P.E. Newstead, Introduction to moduli problems and orbit spaces, Tata Institute, 1978.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....are stressed. Furthermore, a similar treatment is given for coherent sheaves. 1 Introduction It is an interesting mathematical problem to determine the space of deformations of a vector bundle or sheaf on some xed manifold, having a remarkable history and by now a huge literature, see, e.g. [HL97, Nar85, New78, Pot97, Ses82] and the references given therein. During the last years these kind of problems have shown to have applications in physics centered around string theory and twistor theory, see, e.g. GSW87, Hue92, Pol98, WW90] Probably the most famous example appearing in conformal eld theory (in the ....

P.E.Newstead, Introduction to moduli problems and orbit spaces (Springer, 1978) 15

....and (D 1 ) D 2 ) D r ) Proof. It follows from Theorem 5.1.2 just as in the case of vector bundles. # 5.2 Universal bundles Nitsure showed that is a coarse moduli space. Here we show that is in fact a fine moduli space. We closely follow the proof of Theorem 5. 12 in [New1] and (1.19) of [Tha3] All the ingredients have already appeared in the unpublished [Tha1] Definition 5.2.1 Two families T and # T of stable Higgs bundles over T # are said to be equivalent, in symbols # T ) if there exists a line bundle L on T such that # T # pr # T (L) The ....

....It follows that we have a tautological Higgs bundle EB# K over (B) #, which is a priori a equivariant complex but reduces to a equivariant complex as proved above. It follows that it reduces to to give a universal Higgs bundle E K. # As in Theorem 5. 12 of [New1] and (1.19) of [Tha3] our Lemma 5.2.2 and Proposition 5.2.3 gives: is a fine moduli space for rank 2 stable Higgs bundles of degree 1 with respect to the equivalence of families of stable Higgs bundles. As another consequence of Proposition 5.2.3 and Lemma 5.2.2, we see that although E ....

P.E. Newstead. Introduction to moduli problems and orbit spaces, Tata Inst. Bombay, 1978

....a C a is an affine morphism which is G equivariant, and by Simpson s construction of moduli for modules, the action of G on C Theta a C a admits a good quotient in the sense of geometric invariant theory. Hence a good quotient H= G exists by Ramanathan s lemma (see Proposition 3. 12 in [Ne]) which by construction and universal properties of good quotients is the coarse moduli scheme of semistable pre D modules with given Hilbert polynomials. Note that under a good quotient in the sense of geometric invariant theory, two different orbits can in some cases get mapped to the same ....

Newstead, P.E. : Introduction to moduli problems and orbit spaces, TIFR lecture notes, Bombay (1978).

....is TOPOLOGICAL DYNAMICS ON MODULI SPACES, I 399 infinite. The second step deals with the cases where the Gamma orbits are potentially finite and involves the theory of trigonometric Diophantine equations. All in all, the proof is a delicate interplay of ideas in geometric invariant theory [6, 7], topological dynamics, and Diophantine equations. Incidentally, the proof also yields the well known result that the only proper closed subgroups SU(2) are the closed subgroups of Pin(2) and the double covers of the automorphism groups of the Platonic solids. The following conjecture is the ....

....fundamental group 1 (M) has a presentation 1 (M) hX; Y; KjK = XYX Gamma1 Y Gamma1 i where K represents the element generated by the boundary component. In particular, 1 (M) is the free group generated by X and Y . Note E = Hom( 1 (M) SU(2) SU(2) The SU(2) invariant polynomials [7] on Hom( 1 (M) SU(2) are generated by the traces of the representations. In particular, a point [oe] 2 E is determined by x = tr(oe(X) y = tr(oe(Y ) z = tr(oe(XY ) This provides a global coordinate chart: F : E 7 Gamma R 3 [oe] F 7 Gamma (tr(oe(X) tr(oe(Y ) tr(oe(XY ) 400 ....

P.E. Newstead, Introductions to Moduli Problems and Orbit Spaces, Springer-Verlag,

....k algebra A. This is a particularly nice case when G is a reductive group (in the sense of the first lecture if char k = 0, in general the definition is different) indeed, it was proved by Weyl for char k = 0, and by Nagata in general, that the algebra A G of G invariants is finitely generated ([23], Theorem AROUND THE HORN CONJECTURE 7 3.4) One can then define the affine variety Y = SpecA G , and the natural morphism f : X Gamma Y has the following nice properties ( 23] Theorem 3.5) 1. O Y (f OX ) G ; 2. the image by f of any closed invariant subset is closed; 3. f separates ....

....by Weyl for char k = 0, and by Nagata in general, that the algebra A G of G invariants is finitely generated ( 23] Theorem AROUND THE HORN CONJECTURE 7 3. 4) One can then define the affine variety Y = SpecA G , and the natural morphism f : X Gamma Y has the following nice properties ([23], Theorem 3.5) 1. O Y (f OX ) G ; 2. the image by f of any closed invariant subset is closed; 3. f separates disjoint closed invariant subsets. In particular, two points in X have the same image iff their orbit closures meet. A consequence of these properties is that (Y; f) is a categorical ....

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Newstead P.E.: Introduction to moduli problems and orbit spaces, Lectures of the Tata Institute, Bombay 1978.

....11 12 CHAPTER 1. PRELIMINARIES is a direct summand of C r a ; a : a 1 r(b 1 b) a c r(b c b) and we are done. 1.2 Basic concepts from GIT 1.2.1 The GIT process We brie y summarize the main steps in Geometric Invariant Theory to x the notation. References are [26] and [30]. Let G be a reductive algebraic group and G F F an action of G on the projective scheme F . Let L be an ample line bundle on F . A linearization of the given action in L is a lifting of that action to an action : G L L, such that for every g 2 G and x 2 F the induced map L x L g x is ....

P.E. Newstead, Introduction to Moduli Problems and Orbit Spaces, Springer 1978.

....Thus the first part gives the second. Likewise, the third statement follows by referring to the last part of Theorem 4.3. 5 Universal bundles Nitsure showed that M is a coarse moduli space. Here we show that M is in fact a fine moduli space. We closely follow the proof of Theorem 5. 12 in [New] and (1.19) of [Tha3] All the ingredients have already appeared in the unpublished [Tha1] Definition 5.1 Two families E T and E 0 T of stable Higgs bundles over T Theta Sigma are said to be equivalent, in symbols E T E 0 T ) if there exists a line bundle L on T such that E 0 T = ....

....4.3 an isomorphism. The result follows. Now we prove the existence of universal Higgs bundles (cf. Tha1] Proposition 5.3 Universal Higgs bundles E M = E M Phi E M Omega K Sigma over M Theta Sigma do exist. Proof. The proof is analogous to the proof of Theorem 5. 12 of [New] using the GIT construction of Nitsure [Nit] cf. also (1.19) of [Tha3] First we recall the construction of M 2k Gamma1 from [Nit] Let n = 2k Gamma 1 2(1 Gamma g) with k large enough. Then by Corollary 3.4 of [Nit] for any stable Higgs bundle E Phi E Omega K, E is a quotient of O ....

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P.E. Newstead. Introduction to moduli problems and orbit spaces, Tata Inst. Bombay, 1978

....space such that each point of this space corresponds to a unique equivalence class. In algebraic geometry this classifying space should again be an algebraic variety, together with certain functorial properties. These ideas lead to the notion of a fine, respectively coarse, moduli space ( MuF] [Ne]) Classically, moduli spaces have been constructed for global algebraic objects such as projective varieties, or for vector bundles on a fixed projective variety. During the past years there has also been some progress in constructing moduli spaces for singularities (cf. GHP] and for ....

P.E. Newstead, Introduction to moduli problems and orbit spaces, Lecture Notes, Tata Institute of Fundamental Research, Springer 1978.

....exists a universal family (E; over H n C , and a lifting of the C action on H n to E whose induced action on is Ad( 1 . That is, H n is a ne moduli space for the Higgs bundles of degree d and rank r with values in K(n) Proof. This follows in a standard way, cf. Newstead [35], from the geometric invariant theory construction of H n due to Nitsure [36] Alternatively, the universal pair can be constructed gauge theoretically just as in Atiyah Bott [2, x9] In the rank 2 case, both methods are explained in detail by the rst author [21, 5.3] 19, 5.2.3] 2 5 Equivalence ....

P.E. Newstead, Introduction to moduli problems and orbit spaces, Tata Institute, 1978.

.... is a vector bundle V Y on X Theta Y together with a section Phi Y 2 Gamma(Y; Y ) X Omega Omega End(V Y ) 18] C M Dol being a moduli space implies that if Y is a family of stable (poly stable or S equivalence classes of semi stable) Higgs bundles, then there is a natural morphism [15, 17] t : Y Gamma C M Dol : Moreover t takes every point y 2 Y to the point of C M Dol that corresponds to the Higgs bundle in the family over y [15, 17, 18] The space M Dol (c) is a subvariety of C M Dol (c) hence, to show that two stable (poly stable or S equivalence classes of semi stable) ....

.... implies that if Y is a family of stable (poly stable or S equivalence classes of semi stable) Higgs bundles, then there is a natural morphism [15, 17] t : Y Gamma C M Dol : Moreover t takes every point y 2 Y to the point of C M Dol that corresponds to the Higgs bundle in the family over y [15, 17, 18]. The space M Dol (c) is a subvariety of C M Dol (c) hence, to show that two stable (poly stable or S equivalence classes of semi stable) Higgs bundles (V 1 ; Phi 1 ) and (V 2 ; Phi 2 ) belong to the same component of M Dol (c) it suffices to exhibit a connected family Y (within M Dol (c) of ....

P.E. Newstead, Introductions to Moduli Problems and Orbit Spaces, Springer-Verlag, 1978.

....[2, 8, 9] In the problem at hand the number of feedback orbits is in general infinite and this makes the problem difficult. In order to classify all orbits it will therefore be necessary to derive a continuous set of invariants . The application of tools from geometric invariant theory (see e.g. [13, 14]) often enables one to derive for a given group action a set of invariants in a systematic way. 2 From a geometric point of view this amounts to describing an algebraic variety whose points parameterize uniquely the closed feedback orbits. In this paper we construct, using tools from geometric ....

....2 Glm p : 2.4) 3 Basic notions from geometric invariant theory Geometric invariant theory constitutes an active research area of algebraic geometry. One of the main references is the book by Mumford and Fogarty [13] The nonspecialists among the interested readers will find the book by Newstead [14] a good introductory book. 4 In this section we explain an important result from geometric invariant theory which we will use later in the paper to derive a set of continuous feedback invariants. Let X be a projective variety, i.e. X is the zero locus of a finite set of homogeneous polynomial ....

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P. E. Newstead. Introduction to Moduli Problems and Orbit Spaces. Tata Inst. of Fund. Research, Bombay, 1978.

....that M(V; q) is locally factorial whereas the completions of the local rings of its singular points may be non factorial. LUNA S SLICE THEOREM 5 1.4. Prerequisites and references For basic results of algebraic geometry, see [6] For basic results on reductive groups and algebraic quotients, see [18], 17] Some results of algebra that will be used here can be found in [21] 14] 20] Some results on etale morphisms are taken from [5] 6] The study of semi stable vector bundles on curves and of their moduli spaces is done in [11] 22] 1.5. Notations and recall of some basic results ....

....subspace of R. If G acts on R, R G will denote the subalgebra of G invariant elements of R. Example : If G acts on an algebraic variety X then there is a natural action of G on A(X) given by (g; # (x # g (x) gx) The natural action of G on A(X) is a rational action (cf. [18], lemma 3.1) De nition 2.6. Let G be an algebraic group. A rational representation of G is a morphism of groups G GL(V ) where V is a nite dimensional vector space. It induces an action of G on V . Lemma 2.7. Let G be an algebraic group acting on an algebraic variety X, and W a nite ....

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Newstead, P. Introduction to Moduli Problems and Orbit spaces. Tata Inst. of Fund. Research, Bombay. Springer-Verlag (1978)

....it is clear that if ffl is small enough the calculation of stable and semistable points will reduce to the corresponding calculation for PGL(3; C ) acting on the product of four copies of P 2 (C ) corresponding to the four point vertices described above. This calculation is well known (see [N]) A configuration is stable (resp. semistable) iff less than (resp. no more than) 1=3 of the total weight is concentrated on any point and less than 2=3 (resp. no more than 2=3) of the total weight is concentrated on any line. We obtain Lemma 8.18 All semistable configurations in R(A; P 2 (C ....

P. E. Newstead, Introduction to Moduli Problems and Orbit Spaces, Tata Institute Lecture Notes.

....on the precise form of the decomposition of the bundle under consideration. 1 Introduction It is an interesting mathematical problem to determine the space of deformations of a vector bundle (or sheaf) on some fixed manifold, having a remarkable history and by now a huge literature, see e.g. [HL97, Nar85, New78, Pot97, Ses82] and the references given there. During the last years these kind of problems have shown to have applications in physics centered around string theory, see e.g. GSW87, Pol98, WW90] Probably the most famous example appearing also in conformal field theory is the one where the underlying ....

P.E.Newstead, Introduction to moduli problems and orbit spaces (Springer, 1978)

....it is clear that if ffl is small enough the calculation of stable and semistable points will reduce to the corresponding calculation for PGL(3; C ) acting on the product of four copies of P 2 (C ) corresponding to the four point vertices described above. This calculation is well known (see [N]) A configuration is stable (resp. semistable) iff less than (resp. no more than) 1=3 of the total weight is concentrated on any point and less than 2=3 (resp. no more than 2=3) of the total weight is concentrated on any line. We obtain Lemma 8.19 All semistable configurations in R(A; P 2 (C ....

P. E. Newstead, Introduction to Moduli Problems and Orbit Spaces, Tata Institute Lecture Notes. 73

....d) over S whose generic fibre is the corresponding moduli space over the generic fibre of C and a natural morphism N C (n; d) Theta S Spec(k) Gamma N C (n; d) that induces a bijection on the set of k points. Proof. The proof runs along the lines of the corresponding proof over a field (see [33]) combined with the results of Seshadri on Geometric Invariant Quotients over an arbitrary base (see appendix 1G in [31] 2. Localisation of G m actions The purpose of this section is prove a localisation theorem relating the Chow groups of a variety acted on by G m to the Chow groups of the ....

P.E. Newstead. Introduction to moduli problems and orbit spaces. Tata Institute of Fundamental Research, Bombay, 1978.

....of very particular examples of tilting bundles on toric quiver varieties is known ( Hi2] Theorem 3.9) 1. 6) For an introduction to quivers and path algebras we refer the reader to [Ri1] and [ARS] the theory of localizations may be found in [S] For an introduction to moduli spaces we mention [N] and for moduli of representations of quivers we refer to the work of King [Ki] For results on triangulated categories we refer to [Hap] and [Har] Our standard reference for toric geometry is [Ke] for a short introduction to this area we also mention [F] We would like to thank G. Hein, A. ....

.... t(ff) 2 Q Gamma 0 [by fi the unique arrow with s(fi) 2 Q Gamma 0 and t(fi) 2 Q 0 ] We show that ff is not in Q 1 ( c ) ff and fi are not both in Q 1 ( c ) The set Q 0 is closed under successors in Q n fffg, but P q2Q 0 c q = 1 [ 0] Thus, Q n fffg is not stable [both Q n fffg and Q n ffig are not stable]. It remains to show the converse. Assume we have no (1; 0) wall and no (1; 1) wall. Thus, t 2 for each wall W . Assume further that W is a (t ; t Gamma ) wall with t t Gamma . We define the open halfspace W : f 2 IH j P q2Q 0 q 0g, i.e. c 2 W . Using ....

Newstead, P. E.: Introduction to Moduli Problems and Orbit Spaces. Tata Institute of Fund. Research, Springer Verlag, Berlin-Heidelberg-New York 1978.

....to [2, 10, 11] In the problem at hand the number of feedback orbits is in general infinite and this makes the problem difficult. In order to classify all orbits it will thus be necessary to derive a continuous set of invariants . The application of tools from geometric invariant theory (see e.g. [15, 16]) often enables one to derive for a given group action a set of invariants in a systematic way. From a geometric point of view this amounts to describing an algebraic variety whose points parameterize uniquely the closed feedback orbits. In this paper we construct, using tools from geometric ....

....from geometric invariant theory In this section we explain an important result from geometric invariant theory which we will use later in the paper to derive a set of continuous feedback invariants. The interested reader will find details on geometric invariant theory in the book of Newstead [16]. Let X be a projective variety, i.e. X is the zero locus of a finite set of homogeneous polynomial equations. We will assume that X is embedded into the projective space P N . Let G ae GlN 1 be a reductive group (e.g. a group isomorphic to a general linear group) which acts on the projective ....

P. E. Newstead. Introduction to Moduli Problems and Orbit Spaces. Tata Inst. of Fund. Research, Bombay, 1978.

.... W ss (G; then there exists a good quotient W ss (G; Gamma M , such that M is a normal projective variety, M s is an open subset of M , and W s (G; M s is the restriction of . We recall here the definition of a good and a geometric quotient of C.S. Seshadri, see [22], 20] Let an algebraic group G act on an algebraic variety or algebraic scheme X. Then a pair ( Y ) of a variety and a morphism X Gamma Y is called a good quotient if (i) is G equivariant (for the trivial action of G on Y ) ii) is affine, open and surjective, iii) If U is an open ....

Newstead, P.E., Introduction to moduli problems and orbit spaces. TIFR Lect. Notes. Math. vol. 51. Berlin Heidelberg New-York : Springer (1978)

.... theory include Weyl s work on semi simple groups in characteristic zero, Nagata s counterexample to Hilbert s 14th problem, Mumford s geometric invariant theory, and work of Nagata, Haboush and others, who proved the finiteness theorem for rational representations of reductive groups (compare Newstead (1978, pp. 90 92) for a short history of Hilbert s 14th problem and precise references) For an overview concerning degree bounds we refer to Derksen et al. 1995) Nowadays, with powerful computers available, one is tempted to come 4 Decker de Jong back to the explicit computation of invariants ....

Newstead, P. (1978) `Introduction to Moduli Problems and Orbit Spaces', Springer, Berlin, Heidelberg, New York.

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P.E. Newstead, Introduction to moduli problems and orbit spaces, Tata Institute, 1978.

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P. E. Newstead, Introduction to Moduli Problems and Orbit Spaces, Tata Institute, Bombay, 1978.

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Newstead, P.E: Introduction to Moduli Problems and Orbit Spaces. Tata Inst. Fund. Res. Lecture Notes 51. Berlin--Heidelberg--New York: Springer 1978.

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Newstead, P.E: Introduction to Moduli Problems and Orbit Spaces. Tata Inst. Fund. Res. Lecture Notes 51. Berlin--Heidelberg--New York: Springer 1978.

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