Rigid Schubert classes in compact Hermitian symmetric spaces

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Rigid Schubert classes in compact Hermitian symmetric spaces Colleen Robles joint work with Dennis The Texas A&M University April 10, 2011

Part A: Question of Borel & Haefliger 1. Compact Hermitian symmetric spaces. 2. Schubert varieties. 3. The motivating question.

CHSS X = G/P a compact Hermitian symmetric space (CHSS). Example (Projective space) CP m 1

CHSS X = G/P a compact Hermitian symmetric space (CHSS). Example (Projective space) CP m 1 = Gr(1, m) Example (Grassmannians) The space Gr(k, m) of k dimensional linear subspaces in C m. G = SL(m, C), P = stabilizer of fixed k plane E C m.

Irreducible CHSS Classical Grassmannian of E k C n+1 Gr(k, n + 1) = A n /P k. Quadric hypersurfaces Q 2n 1 = B n /P 1, n 2; Q 2n 2 = D n /P 1, n 4. Lagrangian grassmannians Gr ω (n, C 2n ) = C n /P n, n 3. Spinor varieties D n /P n, n 4.

Irreducible CHSS Classical Grassmannian of E k C n+1 Gr(k, n + 1) = A n /P k. Quadric hypersurfaces Q 2n 1 = B n /P 1, n 2; Q 2n 2 = D n /P 1, n 4. Lagrangian grassmannians Gr ω (n, C 2n ) = C n /P n, n 3. Spinor varieties D n /P n, n 4. Exceptional Cayley plane X 16 = E 6 /P 6. Freudenthal variety X 27 = E 7 /P 7.

CHSS as algebraic varieties Definition G a complex semisimple Lie group. V an irreducible G representation. That is, G GL(V ). Fact There is a unique compact G orbit X PV. Definition P the stabilizer of a point o X. Fact X G/P is a (smooth) homogeneous variety. Fact X is CHSS if and only if the isotropy representation of P on T o X is irreducible.

Schubert varieties Theorem (B. Kostant 1963) The integral homology H (X ) of a CHSS is generated by the classes [X w ] of the Schubert varieties X w X. Example (Grassmannian) Fix 0 < l m k and a subspace W C m of codimension k + l 1. X (l) = {E Gr(k, m) dim(e W ) 0} is a singular Schubert variety of codimension l. Fact Most Schubert varieties in a CHSS are singular. X w is the most singular variety representing [X w ].

Motivating Question (Borel and Haefliger 1961) Which Schubert classes [X w ] can be represented by a smooth variety Y X? Definition When smooth Y representing [X w ], we say [X w ] is smoothable. Example The hypersurface X (1) Gr(k, m) is a hyperplane section, = [X (1) ] is smoothable. Theorem (Hartshorne, Rees & Thomas 1974) 1. [X (2) ] H 14 (Gr(3, 6)) can not be represented by any integral linear combination of smooth, oriented submanifolds of (real) codimension four. 2. [X (2) ] = [Y 1 ] [Y 2 ] H 8 (Gr(2, 5)), but cannot be smoothed.

Part B: What is known 1. Izzet Coskun 2. A differential geometric approach 2.a Maria Walters and Robert Bryant 2.b Jaehyun Hong

Work of Izzet Coskun 2010 Theorem A (nearly sharp) description of the smoothable Schubert classes in the Grassmannian. Definition A Schubert class [X w ] is rigid if the only varieties Y representing the class are the Schubert varieties g X w, g G. Theorem Identified all rigid Schubert classes in the Grassmannian.

A differential geometric approach Theorem (Maria Walters 1997 & Robert Bryant 2001) The varieties Y with the property that [Y ] = r[x w ], for some r Z, (1) are characterized by the Schur differential system (to be defined). Definition The Schubert variety X w is Schur rigid if every irreducible variety Y satisfying (1) is a Schubert variety g X w, g G. Otherwise, X w is Schur flexible. X w Schur rigid = [X w ] rigid. X w singular and Schur rigid = [X w ] not smoothable. BH question can be studied via differential geometry.

Schur flexibility for trivial topological reasons When H 2k (X ) is generated by a single Schubert class [X w ], H 2k (X ) = Z, every Y k satisfies [Y ] = r[x w ]. = X w is Schur flexible. Examples The following are Schur flexible for trivial topological reasons. 1a. Every Schubert variety of P n. 1b. Any X w P m X. 2a. Every Schubert variety of Q 2n 1. 2b. Any Schubert variety of Q 2n 2, with dim X w n 1. Remark There are two Schubert varieties X w Q 2n 2 of dimension n 1; they are maximal linear spaces P n 1.

Work of Walters and Bryant Maria Walters, Ph.D. Thesis 1997 Identified first-order obstructions to Schur flexibility for (A) smooth Schubert varieties in Gr(k, m), and (B) codimension two Schubert varieties in Gr(2, m). Robert Bryant 2001 Identified first-order obstructions to Schur flexibility for (1) smooth Schubert varieties in Gr(k, m) and C n /P n, (2) maximal linear subspaces in the classical CHSS, (3) singular Schubert varieties of low (co)dimension in Gr(k, m).

Work of Jaehyun Hong Theorem (Hong 2007) Let X be an irreducible CHSS, excluding the quadrics of odd dimension. Let X w X be a smooth Schubert variety, excluding the non-maximal linear subspaces, and P 1 C n /P n. Then X w is Schur rigid. Remark Omissions above are for trivial topological reasons. Theorem (Hong 2005) Identified a large class of singular Schubert varieties in Gr(k, m) for which there exist first-order obstructions to Schur flexibility.

Part C: Main Result joint with Dennis The 1. Statement and examples. 2. Key tools in Hong s approach. 3. Obstructions to extending Hong s strategy to the general case. 4. Outline of our solution.

Theorem (Robles - The 2011) A complete list of the Schubert varieties in a CHSS for which there exist first-order obstructions to Schur flexibility. Remark 1. List includes all (singular) X w for which there exist first order obstructions to the existence of Y (BH s question). 2. A Schubert class appears on this list if and only if its Poincaré dual does. 3. Theorem recovers the results of Walters, Bryant and Hong. 4. Theorem need not be a complete list of Schur rigid Schubert varieties: there may be higher-order obstructions. 5. Hartshorne, Rees & Thomas examples are not on the list.

Example: the Lagrangian Grassmannian X 15 = C 5 /P 5 Schubert variety Schur rigid by Thm: 1 st order obstruction to flexibility. 1 2 3 3 1 2 1 1 3 1 3 2 3 0 0 1 2 1 4 2 1 0 0 2 2 2 1 0 2 2 1 0 smooth: a = 0 singular: a > 0 Schur flexible for trivial topological reasons.

Example: the Spinor variety X 15 = D 6 /P 6 Schubert variety Schur rigid by Thm: 1 st order obstruction to flexibility. 0 1 2 2 1 2 0 0 3 1 2 1 2 0 0 0 1 1 3 2 1 0 0 1 1 2 1 0 1 2 1 0 smooth: a = 0 singular: a > 0 Schur flexible for trivial topological reasons.

Example: the Cayley plane X 16 = E 6 /P 6 Schubert variety Schur rigid by Thm: 1 st order obstruction to flexibility. 0 1 1 1 0 1 2 3 1 2 0 0 0 0 3 2 1 0 1 0 1 1 1 3 2 1 2 smooth: a = 0 singular: a > 0 Schur flexible for trivial topological reasons.

Hong s approach for smooth X w Two key tools 1. Well-known description of smooth X w by connected sub-diagrams of the Dynkin diagram of G. 2. A Lie algebra cohomology H 1 (w) arises in analysis. First-order obstruction to flexibility is equivalent to the vanishing of a subspace of H 1 (w). Theorem of Kostant (1961) reduces computation of H 1 (w) to Weyl group combinatorics. Obstructions to generalizing to singular case 1. No analogous description of the singular X w. 2. Kostant s Theorem does not apply to H 1 (w).

The sine quibus non of our approach 1. Characterization of the Schubert varieties by an integer 0 a and a marking J of the Dynkin diagram of G. This generalizes the descriptions of both the smooth Schubert varieties (a = 0), and the Schubert varieties in Gr(k, m) by partitions. 2. Construction of an algebraic Laplacian (à la Kostant) with the property that ker Lie algebra cohomology.

Characterization of Schubert varieties Theorem The Schubert varieties X w are characterized by an integer a w and a marking J w of the Dynkin diagram of G. The smooth Schubert varieties are characterized by a = 0. Example (In C 8 /P 8 ) < Dynkin diagram of C 8. J = {2, 5, 6}. J E w p Marking for parabolic P 8. E w eigenspace decomp. T o X = n 0 n 1 n A n 0 n a = T o X w, for some w.

Precise statements for C n /P n - Lagrangian grassmannian Theorem (Characterization of Schubert varieties by (a, J)) a bijection between Schubert varieties and pairs (J, a) satisfying J = { j 1,..., j p } {1,..., n 1} and J = a, a + 1. Theorem (First-order obstructions to Schur flexibility) The X w for which first-order obstructions to Schur flexibility are 1. J = a, any J with 1 < j l j l 1 for all 1 l p; 2. J = a + 1, any J with 1 < j l j l 1 for all 2 l p + 1. For the singular X w above (a > 0), [X w ] is not smoothable.

The Schur system R w n := T o X, n w := T o X w and w := dim C n w. Theorem (Kostant 1961) a p module indecomposition k n = w =k I w into (inequivalent) irreducible modules I w with highest weight line n w Gr(k, n) P( k n). Definitions 1. R w := PI w Gr(k, n) P n w. 2. R w := G R w Gr(k, TX ). 3. Y is an integral variety of R w if TY 0 R w.

Analysis of the Schur system The Schur system R w may be lifted to an exterior differential system (EDS) on a frame bundle G. To this system there is associated a Lie algebra cohomology H 1 (w), which admits a P induced graded decomposition G GL(V ) R w X PV H 1 (w) = H0 1 (w) H1 1 (w) H2 1 (w) = H0 1 (w) H+(w) 1 ; When P n w R w the system R w is flexible (Bryant 2001); H0 1(w) plays role in determining when P n w = R w. Given P n w = R w, H+(w) 1 = 0 = first-order obstructions to flexibility.

Computing the Lie algebra cohomology H 1 (w) Step 1 Construction of an algebraic Laplacian with the property ker := H 1 (w) H 1 (w). Step 2 Analyze H+(w) 1 to determine first-order obstructions to flexiblity. Tools: representation theory, including (a, J) characterization. EDS machinery. Step 3 Determine when R w = P n w. Tools: representation theory; spectral sequence of filtered complex.

Thank you.

(a, J) for the classical G/P i J = {j 1 < j 2 < < j p } G/P i (a, J) A n /P i Defining 0 q p by j q < i < j q+1, we have: (p, q) {(2a, a), (2a + 1, a), (2a + 1, a + 1), (2a + 2, a + 1)}. B n /P 1 J = {j} D n /P 1 a = 0, J = {j} or {n 1, n}; a = 1, J = {j n 2} or {n 1, n}. C n /P n p = a or a + 1. p = a and n 1 J, or j s j s 1 2 for s = p+1 2 ; D n /P n p = a + 1 and n 1 J p = a + 1 and n 1 J, or p = a + 2 and n 1 J j s+1 j s 2 for s = p 2.

Proper X w in classical CHSS for which 1 st order obstructions to Schur flexibility. In A n /P i with 1 < i < n. Define q Z by j q < i < j q+1. : (p, q) = (2a + 2, a + 1) with 1 < j l j l 1, for all 1 < l p, and 1 < i j q, j q+1 i; : (p, q) = (2a + 1, a + 1) with 1 < j l j l 1, for all 1 < l p + 1, and 1 < j q+1 i; : (p, q) = (2a + 1, a) with 1 < j l j l 1, for all 1 l p, and 1 < i j q ; : (p, q) = (2a, a) with 1 < j l j l 1 for all 1 l p + 1. In D n /P 1 : the X w with a = 0 and J = {n 1} or J = {n}.

In C n /P n : (1) when p = a, any J with 1 < j l j l 1 for all 1 l p; (2) when p = a + 1, any J with 1 < j l j l 1 for all 2 l p + 1. In D n /P n : (1) Either p = a and n 1 J, or p = a + 1 and n 1 J; in both cases 1 < j l j l 1 for all 1 l p and 2 < j s j s 1 with s = p+1 2. (2) p = a + 1 and n 1 J with 1 < j l j l 1 for all 2 l p, and 2 < j s+1 j s with s = p 2.

Proper X w E 6 /P 6 with 1 st order obstructions to Schur flexibility. w a J dim X w a J (6542) 0 {3} 4 1 {3} (65431) 0 {2} 5 1 {5} (65432413) 1 {4} 8 1 {4} (65432456) 0 {1} 8 0 {1} (65432451342) 1 {5} 11 0 {2} (654324561345) 1 {3} 12 0 {3}

Proper X w E 7 /P 7 with 1 st order to obstructions Schur flexibility. w a J dim X w a J (76542) 0 {3} 5 2 {5} (765431) 0 {2} 6 1 {2} (765432413) 1 {4} 9 2 {4} (7654324567) 0 {1} 10 1 {6} (765432451342) 1 {5} 12 1 {3} (7654324561345) 2 {3, 6} 13 2 {1, 5} (76543245671342) 2 {1, 5} 14 2 {3, 6} (765432456713456) 1 {3} 15 1 {5} (76543245613452431) 1 {6} 17 0 {1} (765432456713456245) 2 {4} 18 1 {4} (765432456713456245342) 1 {2} 21 0 {2} (7654324567134562453413) 2 {5} 22 0 {3}

(a, J) versus Young tableau for Gr(k, m) Tableau π = (p q 1 1,..., pqr r ) sits in k (m k) rectangle: m k p 1 > p 2 > > p r > 0, qi k. Recipe is: a(π) = r 1 = no. interior corners in π, J(π) = {q 1, q 1 + q 2,..., q i, m p 1, m p 2,..., m p r }\{k}. Dual tableau π is the complement of π, rotated 180 o.

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