leq 2n, où mathrm{Amp}_k(X) est le cône convexe des classes des diviseurs big ayant le lieu base augmenté de dimension strictement plus petite que k. En utilisant la décomposition divisorielle de Zariski, la forme de Beauville-Bogomolov-Fujiki (BBF) et la décomposition du cône big de X en chambres de Boucksom-Zariski, nous associons à toute classe de diviseurs big alpha et à un diviseur premier E sur X un polygone Delta_E(alpha), dont la géométrie est liée à la variation de la décomposition divisorielle de alpha dans le cône big de X. Le volume euclidien est exprimé en termes de la forme BBF et est indépendant du choix de E. Nous montrons que ces polygones s'inscrivent dans un cône convexe Delta_E(X) sous forme de tranches, globalisant ainsi la construction. En conclusion, nous montrons que sous certaines hypothèses, les polygones Delta_E(alpha) peuvent être écrits comme une somme de Minkowski de certains polygones {Delta_E(Beta_i)}_{iin I}, pour certaines classes big {Beta_i}_{i in I}. Il est remarquable que ces polygones se comportent comme les corps de Newton-Okounkov des diviseurs big sur les surfaces projectives lisses
In this thesis, we study some aspects of the positivity of divisors on irreducible holomorphic symplectic (IHS) manifolds. Fix a projective IHS manifold X of complex dimension 2n. Inspired by the work of Bauer, Küronya, and Szemberg, we show that the big cone of X has a locally finite decomposition into locally rational polyhedral subcones, called Boucksom-Zariski chambers. These subcones have a geometric meaning: on any of them, the volume function is expressed by a homogeneous polynomial of degree 2n. Furthermore, in the interior of any Boucksom-Zariski chamber, the divisorial part of the augmented base locus of big divisors stays the same. After analyzing the big cone, we determine the structure of the pseudo-effective cone of X, generalizing a well-known result due to Kovács for K3 surfaces. In particular, we show that if the Picard number of X is at least 3, the pseudo-effective cone either is circular or does not contain circular parts and is equal to the closure of the cone generated by the prime exceptional divisor classes. From this result in convex geometry, we deduce some geometric properties of X and show the existence of rigid uniruled divisors on some singular symplectic varieties. We study the behaviour of the asymptotic base loci of big divisors on X, and we provide a numerical characterization for them. As a consequence of this numerical characterization, we obtain a description for the dual of the cones mathrm{Amp}_k(X), for any 1leq k leq 2n, where mathrm{Amp}_k(X) is the convex cone of big divisor classes having the augmented base locus of dimension strictly smaller than k. Using the divisorial Zariski decomposition, the Beauville-Bogomolov-Fujiki (BBF) form, and the decomposition of the big cone of X into Boucksom-Zariski chambers, we associate to any big divisor class alpha and a prime divisor E on X a polygon Delta_E(alpha) whose geometry is related to the variation of the divisorial Zariski decomposition of alpha in the big cone. Its euclidean volume is expressed in terms of the BBF form and is independent of the choice of E. We show that these polygons fit in a convex cone Delta_E(X) as slices, globalizing in this way the construction. To conclude, we show that under some hypothesis, the polygons Delta_E(alpha) can be expressed as a Minkowski sum of some polygons {Delta_E(Beta_i)}_{i in I}, for some big classes {Beta_i}_{_ iin I}. Remarkably, these polygons behave like the Newton-Okounkov bodies of big divisors on smooth projective surfaces
6

Joumaah, Malek [Verfasser]. "Automorphisms of irreducible symplectic manifolds / Malek Joumaah." Hannover : Technische Informationsbibliothek und Universitätsbibliothek Hannover (TIB), 2015. http://d-nb.info/1068920580/34.

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7

Bertini, Valeria. "Rational curves on irreducible symplectic varieties of OG10 type." Thesis, Strasbourg, 2019. https://publication-theses.unistra.fr/public/theses_doctorat/2019/Bertini_Valeria_2019_ED269.pdf.

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Анотація:
Les variétés holomorphes symplectiques irréductibles (VHSI) sont l'analogue algébrique des variétés Riemannienne hyperkähler. Une VHSI X avec dimension 2 est une surface K3, et dans ce cas, si de plus X est projective, chaque courbe ample sur X est linéairement équivalente à une somme de courbes rationnelles (Bogomolov, Mumford). Charles, Mongardi et Pacienza ont démontré l'existence de diviseurs uniréglés dans (presque) tous les systèmes linéaires amples sur une VHSI qui est déformation d'un schéma de Hilbert sur une surface K3 ou d'une variété de Kummer generalisée. La présence de nombreuses courbes rationnelles dans X simplifie la structure du 0-group de Chow de X. Dans ma thése, j'ai travaillé sur le cas OG10, la VHSI définie par O'Grady; la variété OG10 est importantes et très activement étudiées. Le résultat principal de ma thèse démontre l'existence de diviseurs uniréglés amples sur chaque VHSI projectives appartenant à trois composantes connexes de l'espace de modules des OG10
Irreducible holomorphic symplectic varieties (IHSV) are the algebraic analogue of the hyperkähler Riemannian manifolds. An IHSV of dimension 2 is a K3 surface; in this case, if furthermore X is projective, any ample curve on X is linearly equivalent to a sum of rational curves (Bogomolov, Mumford). Charles, Mongardi and Pacienza proved the existence of uniruled divisors on (almost) any ample linear system on a IHSV that is deformation equivalent to an Hilbert scheme on a K3 surface, or to a generalized Kummer variety. The existence of many rational curves on X semplifies the structure of the 0-Chow group of X. In my thesis, I worked on the OG10 case, the IHSV defined by O’Grady; the variety OG10 isimportant and very actively studied. The main result of my thesis proves the existence of ample uniruled divisors on any IHSV inside three connected components of the moduli space of OG10 varieties
8

Istrati, Nicolina. "Conformal structures on compact complex manifolds." Thesis, Sorbonne Paris Cité, 2018. http://www.theses.fr/2018USPCC054/document.

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Dans cette thèse on s’intéresse à deux types de structures conformes non-dégénérées sur une variété complexe compacte donnée. La première c’est une forme holomorphe symplectique twistée (THS), i.e. une deux-forme holomorphe non-dégénérée à valeurs dans un fibré en droites. Dans le deuxième contexte, il s’agit des métriques localement conformément kähleriennes (LCK). Dans la première partie, on se place sur un variété de type Kähler. Les formes THS généralisent les formes holomorphes symplectiques, dont l’existence équivaut à ce que la variété admet une structure hyperkählerienne, par un théorème de Beauville. On montre un résultat similaire dans le cas twisté, plus précisément: une variété compacte de type kählerien qui admet une structure THS est un quotient fini cyclique d’une variété hyperkählerienne. De plus, on étudie sous quelles conditions une variété localement hyperkählerienne admet une structure THS. Dans la deuxième partie, les variétés sont supposées de type non-kählerien. Nous présentons quelques critères pour l’existence ou non-existence de métriques LCK spéciales, en terme du groupe de biholomorphismes de la variété. En outre, on étudie le problème d’irréductibilité analytique des variétés LCK, ainsi que l’irréductibilité de la connexion de Weyl associée. Dans un troisième temps, nous étudions les variétés LCK toriques, qui peuvent être définies en analogie avec les variétés de Kähler toriques. Nous montrons qu’une variété LCK torique compacte admet une métrique de Vaisman torique, ce qui mène à une classification de ces variétés par le travail de Lerman. Dans la dernière partie, on s’intéresse aux propriétés cohomologiques des variétés d’Oeljeklaus-Toma (OT). Plus précisément, nous calculons leur cohomologie de de Rham et celle twistée. De plus, on démontre qu’il existe au plus une classe de de Rham qui représente la forme de Lee d’une métrique LCK sur un variété OT. Finalement, on détermine toutes les classes de cohomologie twistée des métriques LCK sur ces variétés
In this thesis, we are concerned with two types of non-degenerate conformal structures on a given compact complex manifold. The first structure we are interested in is a twisted holomorphic symplectic (THS) form, i.e. a holomorphic non-degenerate two-form valued in a line bundle. In the second context, we study locally conformally Kähler (LCK) metrics. In the first part, we deal with manifolds of Kähler type. THS forms generalise the well-known holomorphic symplectic forms, the existence of which is equivalent to the manifold admitting a hyperkähler structure, by a theorem of Beauville. We show a similar result in the twisted case, namely: a compact manifold of Kähler type admitting a THS structure is a finite cyclic quotient of a hyperkähler manifold. Moreover, we study under which conditions a locally hyperkähler manifold admits a THS structure. In the second part, manifolds are supposed to be of non-Kähler type. We present a few criteria for the existence or non-existence for special LCK metrics, in terms of the group of biholomorphisms of the manifold. Moreover, we investigate the analytic irreducibility issue for LCK manifolds, as well as the irreducibility of the associated Weyl connection. Thirdly, we study toric LCK manifolds, which can be defined in analogy with toric Kähler manifolds. We show that a compact toric LCK manifold always admits a toric Vaisman metric, which leads to a classification of such manifolds by the work of Lerman. In the last part, we study the cohomological properties of Oeljeklaus-Toma (OT) manifolds. Namely, we compute their de Rham and twisted cohomology. Moreover, we prove that there exists at most one de Rham class which represents the Lee form of an LCK metric on an OT manifold. Finally, we determine all the twisted cohomology classes of LCK metrics on these manifolds
9

Torres, Ruiz Rafael. "Geography and Botany of Irreducible Symplectic 4-Manifolds with Abelian Fundamental Group." Thesis, 2010. https://thesis.library.caltech.edu/5941/1/thesis_template.pdf.

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In this thesis the geography and botany of irreducible symplectic 4-manifolds with abelian fundamental group of small rank are studied. It resembles an anthology of the contribution obtained by the author during his infatuation with 4-dimensional topology by studying its recent developments. As such, each chapter is independent from each other and the reader is welcomed to start reading whichever one seems more appealing. We now give an outline for the sake of convenience.

The first chapter of the thesis deals with the existence and (lack of) uniqueness of smooth irreducible symplectic non-spin 4-manifolds with cyclic fundamental group (both finite and infinite). Chapter 2 does the same for 4-manifolds with abelian, yet non-cyclic π1; the use of the homeomorphism criteria on these manifolds due to I. Hambleton and M. Kreck is of interest. In Chapter 3, the Spin geography for abelian fundamental groups of small rank is studied. A couple of subtle relations between simply connected and non-simply connected exotic 4-manifolds are explored through out the fourth chapter.

Chapter 5 gives closure to a question raised in Chapter 4, and describes current research projects pursued by the author. These projects came naturally through the results presented in previous chapters. The thesis ends by describing two research progress that are being pursued. Chapter 6 contains the current situation regarding the geography and botany of spin manifolds with zero signature.

The current state of the joint work of the author with Jonathan Yazinski (at McMaster University at the time of writing) is described in the seventh and final chapter.

Книги з теми "Irreducible holomorphic symplectic manifolds":

1

Michèle, Audin, and Lafontaine, J. 1944 Mar. 10-, eds. Holomorphic curves in symplectic geometry. Basel: Birkhäuser, 1994.

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2

McDuff, Dusa. J-holomorphic curves and symplectic topology. Providence, R.I: American Mathematical Society, 2004.

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3

McDuff, Dusa. J-holomorphic curves and symplectic topology. 2nd ed. Providence, R.I: American Mathematical Society, 2012.

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4

McDuff, Dusa. J-holomorphic curves and quantum cohomology. Providence, R.I: American Mathematical Society, 1994.

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5

Ibragimov, Zair. Topics in several complex variables: First USA-Uzbekistan Conference on Analysis and Mathematical Physics, May 20-23, 2014, California State University, Fullerton, California. Providence, Rhode Island: American Mathematical Society, 2016.

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6

Audin, Michèle, and Jacques Lafontaine. Holomorphic Curves in Symplectic Geometry. Springer Basel AG, 2012.

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7

Wendl, Chris. Holomorphic Curves in Low Dimensions: From Symplectic Ruled Surfaces to Planar Contact Manifolds. Springer, 2018.

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8

Ma, Xiaonan, and George Marinescu. Holomorphic Morse Inequalities and Bergman Kernels. Springer London, Limited, 2007.

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9

Ma, Xiaonan, and George Marinescu. Holomorphic Morse Inequalities and Bergman Kernels (Progress in Mathematics). Birkhäuser Basel, 2007.

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10

McDuff, Dusa, and Dietmar Salamon. Introduction to Symplectic Topology. Oxford University Press, 2017. http://dx.doi.org/10.1093/oso/9780198794899.001.0001.

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Over the past number of years powerful new methods in analysis and topology have led to the development of the modern global theory of symplectic topology, including several striking and important results. The first edition of Introduction to Symplectic Topology was published in 1995. The book was the first comprehensive introduction to the subject and became a key text in the area. In 1998, a significantly revised second edition contained new sections and updates. This third edition includes both further updates and new material on this fast-developing area. All chapters have been revised to improve the exposition, new material has been added in many places, and various proofs have been tightened up. Copious new references to key papers have been added to the bibliography. In particular, the material on contact geometry has been significantly expanded, many more details on linear complex structures and on the symplectic blowup and blowdown have been added, the section on J-holomorphic curves in Chapter 4 has been thoroughly revised, there are new sections on GIT and on the topology of symplectomorphism groups, and the section on Floer homology has been revised and updated. Chapter 13 has been completely rewritten and has a new title (Questions of Existence and Uniqueness). It now contains an introduction to existence and uniqueness problems in symplectic topology, a section describing various examples, an overview of Taubes–Seiberg–Witten theory and its applications to symplectic topology, and a section on symplectic 4-manifolds. Chapter 14 on open problems has been added.

Частини книг з теми "Irreducible holomorphic symplectic manifolds":

1

Camere, Chiara. "Moduli Spaces of Cubic Threefolds and of Irreducible Holomorphic Symplectic Manifolds." In Birational Geometry and Moduli Spaces, 13–27. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-37114-2_2.

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2

Audin, Michèle. "Symplectic and almost complex manifolds." In Holomorphic Curves in Symplectic Geometry, 41–74. Basel: Birkhäuser Basel, 1994. http://dx.doi.org/10.1007/978-3-0348-8508-9_3.

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3

Forstnerič, Franc. "Surjective Holomorphic Maps onto Oka Manifolds." In Complex and Symplectic Geometry, 73–84. Cham: Springer International Publishing, 2017. http://dx.doi.org/10.1007/978-3-319-62914-8_6.

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4

Sawon, Justin. "Derived equivalence of holomorphic symplectic manifolds." In CRM Proceedings and Lecture Notes, 193–211. Providence, Rhode Island: American Mathematical Society, 2004. http://dx.doi.org/10.1090/crmp/038/09.

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5

Siebert, Bernd, and Gang Tian. "Lectures on Pseudo-Holomorphic Curves and the Symplectic Isotopy Problem." In Symplectic 4-Manifolds and Algebraic Surfaces, 269–341. Berlin, Heidelberg: Springer Berlin Heidelberg, 2008. http://dx.doi.org/10.1007/978-3-540-78279-7_5.

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6

de Bartolomeis, Paolo. "ℤ2 and ℤ-Deformation Theory for Holomorphic and Symplectic Manifolds." In Complex, Contact and Symmetric Manifolds, 75–103. Boston, MA: Birkhäuser Boston, 2005. http://dx.doi.org/10.1007/0-8176-4424-5_6.

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7

"Compact hyper-Kähler manifolds and holomorphic symplectic manifolds." In Chern Numbers and Rozansky–Witten Invariants of Compact Hyper-Kähler Manifolds, 1–38. WORLD SCIENTIFIC, 2004. http://dx.doi.org/10.1142/9789812562357_0001.

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8

"Closed Holomorphic Curves in Symplectic 4-Manifolds." In Lectures on Contact 3-Manifolds, Holomorphic Curves and Intersection Theory, 11–25. Cambridge University Press, 2020. http://dx.doi.org/10.1017/9781108608954.003.

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9

"Symplectic Fillings of Planar Contact 3-Manifolds." In Lectures on Contact 3-Manifolds, Holomorphic Curves and Intersection Theory, 77–93. Cambridge University Press, 2020. http://dx.doi.org/10.1017/9781108608954.007.

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10

GUAN, DANIEL (ZHUANG-DAN). "EXAMPLES OF COMPACT HOLOMORPHIC SYMPLECTIC MANIFOLDS WHICH ADMIT NO KÄHLER STRUCTURE." In Geometry and Analysis on Complex Manifolds, 63–74. WORLD SCIENTIFIC, 1994. http://dx.doi.org/10.1142/9789814350112_0004.

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