Exact Formulas for Invariants of Hilbert Schemes

Transcription

1 Exact Formulas for Invariants of Hilbert Schemes N. Gillman, X. Gonzalez, M. Schoenbauer, advised by John Duncan, Larry Rolen, and Ken Ono September 2, / 36

2 Paper See the related paper at 2 / 36

3 Real Cohomology: Two Perspectives 1: Holes in topological spaces 3 / 36

4 Real Cohomology: Two Perspectives 2: Classes of differential forms 4 / 36

5 Real Cohomological Invariants Definition (Betti numbers) b i (M) := dim R (H i (M; R)). 5 / 36

6 Real Cohomological Invariants Definition (Betti numbers) b i (M) := dim R (H i (M; R)). Definition (Euler characteristic χ(m)) χ(m) := dim(m) i=0 ( 1) i b i (M). 5 / 36

7 Real Cohomological Invariants For M a 2d-dimensional oriented manifold, we have an intersection pairing : H d (M) H d (M) Z. Definition The signature σ(m) of M is σ(m) := { eigenvalues > 0} { eigenvalues < 0}. 6 / 36

8 Complex Cohomology 7 / 36

9 Complex Cohomology Definition (Hodge numbers) h s,t := dim C (H s,t (M)). 7 / 36

10 Complex Cohomology Definition (Hodge numbers) h s,t := dim C (H s,t (M)). Connecting Invariants For Kähler surfaces S: χ(s) =2h 2,0 + 2h 0,0 + h 1,1 4h 1,0, σ(s) =2h 2,0 + 2 h 1,1. 7 / 36

11 Cohomological Invariant Polynomials Definition The Hodge polynomial of a complex manifold M is χ Hodge (M)(x, y) := x d/2 y d/2 s,t h s,t (M)( x) s ( y) t. 8 / 36

12 Cohomological Invariant Polyonomials Specializing to ±1 χ Hodge (M)(1, 1) =χ(m). χ Hodge (M)( 1, 1) =( 1) n σ(m). 9 / 36

13 Hilbert Scheme Intuition A Hilbert Scheme of n points on a manifold M is a smoothed version" of the n th symmetric product of M. 10 / 36

14 Göttsche s Generating function Theorem (Göttsche, 1994) χ Hodge (Hilb n (S))q n = n 0 n=1 s+t odd (1 xs 1 y t 1 q n ) hs,t s+t even (1 xs 1 y t 1 q n ) hs,t. 11 / 36

15 Göttsche s Generating function Theorem (Göttsche, 1994) χ Hodge (Hilb n (S))q n = n 0 n=1 s+t odd (1 xs 1 y t 1 q n ) hs,t s+t even (1 xs 1 y t 1 q n ) hs,t. Notation Z S (x, y; n) := n 0 χ Hodge (Hilb n (S))q n. 11 / 36

16 Example: K3 surfaces If S is a K3 surface, n 0 χ Hodge(Hilb n (S))q n is given by n=1 1 (1 xyq n )(1 x 1 y 1 q n )(1 q n ) 20 (1 xy 1 q n )(1 xy 1 q n ). 12 / 36

17 Specialization of Göttsche s formula to ±1 n=0 χ(hilb n (S))q n = (1 q n ) χ(s) n=0 n=1 ( 1) n σ(hilb n (S))q n (1 q n ) σ(s) = (1 q 2n ) (σ(s)+χ(s))/2 n=1 13 / 36

18 The Dedekind Eta Function Definition The Dedekind eta function η(τ) is defined by where q := e 2πiτ. η(τ) = q 1/24 n=1 (1 q n ), 14 / 36

19 Specialization of Göttsche s formula to ±1 χ(hilb n (S))q n =q χ(s)/24 η(τ) χ(s) n=0 ( 1) n σ(hilb n (S))q n =q χ(s)/24 n=0 η(τ) σ(s) η(2τ) (σ(s)+χ(s))/2 15 / 36

20 Exact Formulas Theorem (Exact Formula for Euler Characteristic) If 0 χ(s) < 24n, then we have χ(hilb n (S)) =2π j< χ(s) 24 k χ(s)/2 A k ( χ(s), 0, j; n)χ(hilb j (S))L (0, j, k; n). k=1 16 / 36

21 Exact Formulas Theorem (Exact Formula for Euler Characteristic) If 0 χ(s) < 24n, then we have χ(hilb n (S)) =2π j< χ(s) 24 k χ(s)/2 A k ( χ(s), 0, j; n)χ(hilb j (S))L (0, j, k; n). k=1 A k ( χ(s), 0, j; n) is a Kloosterman sum. 16 / 36

22 Exact Formulas Theorem (Exact Formula for Euler Characteristic) If 0 χ(s) < 24n, then we have χ(hilb n (S)) =2π j< χ(s) 24 k χ(s)/2 A k ( χ(s), 0, j; n)χ(hilb j (S))L (0, j, k; n). k=1 A k ( χ(s), 0, j; n) is a Kloosterman sum. L (0, j, k; n) is a scaled modified Bessel function of the first kind. 16 / 36

23 Exact Formulas Theorem (Exact Formula for Signature) If σ(s) χ(s) < 24n, then we have σ(hilb n (S)) = 2π +2π j< χ(s) 24 k=2 k even j< 3σ(S) χ(s) 48 A k (σ(s), Λ(S), j; n)σ(hilb j (S)) k Λ L (0, j, k; n) (S)/2 k=1 k odd ( 1) n B k (σ(s), Λ(S), j; n)a(λ(s), σ(s); j) 2 Λ(S)/2 k Λ L (1, j, k; n). (S)/2 where Λ(S) := (σ(s) + χ(s))/2 and Λ (S) := (σ(s) χ(s))/2. 17 / 36

24 Asymptotic Properties Theorem (Asymptotic for signature) Suppose χ(s) σ(s), χ(s) 0. As n, the following are true: ( ) n σ(s) χ(s) 6 χ(s) 8 exp π 6 n σ(s) > 0 ( ) σ(hilb n (S)) S n (σ(s) χ(s))/2 3 χ(s) 3σ(S) 4 exp π 6 n σ(s) < 0 ( ) n χ(s) 6 χ(s) 8 exp π 3 n σ(s) = / 36

25 Asymptotic Properties Theorem (Asymptotic for Euler Characteristic) Suppose χ(s) 0. As n, ( ) χ(hilb n (S)) S n χ(s) 3 2χ(S) 4 exp π n / 36

26 Main Takeaways For certain S, these theorems give exact formulas for χ(hilb n (S)) and σ(hilb n (S)). 20 / 36

27 Main Takeaways For certain S, these theorems give exact formulas for χ(hilb n (S)) and σ(hilb n (S)). We can obtain asymptotic expressions for the values of χ(hilb n (S)) and σ(hilb n (S)) and n. 20 / 36

28 Set-up 1 Z S (x, y; τ) := n 0 χ Hodge(Hilb n (S))q n 21 / 36

29 Set-up 1 Z S (x, y; τ) := n 0 χ Hodge(Hilb n (S))q n 2 Our theorems relate to specializations of Z S (x, y; τ) for choices x, y = ±1. 21 / 36

30 Set-up 1 Z S (x, y; τ) := n 0 χ Hodge(Hilb n (S))q n 2 Our theorems relate to specializations of Z S (x, y; τ) for choices x, y = ±1. 3 Express Z S (x, y; τ) as functions, H α,β (q) := q α+2β 24 η(τ) α η(2τ) β =: n 0 a(α, β; n)q n. 21 / 36

31 Circle method: main idea Cauchy s Integral Formula For any contour C surrounding the origin, f(q) = a(n)q n = a(n) = n=0 C f(q) dq. qn+1 22 / 36

32 η(τ) = q 1/24 n 1 (1 qn ) in the unit disk 23 / 36

33 Conclusion: The rate of decay of η(τ) near the boundary of the unit circle is distinct at different roots of unity e 2πih/k, and the stronger decay corresponds to small k. 24 / 36

34 Conclusion: The rate of decay of η(τ) near the boundary of the unit circle is distinct at different roots of unity e 2πih/k, and the stronger decay corresponds to small k. We use this idea to evaluate our integral of H α,β (q) = q α+2β 24 η(τ) α η(2τ) β. 24 / 36

35 The Path of Integration Circle of radius e 2πN 2, where N. 25 / 36

36 The Path of Integration Circle of radius e 2πN 2, where N. We break this circle into Farey arcs ξ h,k, which zoom in" on e 2πih/k as N. 25 / 36

37 Farey Arcs Figure: Farey arcs ξ 1,3 26 / 36

38 Modularity "Definition" (Modular Form) A meromorphic function f : Im(τ) > 0 C is called a modular form of weight k 1 Z if for all integer matrices 2 a b c d with determinant 1, we have ( ) aτ + b f = ε(cτ + d) k f(τ) cτ + d for some root of unity ε. 27 / 36

39 Modularity Modularity properties encode the behavior of the function f at the points e 2πih/k. 28 / 36

40 Modularity Modularity properties encode the behavior of the function f at the points e 2πih/k. η(τ) is a modular form of weight 1/2. 28 / 36

41 Bessel functions Definition (I-Bessel) ( 1 2 I v (z) := z) v 2πi (0+) t v 1 exp ] [t + z2 dt. 4t 29 / 36

42 30 / 36

43 Asymptotic behavior for I-Bessel As z, I v (z) ( ) ez 1 4v2 1 + (4v2 1)(4v 2 9) + 2πz 8z 2!(8z) 2 ez 2πz. 31 / 36

44 The Circle Method Recap 1 Use Cauchy s Integral Theorem 32 / 36

45 The Circle Method Recap 1 Use Cauchy s Integral Theorem 2 Expand the circle to the boundary of the disk 32 / 36

46 The Circle Method Recap 1 Use Cauchy s Integral Theorem 2 Expand the circle to the boundary of the disk 3 Approximate the behvior near roots of unity 32 / 36

47 The Circle Method Recap 1 Use Cauchy s Integral Theorem 2 Expand the circle to the boundary of the disk 3 Approximate the behvior near roots of unity 4 Add up, and enjoy obtain formulas! 32 / 36

48 The Circle Method Recap 1 Use Cauchy s Integral Theorem 2 Expand the circle to the boundary of the disk 3 Approximate the behvior near roots of unity 4 Add up, and enjoy obtain formulas! 5 Obtain asymptotic properties!! 32 / 36

49 Numerics: Exact Values n H 24,0(n) , H 0, 2(n) H 1, 2(n) Table: Exact formulas, including two summands 33 / 36

50 Numerics: Exact Values n Λ 2 24,0 (n) Λ 2 0, 2 (n) Λ 2 1, 2 (n) Table: Ratios of approximate and actual values, including two summands n Λ 5 0, 2 (n) Λ 5 1, 2 (n) Table: Ratios of approximate and actual values, including five summands 34 / 36

51 Numerics: Asymptotics n , , , 000 Ψ 24,0(n) Ψ 0, 2(n) Ψ 1, 2(n) Table: Ratios of asymptotic predictions and exact values 35 / 36

52 Thank you for listening! 36 / 36

53 Thank you for listening! See the related paper at 36 / 36