The tangent space to an enumerative problem

Transcription

1 The tangent space to an enumerative problem Prakash Belkale Department of Mathematics University of North Carolina at Chapel Hill North Carolina, USA ICM, Hyderabad 2010.

2 Enumerative problems An enumerative problem is a problem of counting the number of points in a space that satisfy certain geometrically defined conditions. For example, The classical problem of intersecting Schubert varieties (in general position) in a Grassmannian or on a general flag variety. The more recent problem in quantum cohomology of counting maps from the projective line to a homogeneous space satisfying certain incidence conditions, Of counting subbundles of a fixed (general) vector bundle of a given degree and rank on an algebraic curve.

3 The following questions will be considered. (a) Can we obtain (hopefully simpler ) consequences of a (non-empty) transversal intersection which are equivalent to the enumerative problem having a non-empty solution? Is there an inductive framework to understand enumerative problems which have non-empty solutions? (b) Transversality can be immediately translated as the non-zeroness of suitable determinants. This leads one to sections ( the theta sections ) of line bundles over suitable moduli-spaces which have representation theoretic significance. How effective is this link between intersection theory and representation theory?

4 General formulations For (a) consider the following strategy: Let us choose parameters so that the enumerative problem has a solution, for example choosing Schubert varieties which pass through a chosen point in a Grassmannian (but otherwise general). The main effort will then be to decide whether the intersection is transverse at the chosen point. The problem now is a smaller problem in suitable tangent spaces.

5 For (b): Consider an enumerative problem, and form the moduli stack of the universal intersection, for example: The stack parameterizing triples of Schubert varieties in some Grassmannian, together with a point on the intersection of these varieties. The stack parameterizing pairs (S, V ) where S V is a subbundle of a vector bundle V on an algebraic curve (which may also be allowed to vary). The enumerative problem is one of counting subbundles of V of a given degree and rank. We assume that the expected dimension is zero. There is a line bundle L on this stack M together with a global section, this global section θ vanishes exactly at points where the intersection (at the given point) is non-transverse. Therefore the section θ does not vanish identically if and only if the enumerative problem has a solution (is it an equivalent condition to require h 0 (M, L) 0, or even weaker M ss 0?).

6 Note: Frequently, deformation theory is controlled by two term complexes (with H 0 the formal tangent space, and H 1 the space of obstructions). In such cases θ is the determinant. Even if there is no transversality available, the determinant line bundle makes sense. Naive hope: This construction produces all global sections of line bundles on moduli spaces (by suitable specializations, functoriality an deformations). So far this seems to be mostly apply to the general linear group (the above, and the strange duality conjecture for vector bundles), although there are some generalizations to other situations. All known geometrically defined invariants seem to arise from θ or square-roots (Pfaffian type).

7 We will review some of the applications of (a) and (b). In each of the contexts presented below there is 1. An enumerative problem. 2. An invariant theory problem. 3. A topological problem, which should be seen as (related to) a problem on representation theory of the fundamental group of a curve. Note that (3) relates to (2), because moduli spaces of algebraic objects frequently have a topological meaning (Narasimhan-Seshadri, Mehta-Seshadri).

8 We will consider Ordinary Grassmannians and representations of the general linear group. Quantum cohomology. In a G/P. On higher genus curves: Strange duality.

9 Ordinary Grassmannians and representations of the general linear group The enumerative problem is one of intersecting Schubert varieties in an ordinary Grassmannian Gr(r, n). The invariant theoretic problem is representations of general linear groups and the topological problem is of characterizing possible eigenvalues of a sum of hermitian matrices (lie algebra version of unitary representations of π 1 (P 1 {0, 1, })). Recall that the cohomology H (X ) of a Grassmannian X = Gr(r, C n ) has an additive basis of cycle classes of Schubert varieties ω I where I is a subset of {1,..., n} of cardinality r. Note that ω I is the cycle class of the Schubert variety Ω I (F ) = {V X : dim(v F il ) l, l = 1,..., r} where 0 F 1 F 2 F n = C n is a complete flag of vector subspaces.

10 The prototypical enumerative problem is the one of counting the number of points m in the intersection Ω I (E ) Ω J (F ) Ω K (H ) (1) (this can also be viewed as the multiplicity of the class of a point in a cohomology cup product ω I ω J ω K )

11 Numerical relations between intersection numbers and invariant theory Irreducible polynomial representations of the GL(r), (or equivalently, the unitary group U(r)) are parameterized by weakly decreasing sequences of non-negative integers λ = (λ 1 λ 2 λ r ). Intersection theory of Grassmannians and invariant theory of the special linear group GL(r) are related, and this has been known for a long time. To describe this, note that sequences I, J and K as above also parameterize some irreducible (polynomial) representations of GL(r). The association takes I λ I = (λ 1 λ 2 λ r ), λ a = n r + a i a, 1 a r. Denote the corresponding representation of GL(r) by V (λ I ). Then, the intersection multiplicity m equals the dimension of the space of invariants ( V (λi ) V (λ J ) V (λ K ) ) SL(r) (2)

12 In 2002, I gave a geometric explanation for the coincidence of multiplicities: Each point of intersection V of the Schubert varieties (1) produces an non-zero invariant θ V in the dual of the vector space (2) (using the general technique described before). [Briefly, M maps to the moduli space of pairs (V, Q) where V and Q are vector spaces each equipped with three complete flags, and θ is pulled pack from this space, we can now specialize V or Q] Recall that the global sections of line bundles on flag varieties give irreducible reps. We will view global section of line bundles on moduli spaces as rep. theoretic objects. Theorem (B, 2002, IMRN 2004) Let V 1,..., V m be the points in (1). Then, θ V1,..., θ Vm form a basis for the space of invariants (2).

13 The work of Klyachko, Knutson-Tao and the numerical relation between intersection numbers and invariant theory implies the following geometric Horn theorem. Theorem Let λ = λ I, µ = λ J and ν = λ K.The following are equivalent 1. m For every 1 r < r and choice of subsets A, B, C of [r] each of cardinality r so that ω A ω B ω C 0 H (Gr( r, r)) the following inequality holds ν c r(n r) a A λ a + b B µ b + c C

14 Let V be the point of intersection and Q = C n /V. If the intersection is non transverse, pick φ Hom(V, Q) in the tangent space of the intersection. Let S be the kernel of φ. S corresponds to the obstruction in the smaller Grassmannian (this has to be modified). The game is then in Hom(V, Q), a space which is almost homogeneous (finitely many orbits) (similar ideas where applied simultaneously by Purbhoo in related problems).

15 I gave a direct geometric proof of this theorem (Fulton posed this as a challenge) using the methods (a) in This theorem implies that the problem of characterizing possible eigenvalues of a sum of Hermitian matrices is inductive (Horn s conjecture). The original proof used the Saturation conjecture (which is implied by the above theorem) in representation theory. The geometric understanding of Theorems 1 and 2 was the starting point for the generalizations.

16 Quantum cohomology Let I, J and K be subsets of [n] each of cardinality r and d a non-negative integer. The Gromov-Witten number ω I, ω J, ω K d is the number of maps P 1 Gr(r, n) of degree d such that f (0) Ω I (E ), f (1) Ω J (F ), f ( ) Ω K (G ). The (small) quantum cohomology ring, a generalization of ordinary cohomology of Grassmannians encapsulates the Gromov-Witten numbers as structure coefficients. It is easy to see that sub-bundles of the trivial rank n bundle O n P 1, of degree rank r and degree d correspond bijectively to maps P 1 Gr(r, n) of degree d. The conditions at 0, 1 and above, can be interpreted in terms of the corresponding sub-bundle.

17 The analogue of the ring of invariants is the fusion ring of the special unitary group SU(r) (which incorporate an additional parameter of a nonnegative level ). The structure coefficients in the fusion ring are dimensions of spaces of sections of suitable line bundles on moduli stacks of parabolic bundles on P 1. These structure coefficients are related to the structure coefficients of quantum cohomology (generalizing the earlier numerical relations) The topological problem is one of determining the possible local monodromies in a unitary representation of the fundamental group of P 1 {0, 1, }.

18 The geometric horn property generalizes to the following theorem. This says that the non-zeroness of Gromov-Witten invariants of Grassmannians has an inductive structure. Theorem (B 2003) The following are equivalent: (a) ω I, ω J, ω K d 0. (b) For any integers d and r with 0 < r < r, d 0, and A, B, C subsets of {1,..., r} each of cardinality r, such that ω A, ω B, ω C d = 1, the following inequality holds: α A λ a + b B µ b + c C ν c d(n r) + r(qn + i h ) + r(n r).

19 The above theorem implies 1. A saturation theorem for fusion structure coefficients. 2. Shows that the non-zeroness of Gromov-Witten numbers is equivalent to the existence of certain unitary local systems with corresponding monodromies on P 1. Under this correspondence, the GW number is one if and only if the corresponding local system is rigid (generalizing a conjecture of Fulton). 3. Gives an explicit inductive characterization of the possible eigenvalues of a product of unitary matrices. There is a list of inequalities for every n. The list for n is prepared using the lists for smaller n.

20 In a G/P The enumerative problem is the problem of intersections of Schubert varieties in a G/P. The representation theoretic problem is the invariant theory of G, and of the Levi subgroup of P. The analogue of the Hermitian eigenvalue problem is the problem of characterizing the conjugacy class of a sum C = A + B where A, B are in the lie algebra of the maximal compact subgroup of G, given the conjugacy classes of A and B.

21 In this case, the moduli space M (of intersections of open Schubert varieties) is a quotient of (P/B L ) 3 by P. A basic dichotomy appears here which leads to the concept of Levi-moveability and subsequently a deformation of cohomology of G/P (2004, Belkale-Kumar).

22 Let P be a parabolic subgroup of G. Schubert varieties in G/P are parameterized by elements in the relative Weyl group W P = W /W P. For w W, set Λ w = w 1 BwP G/P. The cycle classes of the closures Λ u additively span the cohomology H (G/P, Z). Assume that the codimensions of Λ u, Λ v and Λ w add up to the dimension of G/P. In this case [ Λ u ][ Λ v ][ Λ w ] is a multiple of the class of a point. Fact: For general p 1, p 2, p 3 P, p 1 Λ u, p 2 Λ v and p 3 Λ w meet transversally at e if and only if the cup product is non-zero (using Kleiman transversality).

23 Let T = T (G/P) e. We have a natural map T T /p 1 T (Λ u ) T /p 2 T (Λ v ) T /p 3 T (Λ w ) We may view the above sequence as a morphism of vector bundles on (P/B L ) 3 (a small check shows T (X w ) is B L stable). This morphism is an isomorphism iff p 1 Λ u, p 2 Λ v and p 3 Λ w meet transversally at e. We can then take the determinant of the map and view it as an invariant section θ H 0 ((P/B L ) 3, L u L v L w ) P Here L u is a natural line bundle on P/B L. Note that θ 0 is equivalent to the cup product being non-zero.

24 If P were reductive, we will have H 0 (P/B L, L u ) an irreducible representation of P, and hence we would obtain a connection to Rep theory of P. Definition We call a triple (u, v, w) Levi-movable if, for generic (l 1, l 2, l 3 ) L 3, the intersection l 1 Λ P u l 2 Λ P v l 3 Λ P w is a transverse intersection at e. This definition is equivalent to the non-vanishing of θ when restricted to (L/B L ) 3.

25 Properties of Levi-moveability 1. In the case of cominiscule flag varieties, the above condition is vacuous. In general, the condition of Levi-movability is equivalent to having the intersection number not equal to 0 and a system of linear equalities. 2. Suppose we write the structure coefficients in the usual cup product in H (G/P) by [ Λ P u ] [ Λ P v ] = c w u,v [ Λ P w ] Define the deformed product 0 by the following rule [ Λ P u ] 0 [ Λ P v ] = c w u,v [ Λ P w ] where the sum is restricted to w so that the triple (u, v, w o ww P o ) is Levi-movable. This deformed product is commutative and associative.

26 1. The transversality analysis leads to a deformation of cohomology of G/P. This deformation relates to the representation theory of the Levi subgroup (Belkale-Kumar, Belkale-Kumar-Ressayre). 2. The deformation is finely tuned to representation theory (B-Kumar, Ressayre). 3. The analogue of the Geometric Horn problem is open. 4. Some funtoriality for this product is known (Ressayre-Richmond).

27 On higher genus curves Let X be a connected smooth projective algebraic curve X of genus g 1 over C. Let SU X (r) be the moduli space of semi-stable vector bundles of rank r with trivial determinant over X. The spaces H 0 (SU X (r), L k ) should be considered as a non-abelian generalization of invariant theory. The topological problem is the problem of the moduli of special unitary representations of the fundamental group of X.

28 The enumerative generalization in the higher genus setting is not immediately clear. We may expect that having a corresponding enumerative problem would lead to spanning sets for the spaces H 0 (SU X (r), L k ). The strange duality conjecture predicts a good spanning set for the space H 0 (SU X (r), L k ). Let U X (k) be the moduli space of rank k vector bundles of degree k(g 1). There is a natural divisor Θ on SU X (r) UX (k). This divisor is supported on pairs (E, F ) such that h 0 (E F ) 0. According to the strange duality conjecture, Θ F for F U X (k) span H 0 (SU X (r), L k ). [Equivalently Θ induces a perfect duality between H 0 (SU X (r), L k ) and a related space of global sections on U X (k)]

29 The enumerative problem that I found was not on X but on a nodal degeneration of X. Let T be a general vector bundle of degree k(g 1) and rank r + k on P 1. Let p 1,,... p g and q 1,..., q g be distinct points on P 1. For i = 1,..., g, consider vector space homomorphisms η i : T pi T qi with kernel of rank 1 (but general up to this condition). The enumerative problem is the following: Count the set of subbundles E of T of degree zero and rank r such that the following two conditions are satisfied: 1. η i (E pi ) E qi 2. E pi contains the kernel of η i.

30 Say we get bundles E 1,..., E m above. The enumerative number m above equals h 0 (SU X (r), L k ), where X as before, is an arbitrary connected, smooth and projective curve of genus g. Now consider the nodal curve X obtained by gluing p i to q i, for i = 1,..., g. It can be shown that η i descend to give isomorphisms (T /E i ) pi (T /E i ) qi. These can then be used to glue, and therefore one obtains vector bundles F 1,..., F m of rank k and degree k(g 1) on X. Consider the deformations F 1,..., F m of F 1,..., F m to a general smooth X. The strange duality on the general curve X was proved in in the following form: Theorem (B, 2006) The theta sections θ F1,..., θ Fm form a basis of H 0 (SU X (r), L k ).

31 The equality of the enumerative number with the dimension of H 0 (SU X (r), L k ) uses the Verlinde formula, the degeneration formulas of Tsuchiya-Ueno-Yamada and the determination of the cohomology class of a diagonal in a self product of Grassmannians. 1. Subsequently, Marian and Oprea found a suitable enumerative problem on a smooth curve, and proved strange duality for all curves. 2. Using properties of the Hitchin connnection, one can show that the conjecture for general curves implies it for all curves. The strange duality is, in an appropriate sense, flat for the Hitchin connection. 3. The conjecture originated in physics, in relation to the KZ/Hitchin connection (and conformal embeddings).

32 Questions 1. There are more general contexts of enumerative problems in representation theory. Under an embedding of homogeneous spaces, one would like to know when the pull back of a Schubert class is non-zero. Analogous results are known, but the full picture is unclear. 2. There are other approaches to relations between enumerative geometry and invariant theory (Tamvakis, Mukhin-Tarasov-Varchenko). 3. Relations to KZ/Hitchin connections need to be developed. One may hope that the monodromy of KZ/Hitchin connections links up with enumerative geometry.