On the intersection theory of the moduli space of rank two bundles


 Lauren Blankenship
 1 years ago
 Views:
Transcription
1 On the intersection theory of the moduli space of rank two bundles Alina Marian a, Dragos Oprea b,1 a Yale University, P.O. Box , New Haven, CT, , USA b M.I.T, 77 Massachusetts Avenue, Room 2492, Cambridge, MA, 02139, USA. Abstract We give an algebrogeometric derivation of the known intersection theory on the moduli space of stable rank 2 bundles of odd degree over a smooth curve of genus g. We lift the computation from the moduli space to a Quot scheme, where we obtain the intersections via equivariant localization with respect to a natural torus action. Key words: Moduli spaces of sheaves, Quot schemes, equivariant localization. We compute the intersection numbers on the moduli space of stable rank 2 odd degree bundles over a smooth complex curve of genus g. This problem has been intensely studied in the physics and mathematics literature. Complete and mathematically rigorous answers were obtained by Thaddeus [7] [8], Donaldson [2], Zagier [11], and others in rank 2, and by JeffreyKirwan [3] in arbitrary rank. These answers are in agreement with the formulas first written down by Thaddeus in rank 2 [7], and by Witten in arbitrary rank [9]. We indicate yet another method of calculation which recovers the exact formulas obtained by the aforementioned authors in rank 2. Our approach works in principle in any rank, and we will turn to this general case in future work. To set the stage, we let C be a smooth complex algebraic curve of genus g. We let N g denote the moduli space of stable rank 2 bundles with fixed odd determinant, and we write V for the universal bundle on C N g. N g is a smooth projective variety of dimension 3g 3. We also fix, once and for all, a symplectic basis addresses: Alina Marian), Dragos Oprea). 1 Present address: Stanford University, 450 Serra Mall, Stanford, CA, 94305, USA. Preprint submitted to Elsevier Science 17 October 2005
2 {1, δ 1,..., δ 2g, ω} for the cohomology H C). A result of Newstead [6] shows that the Künneth components of c 2 End V) generate the cohomology ring H N g ). We will use the notation of Newstead and Thaddeus, writing c 2 End V) = β ψ k δ k + 2α ω for classes α H 2 N g ), β H 4 N g ), ψ k H 3 N g ). Thaddeus [7] showed that nonzero top intersections on N g must contain the ψ k s in pairs, which can then be removed using the formula g α m β n ψ k ψ k+g ) p k = α m β n, N g N g p where p = k p k and 2m + 4n + 6p = 6g 6. The top intersections of α and β are further determined: Theorem 1 [7][8][2][11][3] N g α m β n = 1) g 2 2g 2 m! m g + 1)! 2m g+1 2)B m g+1. 1) Here B k are the Bernoulli numbers defined, for instance, by the power series expansion u sinh u = 2 k 2 B k u k. 2) k k! In this paper, we reprove Theorem 1. The idea is to lift the computation from N g to a Quot scheme as indicated in the diagram below. Then, we effectively calculate the needed intersections on Quot by equivariant localization with respect to a natural torus action. The convenience of this approach lies in that the fixed loci are easy to understand; in any rank they are essentially symmetric products of C. In rank 2, their total contribution can be evaluated to the intersection numbers 1). Quot N,d P N,d π M g τ N g J The idea that the intersection theory of the moduli space of rank r bundles on a curve and that of a suitable Quot scheme are related goes back to Witten [10] in the context of the Verlinde formula. Moreover, in [1] the authors have the reverse 2
3 approach of calculating certain intersection numbers on Quot in low rank and genus by using the intersection theory of the moduli space of bundles. In this note however, the translation of the intersection theory of the moduli space of bundles into that of the Quot scheme is really very straightforward. The spaces in the diagram are as follows. M g denotes the moduli space of rank 2, odd degree stable bundles on C. By contrast with N g, the determinant of the bundles is allowed to vary in M g. There is a finite covering map τ : N g J M g of degree 4 g, given by tensoring bundles. Here J is the Jacobian of degree 0 line bundles on C. We will write Ṽ for the universal sheaf on M g C, which is only defined up to twisting with line bundles from M g. For a positive integer N, we let P N,d be the projective bundle π : P N,d M g whose fiber over a stable [V ] M g is PH 0 C, V ) N ). In other words, P N,d = Ppr Ṽ N ). We will take the degree d to be as large as convenient to ensure, for instance, that the fiber dimension of P N,d is constant. One can view P N,d as a fine moduli space of pairs V, φ) consisting of a stable rank 2, degree d bundle V together with a nonzero Ntuple of holomorphic sections φ = φ 1,... φ N ) : O N V considered projectively. As such, there is a universal morphism Φ : O N V on P N,d C. A whole series of moduli spaces of pairs of vector bundles with N sections, each labeled by a parameter τ, were defined and studied in [8] for N = 1 and [1] for arbitrary N. P N,d is one of these moduli spaces, undoubtedly the least exciting one from the point of view of studying a new object, due to its straightforward relationship with M g. One would expect the universal bundle V on the total space of P N,d and the noncanonical universal bundle Ṽ on its base M g to be closely related, and 3
4 indeed they coincide up to a twist of O1) V = π Ṽ O P N,d 1). 3) Finally, Quot N,d is Grothendieck s Quot scheme parametrizing degree d rank 2 subsheaves E of the trivial rank N bundle on C, We let 0 E O N F 0. 0 E O N F 0 be the universal sequence on Quot N,d C. For large d relative to N and g, the Quot scheme is irreducible, generically smooth of the expected dimension Nd 2N 2)g 1) [1]. Then, Quot N,d and P N,d are birational and they agree on the open subscheme corresponding to subbundles E = V where φ : O N V is generically surjective. The universal structures also coincide on this open set. For arbitrary d, Quot N,d may be badly behaved, but intersection numbers can be defined with the aid of the virtual fundamental cycle constructed in [5]. We now define the cohomology classes that we are going to intersect. We consider the Künneth decomposition c 2 EndṼ) = β ψ k δ k + 2 α ω of the universal endomorphism bundle on M g C. In keeping with the notation of [3], we further let c i Ṽ) = ã i bk i δ k + f i ω, 1 i 2, be the Künneth decomposition of the noncanonical) universal bundle Ṽ on M g C. Then f 1 = d, β = ã 2 1 4ã 2, α = 2 f 2 + bk 1 bk+g 1 dã 1. It is an easy exercise, using that N g is simply connected, to see that τ α = α, τ β = β, τ where θ is the class of the theta divisor on J. bk 1 bk+g 1 ) = 4θ, 4) 4
5 On P N,d, we let ζ denote the first Chern class of O1) and let c i V) = ā i bk i δ k + f i ω, 1 i 2, 5) be the Künneth decomposition of the Chern classes of V. Finally, we consider the corresponding a, b, f classes on Quot N,d : c i E ) = a i b k i δ k + f i ω, 1 i 2. We now show how to realize any top intersection of α and β classes on N g as an intersection on Quot N,d. Let m and n be any nonnegative integers such that With the aid of 4) we note M g α + m + 2n = 4g 3, m g. bk 1 bk+g 1 ) m βn = 1 4 g m 4θ) g α g J N m g β n. 6) g A top intersection on M g which is invariant under the normalization of the universal bundle Ṽ on M g C can be readily expressed as a top intersection on P N,d as follows. We assume from now on that N is odd, so that the fiber dimension of π : P N,d M g is Then M g α + bk 1 bk+g 1 2M = Nd 2g 1)) 1. ) m βn = M g = ζ 2M 2 f P N,d = ā M 2 2 f P N,d Here we used 3) to write 2 f ) m bk 1 bk+g 1 dã 1 ã 2 1 4ã 2 ) n = bk 1 bk+g 1 dā 1 bk 1 bk+g 1 dā 1 ) m ā2 ) m ā2 1 4ā 2 ) n = 1 4ā 2 ) n. 7) ā M 2 = ζ 2 + ã 1 ζ + ã 2 ) M = ζ 2M + lower order terms in ζ, 5
6 the latter summand being zero when paired with a top intersection from the base M g. Finally, the last intersection number in 7) can be transfered to Quot N,d using the results of [4]. It is shown there that the equality ā M 2 Rā, b, f) = P N,d a M 2 Ra, b, f) Quot N,d 8) holds for any polynomial R in the a, b, f classes, in the regime that N is large compared to the genus g, and in turn d is large enough relative to N and g, so that Quot N,d is irreducible of the expected dimension. Moreover, the equality [Quot N,d] vir am 2 Ra, b, f) = am N vir 2 Ra, b, f) [Quot N,d 2] 9) established in [5] allows us to assume from this moment on that the degree d is much smaller than N. The tradeoff is that we need to make use of the virtual fundamental classes alluded to above and defined in [5]. Putting 6), 7), 8) together we obtain N g α m g β n = m g)! m! [Quot N,d] vir am 2 To prove 1), we will show that [Quot N,d ] vir a M 2 2f f ) m b k 1b k+g 1 da 1 a 2 n 1 4a 2. ) m b k 1b k+g 1 da 1 a 2 n 1 4a 2 = = 1) g 2 m 1 2 2g 1 m! ) m 2g + 1)! B m 2g+1. 10) Equation 10) will be verified by virtual localization. The torus action we will use and its fixed loci were described in [5]. For the convenience of the reader, we summarize the facts we need below. The torus action on Quot N,d is induced by the fiberwise C action on O N with distinct weights λ 1,..., λ N. On closed points, the action of µ C is [ i E O ] [ ] N E µ i O N. The fixed loci Z correspond to split subbundles E = L 1 L 2 6
7 where L 1 and L 2 are line subbundles of copies of O of degrees d 1 and d 2. Thus Z = Sym d 1 C Sym d 2 C. The fixed loci are labeled by distinct degree splittings d = d 1 + d 2 and the choice of a pair of copies of O from the N available ones. Note further that E Z = L 1 L 2 where we let L 1 and L 2 be the universal line subbundles on Sym d 1 C C and Sym d 2 C C. We write c 1 L i ) = x i 1 + k y k i δ k + d i ω, 1 i 2. We set the weights to be the N th roots of unity. The equivariant Euler class of the virtual normal bundle of Z in Quot N,d was determined in [5] to be 1 e T N vir ) = 1)g λ 1 h + x 1 ) λ 2 h + x 2 )) 2ḡ x i ) di ḡ λ i h + x i ) N h N Nλi h + x i ) N 1 exp θ i λ i h + x i ) N h 1 )). 11) N x i Here ḡ = g 1, h is the equivariant parameter, and θ i are the pullbacks to Sym d i C of the theta divisor from the Jacobian. Moreover, it is clear that on Z a 1 = x 1 + λ 1 h) + x 2 + λ 2 h), a 2 = x 1 + λ 1 h)x 2 + λ 2 h), 12) f 2 = 2 b k 1 = y k 1 + y k 2, 13) y k 1y k+g 2 + d 2 x 1 + λ 1 h) + d 1 x 2 + λ 2 h). 14) In the equation above, the superscripts are considered modulo 2g. We collect 11), 12), 13), 14), and rewrite the left hand side of 10), via the virtual localization theorem, LHS of 10) = 1) g d 1,d 2,λ 1,λ 2 I d1,d 2,λ 1,λ 2 15) where the sum ranges over all degree splittings d = d 1 + d 2 and pairs of distinct roots of unity λ 1, λ 2 ). The summand I d1,d 2,λ 1,λ 2 is defined as the evaluation on Sym d 1 C Sym d 2 C of the expression λ 1 h + x 1 ) λ 2 h + x 2 )) 2n 2ḡ 2θ 1 + 2θ 2 + d 2 d 1 )λ 1 h + x 1 ) λ 2 h + x 2 ))) m ) di ḡ λ i h + x i ) M x i Nλi h + x i ) N 1 exp θ λ i h + x i ) N h N i λ i h + x i ) N h 1 )). N x i 7
8 To carry out this evaluation, we use the following standard facts regarding intersections on Sym d C: x d l θ l = g! g l)! for l g, and xd l θ l = 0 for l > g. 16) We will henceforth replace any θ l appearing in a top intersection on Sym d C by g! g l)! xl. Then θ l Nλh + x) N 1 l! exp θ λh + x) N h 1 )) = N x k θ l+k Nλh + x) N 1 l! k! λh + x) N h 1 ) k 17) N x = g l k=0 g! x l+k Nλh + x) N 1 l! g l k)! k! λh + x) N h 1 N x ) k g = N g l l x g λh + x) N 1)g l) λh + x) N h N ). g l We further set x i = x i λ i h and j = λ 11 + x 1 ) λ x 2 ). In terms of the rescaled variables, I d1,d 2,λ 1,λ 2 becomes, via 17) and the binomial theorem, the residue at x 1 = x 2 = 0 of l 1 +l 2 +s=m We expand m! s! d 2 d 1 ) s j 2n 2ḡ+s 2 l i N g l i g l i λ M+l i ḡ i d j 2n 2ḡ+s = 1) α 2 k λ α 1 1 λ α 2 2 z N s, k) x α 1 1 x α 2 2 k=0 α 1 +α 2 =k α x i ) N 1)g l ) i)+m 1 + x i ) N 1) d i l i. +1 where we set 2n 2ḡ + s z N s, k) = λ 1 λ 2 ) 2n 2ḡ+s k. k In [5], we computed the residue { x a 1 + x) N 1+b } Res x=0 = x) N 1) c+1 N a b+p 1) a p N p=0 c ) a. p 8
9 Therefore, I d1,d 2,λ 1,λ 2 = d l 1 +l 2 +s=m k=0 α 1 +α 2 =k p 1,p 2 2 l i N ḡ l i g l i λ M+l i+α i ḡ i { m! s! d 2 d 1 ) s 1) α 2 k z N s, k) α 1 M+N 1)ḡ li )+p i N d i l i ) αi p i 1) α i p i. 18) Now 15) computes an intersection number on M g, hence it is independent of N. Thus it is enough to make N in the above expression. Writing λ 2 = ζλ 1 we have z N s, k) λ M+l1+α1 ḡ 1 λ M+l 2+α 2 ḡ 2 = N 2n 2ḡ + s λ 1 λ 2 2 k ζ 1 ζ M+l 2+α 2 ḡ 1 ζ) 2ḡ 2n s+k. Lemma 2 below clarifies the N dependence of the sum over ζ. Lemma 3 takes care of all the other terms in 18). Together, the two lemmas show that in the limit N the sum over the roots of unity of the terms 18) equals the coefficient of x α 1 1 x 2 ) α 2 in 1 m! 2 s! d 2 d 1 ) s 2 l i g x1 + d k! z s, k) l ) 2 1 x2 + d l ) 2 2 l i d 1 l 1 d 2 l 2 with 2n 2ḡ + s B 2ḡ 2n s+k z s, k) = k 2ḡ 2n s + k)! 1 22n 2ḡ+s k+1 ). Summing over α 1 + α 2 = k first, we obtain 1 m! 2 s! d 2 d 1 ) s 2 l i g k! z s, k) Coeff l x k i x + d l ) 2 1 x + d l ) 2 2. d 1 l 1 d 2 l 2 We sum next over d 1 + d 2 = d. An easy induction on m shows that generally b 1 +b 2 =a a1 a2 b 1 b 2 t + b 1 b 2 ) m = t + a 1 a 2 ) m 19) holds whenever a 1 + a 2 = a. The base case follows from the following identity which will be used repeatedly below a1 a2 a =. b b 1 +b 2 =b b 1 b 2 9
10 Via equation 19), our expression simplifies to 1 m! 2 s! 2 l i g k! z s, k) Coeff l x k 2x) s = 2 m 1 m! g g 1) s z s, s) i l 1 l 2 = 2 m 1 m! B 2ḡ 2n 2ḡ 2n)! 1 22n 2ḡ+1 ) g 1) s = 2 m 1 m! m 2g + 1)! B m 2g m+2g ) ) g 2n 2ḡ + s s ) m 2g l 1 l 2 g g = 2 m 1 m! m 2g + 1)! B m 2g m+2g ). This last equality and 15) complete the proof of 10), hence of the theorem. l 1 l 2 s Lemma 2 For all integers a and k we have lim N 1 N k ζ 1 ζ N 1 2 +a 1 ζ) k the sum being taken over the N th roots ζ of 1. = 1 2 k+1 ) Bk k!, Proof. When N is large compared to a, the sum to compute is 0 for k < 0, and 1 for k = 0. We may thus assume that k 1. We introduce the auxiliary variable z, and evaluate Setting z = u N z k 1 ζ N 1 2 +a = ζ 1 1 ζ) k ζ 1 u k N k ζ 1 ζ N 1 2 +a 1 z ζ = 1 z and making N we obtain ζ N 1 2 +a The lemma follows. = 1+u 1 ζ) k 1 u/n)n 1 2 +a 1 1 u/n) N 1 N 1 1 z) 2 +a 1 + N 1 z) N ue u/2 e u 1 = 1 u 2 sinh u. 2 Lemma 3 Let b, α be fixed nonnegative integers and z a real number. Then the limit α z + p lim N α N α 1) α p N p=0 b p equals the coefficient of x α in α! x+z b. 10
11 Proof. Let us write f b z) = α p=0 z + p N b ) α 1) α p. p The recursion f b+1 z + 1) = f b+1 z) + f b z) implies by induction that We evaluate N α f j 0) = N α α p=0 p N j f b z) = b j=0 ) α 1) α p = N α p z f j 0). b j where cj, i) are the coefficients defined by the expansion x = j j cj, i) x i. i=0 We use the Euler identities ) α p i α 1) α p 0 if i < α = p=0 p α! if i = α. Making N we obtain Therefore, lim N α f j 0) = α! cj, α). N lim f bz) = α! N b j=0 j α i p α cj, i) 1) α p, i=0 p=0 N p z cj, α). b j This expression can be computed as the coefficient of x α in b z x x + z α! = α!. j=0 b j j b References [1] A. Bertram, G. Daskalopoulos and R. Wentworth, Gromov Invariants for Holomorphic Maps from Riemann Surfaces to Grassmannians, J. Amer. Math. Soc. 11
12 9 1996), no 2, [2] S. K. Donaldson, Gluing techniques in the cohomology of the moduli spaces, Topological methods in modern mathematics Stony Brook, NY, 1991), , Publish or Perish, Houston, TX, [3] L. Jeffrey, F. Kirwan, Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface, Ann. of Math. 2) ), no. 1, [4] A. Marian, Intersection theory on the moduli spaces of stable bundles via morphism spaces, Harvard University Thesis [5] A. Marian, D. Oprea, Virtual intersections on the Quot scheme and the Vafa Intriligator formula, preprint, math.ag/ [6] P. E. Newstead, Characteristic classes of stable bundles over an algebraic curve, Trans. Amer. Math. Soc ), [7] M. Thaddeus, Conformal field theory and the cohomology of the moduli space of stable bundles, J. Differential Geom ), no. 1, [8] M. Thaddeus, Stable pairs, linear systems and the Verlinde formula, Invent. Math ), no. 2, [9] E. Witten, Twodimensional gauge theories revisited, J. Geom. Phys ), no. 4, [10] E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, Geometry, topology, and physics, , Conf. Proc. Lecture Notes Geom. Topology, IV, Internat. Press, Cambridge, MA, [11] D. Zagier, On the cohomology of moduli spaces of rank two vector bundles over curves, The moduli space of curves Texel Island, 1994), , Progr. Math., 129, Birkhäuser Boston, Boston, MA,
ON THE INTERSECTION THEORY OF QUOT SCHEMES AND MODULI OF BUNDLES WITH SECTIONS
ON THE INTERSECTION THEORY OF QUOT SCHEMES AND MODULI OF BUNDLES WITH SECTIONS ALINA MARIAN Abstract. We consider a class of tautological top intersection products on the moduli space of stable pairs consisting
More informationVIRTUAL INTERSECTIONS ON THE QUOT SCHEME AND VAFAINTRILIGATOR FORMULAS
VIRTUAL ITERSECTIOS O THE QUOT SCHEME AD VAFAITRILIGATOR FORMULAS ALIA MARIA AD DRAGOS OPREA Abstract. We construct a virtual fundamental class on the Quot scheme parametrizing quotients of a trivial
More informationIntersections in genus 3 and the Boussinesq hierarchy
ISSN: 14015617 Intersections in genus 3 and the Boussinesq hierarchy S. V. Shadrin Research Reports in Mathematics Number 11, 2003 Department of Mathematics Stockholm University Electronic versions of
More informationALGEBRAIC HYPERBOLICITY OF THE VERY GENERAL QUINTIC SURFACE IN P 3
ALGEBRAIC HYPERBOLICITY OF THE VERY GENERAL QUINTIC SURFACE IN P 3 IZZET COSKUN AND ERIC RIEDL Abstract. We prove that a curve of degree dk on a very general surface of degree d 5 in P 3 has geometric
More informationPROBLEMS FOR VIASM MINICOURSE: GEOMETRY OF MODULI SPACES LAST UPDATED: DEC 25, 2013
PROBLEMS FOR VIASM MINICOURSE: GEOMETRY OF MODULI SPACES LAST UPDATED: DEC 25, 2013 1. Problems on moduli spaces The main text for this material is Harris & Morrison Moduli of curves. (There are djvu files
More informationSegre classes of tautological bundles on Hilbert schemes of surfaces
Segre classes of tautological bundles on Hilbert schemes of surfaces Claire Voisin Abstract We first give an alternative proof, based on a simple geometric argument, of a result of Marian, Oprea and Pandharipande
More informationMODULI OF VECTOR BUNDLES ON CURVES AND GENERALIZED THETA DIVISORS
MODULI OF VECTOR BUNDLES ON CURVES AND GENERALIZED THETA DIVISORS MIHNEA POPA 1. Lecture II: Moduli spaces and generalized theta divisors 1.1. The moduli space. Back to the boundedness problem: we want
More informationarxiv:alggeom/ v2 9 Jun 1993
GROMOV INVARIANTS FOR HOLOMORPHIC MAPS FROM RIEMANN SURFACES TO GRASSMANNIANS arxiv:alggeom/9306005v2 9 Jun 1993 Aaron Bertram 1 Georgios Daskalopoulos Richard Wentworth 2 Dedicated to Professor Raoul
More informationStable maps and Quot schemes
Stable maps and Quot schemes Mihnea Popa and Mike Roth Contents 1. Introduction........................................ 1 2. Basic Setup........................................ 4 3. Dimension Estimates
More informationChern numbers and Hilbert Modular Varieties
Chern numbers and Hilbert Modular Varieties Dylan AttwellDuval Department of Mathematics and Statistics McGill University Montreal, Quebec attwellduval@math.mcgill.ca April 9, 2011 A Topological Point
More informationA GENERALIZED QUOT SCHEME AND MEROMORPHIC VORTICES
A GENERALIZED QUOT SCHEME AND MEROMORPHIC VORTICES INDRANIL BISWAS, AJNEET DHILLON, JACQUES HURTUBISE, AND RICHARD A. WENTWORTH Abstract. Let be a compact connected Riemann surface. Fix a positive integer
More informationA RECONSTRUCTION THEOREM IN QUANTUM COHOMOLOGY AND QUANTUM KTHEORY
A RECONSTRUCTION THEOREM IN QUANTUM COHOMOLOGY AND QUANTUM KTHEORY Y.P. LEE AND R. PANDHARIPANDE Abstract. A reconstruction theorem for genus 0 gravitational quantum cohomology and quantum Ktheory is
More informationPorteous s Formula for Maps between Coherent Sheaves
Michigan Math. J. 52 (2004) Porteous s Formula for Maps between Coherent Sheaves Steven P. Diaz 1. Introduction Recall what the Thom Porteous formula for vector bundles tells us (see [2, Sec. 14.4] for
More informationAsymptotic of Enumerative Invariants in CP 2
Peking Mathematical Journal https://doi.org/.7/s4543844 ORIGINAL ARTICLE Asymptotic of Enumerative Invariants in CP Gang Tian Dongyi Wei Received: 8 March 8 / Revised: 3 July 8 / Accepted: 5 July 8
More informationLECTURE 26: THE CHERNWEIL THEORY
LECTURE 26: THE CHERNWEIL THEORY 1. Invariant Polynomials We start with some necessary backgrounds on invariant polynomials. Let V be a vector space. Recall that a ktensor T k V is called symmetric if
More informationNOTES ON THE MODULI SPACE OF STABLE QUOTIENTS
NOTES ON THE MODULI SPACE OF STABLE QUOTIENTS Dragos Oprea 1 1 Partially supported by NSF grants DMS1001486, DMS 1150675 and the Sloan Foundation. 2 Outline These notes are based on lectures given at
More informationOLIVIER SERMAN. Theorem 1.1. The moduli space of rank 3 vector bundles over a curve of genus 2 is a local complete intersection.
LOCAL STRUCTURE OF SU C (3) FOR A CURVE OF GENUS 2 OLIVIER SERMAN Abstract. The aim of this note is to give a precise description of the local structure of the moduli space SU C (3) of rank 3 vector bundles
More informationFlat connections on 2manifolds Introduction Outline:
Flat connections on 2manifolds Introduction Outline: 1. (a) The Jacobian (the simplest prototype for the class of objects treated throughout the paper) corresponding to the group U(1)). (b) SU(n) character
More informationThe YangMills equations over Klein surfaces
The YangMills equations over Klein surfaces ChiuChu Melissa Liu & Florent Schaffhauser Columbia University (New York City) & Universidad de Los Andes (Bogotá) Seoul ICM 2014 Outline 1 Moduli of real
More informationScalar curvature and the Thurston norm
Scalar curvature and the Thurston norm P. B. Kronheimer 1 andt.s.mrowka 2 Harvard University, CAMBRIDGE MA 02138 Massachusetts Institute of Technology, CAMBRIDGE MA 02139 1. Introduction Let Y be a closed,
More informationarxiv: v1 [math.ag] 13 Mar 2019
THE CONSTRUCTION PROBLEM FOR HODGE NUMBERS MODULO AN INTEGER MATTHIAS PAULSEN AND STEFAN SCHREIEDER arxiv:1903.05430v1 [math.ag] 13 Mar 2019 Abstract. For any integer m 2 and any dimension n 1, we show
More informationThe tangent space to an enumerative problem
The tangent space to an enumerative problem Prakash Belkale Department of Mathematics University of North Carolina at Chapel Hill North Carolina, USA belkale@email.unc.edu ICM, Hyderabad 2010. Enumerative
More informationSEGRE CLASSES AND HILBERT SCHEMES OF POINTS
SEGRE CLASSES AND HILBERT SCHEMES OF POINTS A MARIAN, D OPREA, AND R PANDHARIPANDE Abstract We prove a closed formula for the integrals of the top Segre classes of tautological bundles over the Hilbert
More informationMODULI SPACES OF CURVES
MODULI SPACES OF CURVES SCOTT NOLLET Abstract. My goal is to introduce vocabulary and present examples that will help graduate students to better follow lectures at TAGS 2018. Assuming some background
More informationarxiv:math/ v1 [math.ag] 24 Nov 1998
Hilbert schemes of a surface and Euler characteristics arxiv:math/9811150v1 [math.ag] 24 Nov 1998 Mark Andrea A. de Cataldo September 22, 1998 Abstract We use basic algebraic topology and EllingsrudStromme
More informationRATIONALITY CRITERIA FOR MOTIVIC ZETA FUNCTIONS
RATIONALITY CRITERIA FOR MOTIVIC ZETA FUNCTIONS YUZHOU GU 1. Introduction Work over C. Consider K 0 Var C, the Grothendieck ring of varieties. As an abelian group, K 0 Var C is generated by isomorphism
More informationTheta divisors and the Frobenius morphism
Theta divisors and the Frobenius morphism David A. Madore Abstract We introduce theta divisors for vector bundles and relate them to the ordinariness of curves in characteristic p > 0. We prove, following
More informationON A THEOREM OF CAMPANA AND PĂUN
ON A THEOREM OF CAMPANA AND PĂUN CHRISTIAN SCHNELL Abstract. Let X be a smooth projective variety over the complex numbers, and X a reduced divisor with normal crossings. We present a slightly simplified
More informationON FOURIERMUKAI TRANSFORM ON THE COMPACT VARIETY OF RULED SURFACES
ON FOURIERMUKAI TRANSFORM ON THE COMPACT VARIETY OF RULED SURFACES CRISTINA MARTÍNEZ Abstract. Let C be a projective irreducible nonsingular curve over an algebraic closed field k of characteristic 0.
More informationRes + X F F + is defined below in (1.3). According to [JeKi2, Definition 3.3 and Proposition 3.4], the value of Res + X
Theorem 1.2. For any η HH (N) we have1 (1.1) κ S (η)[n red ] = c η F. Here HH (F) denotes the Hequivariant Euler class of the normal bundle ν(f), c is a nonzero constant 2, and is defined below in (1.3).
More informationNested DonaldsonThomas invariants and the Elliptic genera in String theor
Nested DonaldsonThomas invariants and the Elliptic genera in String theory StringMath 2012, Hausdorff Center for Mathematics July 18, 2012 Elliptic Genus in String theory: 1 Gaiotto, Strominger, Yin
More informationGeometric motivic integration
Université Lille 1 Modnet Workshop 2008 Introduction Motivation: padic integration Kontsevich invented motivic integration to strengthen the following result by Batyrev. Theorem (Batyrev) If two complex
More informationHomotopy and geometric perspectives on string topology
Homotopy and geometric perspectives on string topology Ralph L. Cohen Stanford University August 30, 2005 In these lecture notes I will try to summarize some recent advances in the new area of study known
More informationCOMPLEX ALGEBRAIC SURFACES CLASS 9
COMPLEX ALGEBRAIC SURFACES CLASS 9 RAVI VAKIL CONTENTS 1. Construction of Castelnuovo s contraction map 1 2. Ruled surfaces 3 (At the end of last lecture I discussed the Weak Factorization Theorem, Resolution
More informationThree Descriptions of the Cohomology of Bun G (X) (Lecture 4)
Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) February 5, 2014 Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and
More informationThe Chern character of the Verlinde bundle over M g,n
The Chern character of the Verlinde bundle over M g,n A. Marian D. Oprea R. Pandharipande A. Pixton D. Zvonkine November 2013 Abstract We prove an explicit formula for the total Chern character of the
More informationIf F is a divisor class on the blowing up X of P 2 at n 8 general points p 1,..., p n P 2,
Proc. Amer. Math. Soc. 124, 727733 (1996) Rational Surfaces with K 2 > 0 Brian Harbourne Department of Mathematics and Statistics University of NebraskaLincoln Lincoln, NE 685880323 email: bharbourne@unl.edu
More informationOn the tautological ring of M g,n
Proceedings of 7 th Gökova GeometryTopology Conference, pp, 1 7 On the tautological ring of M g,n Tom Graber and Ravi Vakil 1. Introduction The purpose of this note is to prove: Theorem 1.1. R 0 (M g,n
More informationMathematical Research Letters 2, (1995) A VANISHING THEOREM FOR SEIBERGWITTEN INVARIANTS. Shuguang Wang
Mathematical Research Letters 2, 305 310 (1995) A VANISHING THEOREM FOR SEIBERGWITTEN INVARIANTS Shuguang Wang Abstract. It is shown that the quotients of Kähler surfaces under free antiholomorphic involutions
More informationTAUTOLOGICAL EQUATION IN M 3,1 VIA INVARIANCE THEOREM
TAUTOLOGICAL EQUATION IN M 3, VIA INVARIANCE THEOREM D. ARCARA AND Y.P. LEE This paper is dedicated to Matteo S. Arcara, who was born while the paper was at its last phase of preparation. Abstract. A
More informationLecture VI: Projective varieties
Lecture VI: Projective varieties Jonathan Evans 28th October 2010 Jonathan Evans () Lecture VI: Projective varieties 28th October 2010 1 / 24 I will begin by proving the adjunction formula which we still
More informationF. LAYTIMI AND D.S. NAGARAJ
REMARKS ON RAMANUJAMKAWAMATAVIEHWEG VANISHING THEOREM arxiv:1702.04476v1 [math.ag] 15 Feb 2017 F. LAYTIMI AND D.S. NAGARAJ Abstract. In this article weproveageneralresult on anef vector bundle E on a
More informationPERMUTATIONEQUIVARIANT QUANTUM KTHEORY IV. D q MODULES
PERMUTATIONEQUIVARIANT QUANTUM KTHEORY IV. D q MODULES ALEXANDER GIVENTAL Abstract. In Part II, we saw how permutationequivariant quantum Ktheory of a manifold with isolated fixed points of a torus
More informationMorse theory and stable pairs
Richard A. SCGAS 2010 Joint with Introduction Georgios Daskalopoulos (Brown University) Jonathan Weitsman (Northeastern University) Graeme Wilkin (University of Colorado) Outline Introduction 1 Introduction
More informationOn the Virtual Fundamental Class
On the Virtual Fundamental Class Kai Behrend The University of British Columbia Seoul, August 14, 2014 http://www.math.ubc.ca/~behrend/talks/seoul14.pdf Overview DonaldsonThomas theory: counting invariants
More information2 G. D. DASKALOPOULOS AND R. A. WENTWORTH general, is not true. Thus, unlike the case of divisors, there are situations where k?1 0 and W k?1 = ;. r;d
ON THE BRILLNOETHER PROBLEM FOR VECTOR BUNDLES GEORGIOS D. DASKALOPOULOS AND RICHARD A. WENTWORTH Abstract. On an arbitrary compact Riemann surface, necessary and sucient conditions are found for the
More informationTHE RANKLEVEL DUALITY FOR NONABELIAN THETA FUNCTIONS. 1. Introduction
THE RANKLEVEL DUALITY FOR NONABELIAN THETA FUNCTIONS ALINA MARIAN AND DRAGOS OPREA Abstract. We prove that the strange duality conjecture of BeauvilleDonagiTu holds for all curves. We establish first
More informationh M (T ). The natural isomorphism η : M h M determines an element U = η 1
MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 7 2.3. Fine moduli spaces. The ideal situation is when there is a scheme that represents our given moduli functor. Definition 2.15. Let M : Sch Set be a moduli
More informationTHE TAUTOLOGICAL RINGS OF THE MODULI SPACES OF STABLE MAPS TO FLAG VARIETIES
THE TAUTOLOGICAL RINGS OF THE MODULI SPACES OF STABLE MAPS TO FLAG VARIETIES DRAGOS OPREA Abstract. We show that the rational cohomology classes on the moduli spaces of genus zero stable maps to SL flag
More informationThis theorem gives us a corollary about the geometric height inequality which is originally due to Vojta [V].
694 KEFENG LIU This theorem gives us a corollary about the geometric height inequality which is originally due to Vojta [V]. Corollary 0.3. Given any ε>0, there exists a constant O ε (1) depending on ε,
More informationCharacteristic classes in the Chow ring
arxiv:alggeom/9412008v1 10 Dec 1994 Characteristic classes in the Chow ring Dan Edidin and William Graham Department of Mathematics University of Chicago Chicago IL 60637 Let G be a reductive algebraic
More informationVector bundles in Algebraic Geometry Enrique Arrondo. 1. The notion of vector bundle
Vector bundles in Algebraic Geometry Enrique Arrondo Notes(* prepared for the First Summer School on Complex Geometry (Villarrica, Chile 79 December 2010 1 The notion of vector bundle In affine geometry,
More informationNOTES ON HOLOMORPHIC PRINCIPAL BUNDLES OVER A COMPACT KÄHLER MANIFOLD
NOTES ON HOLOMORPHIC PRINCIPAL BUNDLES OVER A COMPACT KÄHLER MANIFOLD INDRANIL BISWAS Abstract. Our aim is to review some recent results on holomorphic principal bundles over a compact Kähler manifold.
More informationThe Affine Grassmannian
1 The Affine Grassmannian Chris Elliott March 7, 2013 1 Introduction The affine Grassmannian is an important object that comes up when one studies moduli spaces of the form Bun G (X), where X is an algebraic
More informationEnumerative Geometry Steven Finch. February 24, 2014
Enumerative Geometry Steven Finch February 24, 204 Given a complex projective variety V (as defined in []), we wish to count the curves in V that satisfy certain prescribed conditions. Let C f denote complex
More informationConstruction of M B, M Dol, M DR
Construction of M B, M Dol, M DR Hendrik Orem Talbot Workshop, Spring 2011 Contents 1 Some Moduli Space Theory 1 1.1 Moduli of Sheaves: Semistability and Boundedness.............. 1 1.2 Geometric Invariant
More informationTAUTOLOGICAL RELATIONS ON THE STABLE MAP SPACES
TAUTOLOGICAL RELATIONS ON THE STABLE MAP SPACES DRAGOS OPREA Abstract. The cohomology of the spaces of rational stable maps to flag varieties is generated by tautological classes. We study relations between
More informationMath 248B. Applications of base change for coherent cohomology
Math 248B. Applications of base change for coherent cohomology 1. Motivation Recall the following fundamental general theorem, the socalled cohomology and base change theorem: Theorem 1.1 (Grothendieck).
More informationEXAMPLES OF CALABIYAU 3MANIFOLDS WITH COMPLEX MULTIPLICATION
EXAMPLES OF CALABIYAU 3MANIFOLDS WITH COMPLEX MULTIPLICATION JAN CHRISTIAN ROHDE Introduction By string theoretical considerations one is interested in CalabiYau manifolds since CalabiYau 3manifolds
More informationEnumerative Invariants in Algebraic Geometry and String Theory
Dan Abramovich . Marcos Marino Michael Thaddeus Ravi Vakil Enumerative Invariants in Algebraic Geometry and String Theory Lectures given at the C.I.M.E. Summer School held in Cetraro, Italy June 611,
More informationHiggs Bundles and Character Varieties
Higgs Bundles and Character Varieties David Baraglia The University of Adelaide Adelaide, Australia 29 May 2014 GEAR Junior Retreat, University of Michigan David Baraglia (ADL) Higgs Bundles and Character
More informationis a short exact sequence of locally free sheaves then
3. Chern classes We have already seen that the first chern class gives a powerful way to connect line bundles, sections of line bundles and divisors. We want to generalise this to higher rank. Given any
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics STABLE REFLEXIVE SHEAVES ON SMOOTH PROJECTIVE 3FOLDS PETER VERMEIRE Volume 219 No. 2 April 2005 PACIFIC JOURNAL OF MATHEMATICS Vol. 219, No. 2, 2005 STABLE REFLEXIVE SHEAVES
More informationLINKED ALTERNATING FORMS AND LINKED SYMPLECTIC GRASSMANNIANS
LINKED ALTERNATING FORMS AND LINKED SYMPLECTIC GRASSMANNIANS BRIAN OSSERMAN AND MONTSERRAT TEIXIDOR I BIGAS Abstract. Motivated by applications to higherrank BrillNoether theory and the BertramFeinbergMukai
More informationThe diagonal property for abelian varieties
The diagonal property for abelian varieties Olivier Debarre Dedicated to Roy Smith on his 65th birthday. Abstract. We study complex abelian varieties of dimension g that have a vector bundle of rank g
More informationMILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES
MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES NILAY KUMAR In these lectures I want to introduce the ChernWeil approach to characteristic classes on manifolds, and in particular, the Chern classes.
More informationCANONICAL METRICS AND STABILITY OF PROJECTIVE VARIETIES
CANONICAL METRICS AND STABILITY OF PROJECTIVE VARIETIES JULIUS ROSS This short survey aims to introduce some of the ideas and conjectures relating stability of projective varieties to the existence of
More informationHYPERKÄHLER MANIFOLDS
HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly
More informationNonuniruledness results for spaces of rational curves in hypersurfaces
Nonuniruledness results for spaces of rational curves in hypersurfaces Roya Beheshti Abstract We prove that the sweeping components of the space of smooth rational curves in a smooth hypersurface of degree
More informationThe moduli stack of vector bundles on a curve
The moduli stack of vector bundles on a curve Norbert Hoffmann norbert.hoffmann@fuberlin.de Abstract This expository text tries to explain briefly and not too technically the notions of stack and algebraic
More informationON STABILITY OF NONDOMINATION UNDER TAKING PRODUCTS
ON STABILITY OF NONDOMINATION UNDER TAKING PRODUCTS D. KOTSCHICK, C. LÖH, AND C. NEOFYTIDIS ABSTRACT. We show that nondomination results for targets that are not dominated by products are stable under
More informationSPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS
SPACES OF RATIONAL CURVES ON COMPLETE INTERSECTIONS ROYA BEHESHTI AND N. MOHAN KUMAR Abstract. We prove that the space of smooth rational curves of degree e on a general complete intersection of multidegree
More informationTHE MOTIVE OF THE FANO SURFACE OF LINES. 1. Introduction
THE MOTIVE OF THE FANO SURFACE OF LINES HUMBERTO A. DIAZ Abstract. The purpose of this note is to prove that the motive of the Fano surface of lines on a smooth cubic threefold is finitedimensional in
More informationWe can choose generators of this kalgebra: s i H 0 (X, L r i. H 0 (X, L mr )
MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 43 5.3. Linearisations. An abstract projective scheme X does not come with a prespecified embedding in a projective space. However, an ample line bundle
More information1. Algebraic vector bundles. Affine Varieties
0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasiprojective varieties over a field k Affine Varieties 1.
More informationk=0 /D : S + S /D = K 1 2 (3.5) consistently with the relation (1.75) and the RiemannRochHirzebruchAtiyahSinger index formula
20 VASILY PESTUN 3. Lecture: GrothendieckRiemannRochHirzebruchAtiyahSinger Index theorems 3.. Index for a holomorphic vector bundle. For a holomorphic vector bundle E over a complex variety of dim
More informationINTRODUCTION TO THE LANDAUGINZBURG MODEL
INTRODUCTION TO THE LANDAUGINZBURG MODEL EMILY CLADER WEDNESDAY LECTURE SERIES, ETH ZÜRICH, OCTOBER 2014 In this lecture, we will discuss the basic ingredients of the Landau Ginzburg model associated
More informationHYPERKÄHLER FOURFOLDS FIBERED BY ELLIPTIC PRODUCTS
HYPERKÄHLER FOURFOLDS FIBERED BY ELLIPTIC PRODUCTS LJUDMILA KAMENOVA Abstract. Every fibration of a projective hyperkähler fourfold has fibers which are Abelian surfaces. In case the Abelian surface
More informationCohomology of the Mumford Quotient
Cohomology of the Mumford Quotient Maxim Braverman Abstract. Let X be a smooth projective variety acted on by a reductive group G. Let L be a positive Gequivariant line bundle over X. We use a Witten
More informationDERIVED CATEGORIES: LECTURE 6. References
DERIVED CATEGORIES: LECTURE 6 EVGENY SHINDER References [AO] V. Alexeev, D. Orlov, Derived categories of Burniat surfaces and exceptional collections, arxiv:1208.4348v2 [BBS] Christian Böhning, HansChristian
More informationTHE MODULI OF SUBALGEBRAS OF THE FULL MATRIX RING OF DEGREE 3
THE MODULI OF SUBALGEBRAS OF THE FULL MATRIX RING OF DEGREE 3 KAZUNORI NAKAMOTO AND TAKESHI TORII Abstract. There exist 26 equivalence classes of ksubalgebras of M 3 (k) for any algebraically closed field
More informationBetti numbers of abelian covers
Betti numbers of abelian covers Alex Suciu Northeastern University Geometry and Topology Seminar University of Wisconsin May 6, 2011 Alex Suciu (Northeastern University) Betti numbers of abelian covers
More informationSEGRE CLASSES AND HILBERT SCHEMES OF POINTS
SEGRE CLASSES AND HILBERT SCHEMES OF POINTS A. MARIAN, D. OPREA, AND R. PANDHARIPANDE Abstract. We prove a closed formula for the integrals of the top Segre classes of tautological bundles over the Hilbert
More informationPower structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points
arxiv:math/0407204v1 [math.ag] 12 Jul 2004 Power structure over the Grothendieck ring of varieties and generating series of Hilbert schemes of points S.M. GuseinZade I. Luengo A. Melle Hernández Abstract
More informationCoherent sheaves on elliptic curves.
Coherent sheaves on elliptic curves. Aleksei Pakharev April 5, 2017 Abstract We describe the abelian category of coherent sheaves on an elliptic curve, and construct an action of a central extension of
More informationSOME REMARKS ON LINEAR SPACES OF NILPOTENT MATRICES
LE MATEMATICHE Vol. LIII (1998) Supplemento, pp. 2332 SOME REMARKS ON LINEAR SPACES OF NILPOTENT MATRICES ANTONIO CAUSA  RICCARDO RE  TITUS TEODORESCU Introduction. In this paper we study linear spaces
More informationarxiv: v1 [math.ag] 17 Feb 2009
Cohomological Characterization of Vector Bundles on Grassmannians of Lines Enrique Arrondo and Francesco Malaspina arxiv:0902.2897v1 [math.ag] 17 Feb 2009 Universidad Complutense de Madrid Plaza de Ciencias,
More informationCohomological Formulation (Lecture 3)
Cohomological Formulation (Lecture 3) February 5, 204 Let F q be a finite field with q elements, let X be an algebraic curve over F q, and let be a smooth affine group scheme over X with connected fibers.
More informationIntroduction Curves Surfaces Curves on surfaces. Curves and surfaces. Ragni Piene Centre of Mathematics for Applications, University of Oslo, Norway
Curves and surfaces Ragni Piene Centre of Mathematics for Applications, University of Oslo, Norway What is algebraic geometry? IMA, April 13, 2007 Outline Introduction Curves Surfaces Curves on surfaces
More informationGENERALIZED THETA LINEAR SERIES ON MODULI SPACES OF VECTOR BUNDLES ON CURVES
GENERALIZED THETA LINEAR SERIES ON MODULI SPACES OF VECTOR BUNDLES ON CURVES MIHNEA POPA CONTENTS 1. Introduction 1 2. Semistable bundles 2 2.1. Arbitrary vector bundles 2 2.2. Semistable vector bundles
More informationLECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL
LECTURE 7: STABLE RATIONALITY AND DECOMPOSITION OF THE DIAGONAL In this lecture we discuss a criterion for nonstablerationality based on the decomposition of the diagonal in the Chow group. This criterion
More informationGeneralized HitchinKobayashi correspondence and WeilPetersson current
Author, F., and S. Author. (2015) Generalized HitchinKobayashi correspondence and WeilPetersson current, International Mathematics Research Notices, Vol. 2015, Article ID rnn999, 7 pages. doi:10.1093/imrn/rnn999
More informationCOMPLEX ALGEBRAIC SURFACES CLASS 6
COMPLEX ALGEBRAIC SURFACES CLASS 6 RAVI VAKIL CONTENTS 1. The intersection form 1.1. The NeronSeveri group 3 1.. Aside: The Hodge diamond of a complex projective surface 3. RiemannRoch for surfaces 4
More informationTHE SET OF NONSQUARES IN A NUMBER FIELD IS DIOPHANTINE
THE SET OF NONSQUARES IN A NUMBER FIELD IS DIOPHANTINE BJORN POONEN Abstract. Fix a number field k. We prove that k k 2 is diophantine over k. This is deduced from a theorem that for a nonconstant separable
More informationPacific Journal of Mathematics
Pacific Journal of Mathematics THE MODULI OF FLAT PU(2,1) STRUCTURES ON RIEMANN SURFACES Eugene Z. Xia Volume 195 No. 1 September 2000 PACIFIC JOURNAL OF MATHEMATICS Vol. 195, No. 1, 2000 THE MODULI OF
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationInstantons and Donaldson invariants
Instantons and Donaldson invariants George Korpas Trinity College Dublin IFT, November 20, 2015 A problem in mathematics A problem in mathematics Important probem: classify dmanifolds up to diffeomorphisms.
More informationTHE HITCHIN FIBRATION
THE HITCHIN FIBRATION Seminar talk based on part of Ngô Bao Châu s preprint: Le lemme fondamental pour les algèbres de Lie [2]. Here X is a smooth connected projective curve over a field k whose genus
More informationTwo constructions of the moduli space of vector bundles on an algebraic curve
Two constructions of the moduli space of vector bundles on an algebraic curve Georg Hein November 17, 2003 Abstract We present two constructions to obtain the coarse moduli space of semistable vector bundles
More informationORBIFOLDS AND ORBIFOLD COHOMOLOGY
ORBIFOLDS AND ORBIFOLD COHOMOLOGY EMILY CLADER WEDNESDAY LECTURE SERIES, ETH ZÜRICH, OCTOBER 2014 1. What is an orbifold? Roughly speaking, an orbifold is a topological space that is locally homeomorphic
More information