Infinite Dimensional Projective Space Research Articles

Q-deformed rational numbers and the 2-Calabi–Yau category of type

Abstract We describe a family of compactifications of the space of Bridgeland stability conditions of a triangulated category, following earlier work by Bapat, Deopurkar and Licata. We particularly consider the case of the 2-Calabi–Yau category of the $A_2$ quiver. The compactification is the closure of an embedding (depending on q) of the stability space into an infinite-dimensional projective space. In the $A_2$ case, the three-strand braid group $B_3$ acts on this closure. We describe two distinguished braid group orbits in the boundary, points of which can be identified with certain rational functions in q. Points in one of the orbits are exactly the q-deformed rational numbers recently introduced by Morier-Genoud and Ovsienko, while the other orbit gives a new q-deformation of the rational numbers. Specialising q to a positive real number, we obtain a complete description of the boundary of the compactification.

Kähler–Einstein submanifolds of the infinite dimensional projective space

This paper consists of two main results. In the first one we describe all Kaehler immersions of a bounded symmetric domain into the infinite dimensional complex projective space in terms of the Wallach set of the domain. In the second one we exhibit an example of complete and non-homogeneous Kaehler-Einstein metric with negative scalar curvature which admits a Kaehler immersion into the infinite dimensional complex projective space.

Secondary Steenrod operations in cohomology of infinite-dimensional projective spaces

Secondary Steenrod operations in cohomology of infinite-dimensional projective spaces

Holomorphic vector bundles on quadric hypersurfaces of infinite-dimensional projective spaces

Here we prove the following result and a few related statements. Let $V$ be a Banach space with countable unconditional basis and the localizing property, $Q\subset P(V)$ a quadric hypersurface with finite-dimensional singular locus and $E$ a holomorphic vector bundle of finite rank on $Q$. Then $E\cong\oplus_{1\leq i\leq r}0_{Q}(a_{i})$ for some integers $a_{i}$ and $h^{1}(Q, E(t))=0$ for every integer $t$.

QUANTUM MECHANICS IN INFINITE SYMPLECTIC VOLUME

We quantise complex, infinite-dimensional projective space CP(ℋ). We apply the result to quantise a complex, finite-dimensional, classical phase space [Formula: see text] whose symplectic volume is infinite, by holomorphically embedding it into CP(ℋ). The embedding is univocally determined by requiring it to be an isometry between the Bergman metric on [Formula: see text] and the Fubini–Study metric on CP(ℋ). Then the Hilbert-space bundle over [Formula: see text] is the pullback, by the embedding, of the Hilbert-space bundle over CP(ℋ).

Analytic Subsets of Products of Infinite-Dimensional Projective Spaces

Abstract Let 𝑉𝑖, 1 ≤ 𝑖 ≤ 𝑠, be complex topological vector spaces with 𝑉1 infinite-dimensional and 𝑌 a closed analytic subset of finite codimension of 𝐏(𝑉1) × . . . × 𝐏(𝑉𝑠). Here we show that 𝑌 is algebraic (at least if each 𝑉𝑖 is a Banach space) and that any two points of 𝑌 may be connected by a chain of 𝑠 + 3 lines contained in 𝑌.

Union of linear spaces in infinite-dimensional projective spaces and holomorphic vector bundles

Let V be a localizing Banach space with an unconditional countable basis, X an equicodimensional transversal union of finite-codimensional linear subspaces of P(V) and E a holomorphic vector bundle of finite rank on X. Here we prove that Hi(X,E) = 0 for every i > 0 and that E is isomorphic to a direct sum of line bundles OX(t).

The Dolbeault complex in infinite dimensions I

In this paper we introduce certain basic notions concerning infinite dimensional complex manifolds, and prove that the Dolbeault cohomology groups of infinite dimensional projective spaces, with values in finite rank vector bundles, vanish. Some applications of such vanishing theorems are discussed; e.g., we classify vector bundles of finite rank over infinite dimensional projective spaces. Finally, we prove a sharp theorem on solving the inhomogeneous Cauchy–Riemann equations on affine spaces.

Groups Acting on Infinite-Dimensional Projective Spaces II

Groups Acting on Infinite-Dimensional Projective Spaces II

Spazio delle curve semplici chiuse e foliazioni misurate su superfici non compatte, orientabili

In this paper we obtain the following results for non-compact surfaces: 1) the set of isotopy classes of simple, closed, curves is embedded in an infinite dimensional projective space, following the same process which Thurston uses on surfaces of finite type, 2) there exists a measured foliation without singularity.

Groups Acting on Infinite Dimensional Projective Spaces

Groups Acting on Infinite Dimensional Projective Spaces

New Subhierarchies of the KP Hierarchy in the Sato Theory. I. Analysis of the Burgers-Hopf Hierarchy by Sato Theory

It is shown that the Burgers-Hopf(BH) hierarchy consisting of all the n-th BH equations is a subhierarchy of the KP hierarchy in the Sato theory. If is also shown that the totality of solutions of the BH hierarchy forms art infinite dimensional projective space. Since each of the equations of the BH hierarchy possesses the generalized Painlevé property defined by S. Oishi and the author, this result supports their conjecture that the generalized Painlevé property has a connection with the Sato theory.

Compatible relations of modular and orthomodular lattices

Let L be a modular lattice of finite length. L is a projective geometry if and only if L has only trivial tolerances. A binary relation 9 is called a tolerance of an algebra W = (A, Q) if 9 is reflexive, symmetric and compatible with the operations of W. The tolerances D = ((a, a)la E A) and A2 are called the trivial tolerances of W. Obviously every congruence relation of 9I is also a tolerance of W. If L is a lattice then we consider R = ((a, b)la, b E A, a (a, Abl)V(a2A b2) and b, Vb2 > (a, Abl)V(a2 A b2) we have ((a1 V a2) A (b, V b2), b1 V b2) E p. Similarly we prove the three other conditions and have (a1 V a2, b, V b2) E (. In the same way we can show that f is compatible with the operation A. The function s is also order-preserving. We have t o s(p) = t(t) = f n R. If (c, d) E t n R then we have (c, d) = (c A d, d) E p. If (a, b) E p then (a, b) E= R and (a A b, b) E p, (a A b, a) E p, (a, a V b) E p and (b, a V b) E p and therefore (a, b) Et n R. We have t o s = Is and s o t = 1 T is proved similarly. THEOREM 2. Let L be an orthomodular lattice. A binary relation 9 of L is a congruence relation if and only if 9 is reflexive, symmetric and compatible with join and meet. Received by the editors January 21, 1980. 1980 Mathematics Subject Classification. Primary 06C05; Secondary 06C15. ? 1981 American Mathematical Society 0002-9939/81/0000-0126/$01 .75 462 This content downloaded from 207.46.13.189 on Thu, 16 Jun 2016 06:25:00 UTC All use subject to http://about.jstor.org/terms MODULAR AND ORTHOMODULAR LATTICES 463 PROOF. As L is relatively complemented 9 is a lattice congruence of L [4], [7]. It remains to show that from a 9 b we have a' 9 b'. We assume a 0 and a+ = 0 else. In a modular lattice of finite length we have (a V b)+ = a + V b + [3, p. 269, Lemma 6.1(e)]. We consider the following binary relation p = ((a, b)Ia 3. L is isomorphic to the lattice of all subspaces of a vector space over some division ring if and only if L has only trivial tolerances. These results cannot be extended to lattices of infinite length. The restriction is necessary since the relation a p b iff a < b and codimb(a) < oo will generate a proper nontrivial tolerance relation on the subspace lattice of an infinite-dimensional projective space. R-EFERENCES 1. G. Birkhoff, Lattice theory, 3rd ed., Amer. Math. Soc Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, R. I., 1967. 2. I. Chaida, J. Niederle and B. Zelinka, On existence conditions for comWatible tolerances, Czechoslovak Math. J. 26 (1976), 304-311. 3. G. D. Findlay, Reflexive homomorphic relations, Canad. Math. Bull. 3 (1960), 131-132. 4. J. Hashimoto, Congnrence relations and congruence classes in lattices, Osaka J. Math. 15 (1963), 71-86. 5. C. Hermann, S-verklebte Summen von Verbanden, Math. Z. 130 (1973), 255-274. This content downloaded from 207.46.13.189 on Thu, 16 Jun 2016 06:25:00 UTC All use subject to http://about.jstor.org/terms 464 DIETMAR SCHWEIGERT 6. G. Gratzer, General lattice theory, Stuttgart, 1978. 7. G. Gritzer and E. T. Schmidt, On congruence lattices of lattices, Acta Math. Acad. Sci. Hungar. 13 (1962), 178-185. 8. M. Kamara and D. Schweigert, Eine Charakterisierung polynomvollstuidger Polaritatsverbinde, Arch. Math. (Basel) 30 (1978), 661-664. 9. H. Werner, A Mal'cev condtion for admissible relations, Algebra Universalis 3 (1973), 263. 10. R. Wille, Eine Charakterisierung endlicher, ordnungspoIynomvollst?ndiger VerbEinde, Arch. Math. (Basel) 28 (1977), 557-560. FB MATHMATIK, UNvRSrIAT KsSERsLAuTERN, 675 KmsRsLAurERN, WESm GERMANY This content downloaded from 207.46.13.189 on Thu, 16 Jun 2016 06:25:00 UTC All use subject to http://about.jstor.org/terms

VECTOR BUNDLES OF FINITE RANK OVER INFINITE VARIETIES

This article contains a proof of the conjecture of Schwarzenberger that vector bundles on infinite-dimensional projective space split as the sum of line bundles, and a generalization to quasi-homogeneous projective varieties.Bibliography: 7 titles.

A theorem in homological algebra and stable homotopy of projective spaces

Introduction. The paper exhibits a general change of rings theorem in homological algebra and shows how it enables to systematize the computation of the stable homotopy of projective spaces. Chapter I considers the following situation: R and S are rings with unit, h: R -+ S is a ring homomorphism, M is a left S-module. If an S-free resolution of M and an R-free resolution of S are given, Theorem I.1. shows how to construct an R-free resolution of M. Chapter II is devoted to computing the initial stable homotopy groups of projective spaces. Here the results of Chapter I are applied to the homomorphism a: A-+ A of the Steenrod algebra over Z2 (see 1.3). The main tool in computing stable homotopy is the Adams spectral sequence [1]. Let RP?, CP', HP' be the real, complex, and quaternionic infinite-dimensional projective spaces, respectively. If X is a space, let n s(X) denote the mth stable homotopy group of X [1]. Part of the results of Chapter II can be presented as follows: