# Enumerative Geometry: from Classical to Modern

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1 : from Classical to Modern February 28, 2008

2 Summary Classical enumerative geometry: examples Modern tools: Gromov-Witten invariants counts of holomorphic maps Insights from string theory: quantum cohomology: refinement of usual cohomology mirror symmetry formulas duality between symplectic/holomorphic structures integrality predictions for GW-invariants geometric explanation yet to be discovered

3 What is Classical EG about? How many geometric objects satisfy given geometric conditions? objects = curves, surfaces,... conditions = passing through given points, curves,... tangent to given curves, surfaces,... having given shape: genus, singularities, degree

4 Example 0 Q: How many lines pass through 2 distinct points? 1

5 Example 1 Q: How many lines pass thr 1 point and 2 lines in 3-space? 1 lines thr. the point and 1st line form a plane 2nd line intersects the plane in 1 point

6 Example 2 Q: How many lines pass thr 4 general lines in 3-space? bring two of the lines together so that they intersect in a point and form a plane

7 Example 2 Q: How many lines pass thr 4 general lines in 3-space? 2 1 line passes thr the intersection pt and lines #3,4 1 line lies in the plane and passes thr lines #3,4

8 General Line Counting Problems All/most line counting problems in vector space V reduce to computing intersections of cycles on G(2, V ) { 2 dim linear subspaces of V } = { (affine) lines in V } This is a special case of Schubert Calculus (very treatable)

9 Example 0 (a semi-modern view) Q: How many lines pass through 2 distinct points? A line in the plane is described by (A, B, C) 0: Ax + By + C = 0. (A, B, C) and (A, B, C ) describe the same line iff (A, B, C ) = λ(a, B, C) { lines in (x, y)-plane } = { 1dim lin subs of (A, B, C)-space } P 2.

10 Example 0 (a semi-modern view) Q: How many lines pass through 2 distinct points? 1 = # of lines [A, B, C] P 2 solving { Ax 1 + By 1 + C = 0 (x 1, y 1 ), (x 2, y 2 )=fixed points Ax 2 + By 2 + C = 0 The system has 1 1dim lin space of solutions in (A, B, C)

11 Example 0 (higher-degree plane curves) degree d curve in (x, y)-plane 0-set of nonzero degree d polynomial in (x, y) polynomials Q and Q determine same curve iff Q = λq # coefficients of Q is ( ) d+2 2 = ( ) { } { d +2 deg d curves in (x, y)-plane = 1dim lin subs of -dim v.s. } 2 ( ) d + 2 P N(d) N(d) 1 2

12 Example 0 (higher-degree plane curves) { deg d curves in (x, y)-plane } = P N(d) "Passing thr a point" = 1 linear eqn on coefficients of Q = get hyperplane in ( ) d+2 2 -dim v.s. of coefficients { deg d curves in (x, y)-plane thr. (xi, y i ) } P N(d) 1 P N(d) intersection of ( ) ( d HPs in d+2 ) 2 -dim v.s. is 1 1dim lin subs intersection of N(d) HPs in P N(d) is 1 point

13 Example 0 (higher-degree plane curves)! degree d plane curve thr N(d) ( ) d general pts d = 1 :! line thr 2 distinct pts in the plane d = 2 :! conic thr 5 general pts in the plane d = 3 :! cubic thr 9 general pts in the plane

14 Typical Enumerative Problems Count complex curves = (singular) Riemann surfaces Σ of fixed genus g, fixed degree d in C n, CP n = C n C n 1... C 0 in a hypersurface Y C n, CP n (0-set of a polynomial) g(σ) genus of Σ singular points d(σ) # intersections of Σ with a generic hyperplane

15 Adjunction Formula If Σ CP 2 is smooth and of degree d, ( ) d 1 g(σ) = 2 every line, conic is of genus 0 every smooth plane cubic is of genus 1 every smooth plane quartic is of genus 3

16 Classical Problem n d # genus 0 degree d plane curves thr. (3d-1) general pts n 1 = 1: # lines thr 2 pts n 2 = 1: # conics thr 5 pts n 3 = 12: # nodal cubics thr 8 pts = M 1,1 ψ 1 = 1 24 n 3 = # zeros of transverse bundle section over CP 1 CP 2 = euler class of rank 3 vector bundle over CP 1 CP 2 CP 1 = cubics thr. 8 general pts; CP 2 = possibilities for node

17 Genus 0 Plane Quartics thr 11 pts n 4 = # plane quartics thr 11 pts with 3 non-separating nodes Zeuthen 1870s: n 4 = 620 = ! 675 = euler class of rank 9 vector bundle over CP 3 (CP 2 ) 3 minus excess contributions of a certain section CP 3 = quartics thr 11 pts; CP 2 = possibilities for i-th node Details in Counting Rational Plane Curves: Old and New Approaches

18 Kontsevich s Formula (Ruan-Tian 1993) n d # genus 0 degree d plane curves thr. (3d-1) general pts n 1 = 1 n d = 1 6(d 1) d 1 +d 2 =d ( ) (3d d 1 d 2 2 (d 1 d 2 ) 2 ) 2 d 1 d 2 n d1 n d2 3d 2 3d 1 1 n 2 = 1, n 3 = 12, n 4 = 620, n 5 = 87, 304, n 6 = 26, 312, 976,...

19 Gromov s 1985 paper Consider equivalence classes of maps f : (Σ, j) CP n (Σ, j)=connected Riemann surface, possibly with nodes f : (Σ, j) CP n and f : (Σ, j ) CP n are equivalent if f = f τ for some τ : (Σ, j) (Σ, j ) f : (Σ, j) CP n is stable if Aut(f ) { τ : (Σ, j) (Σ, j) f τ = f } is finite non-constant holomorphic f : (Σ, j) CP n is stable iff the restr. of f to any S 2 Σ w. fewer than 3 nodes is not const

20 Gromov s Compactness Theorem genus of f : (Σ, j) CP n is # of holes in Σ ( g(σ)) degree d of f f 1 (H) for a generic hyperplane: f [Σ] = d[cp 1 ] H 2 (CP n ; Z) = Z[CP 1 ] Theorem: With respect to a natural topology, M g (CP n, d) { [f : (Σ, j) CP n ]: g(f )=g, d(f )=d, f holomor } is compact

21 Maps vs. Curves Image of holomorphic f : (Σ, j) CP n is a curve genus of f (Σ) g(f ); degree of f (Σ) d(f ) = n d # genus 0 degree d curves thr. (3d-1) pts in CP 2 = # degree d f : (S 2, j) CP 2 s.t. p i f (CP 1 ) i = 1,..., 3d 1 = # { [f : (Σ, j) CP 2 ] M 0 (CP 2, d): p i f (Σ) }

22 Physics Insight I: Quantum (Co)homology Use counts of genus 0 maps to CP n to deform -product on H, H (CP n ) = Z[x]/x n+1, x a x b = x a+b, to -product on H (CP n )[q 0,..., q n ] x a x b = x a+b + q-corrections counting genus 0 maps thr. CP n a, CP n b Theorem (McDuff-Salamon 93, Ruan-Tian 93,...) The product is associative generalizes to all cmpt algebraic/symplectic manifolds

23 Physics Insight I: Quantum (Co)homology Associativity of quantum multiplication is equivalent to Kontsevich s formula for CP 2, extension to CP n gluing formula for counts of genus 0 maps Remark: Classical proof of Kontsevich s formula for CP 2 only: Z. Ran 95, elaborating on 89

24 Other Enumerative Applications of Stable Maps Genus 0 with singularities: Pandharipande, Vakil, Z.- Genus 1: R. Pandharipande, Ionel, Z.- Genus 2,3: Z.-

25 Gromov-Witten Invariants Y = CP n, = hypersurface in CP n (0-set of a polynomial),... µ 1,..., µ k Y cycles GW Y g,d(µ) # { [f : (Σ, j) Y ] M g (Y, d): f (Σ) µ i } g = 0, Y = CP n : M g (Y, d) is smooth, of expected dim, "#"=# Typically, M g (Y, d) is highly singular, of wrong dim

26 Example: Quintic Threefold Y 5 CP 4 0-set of a degree 5 polynomial Q Schubert Calculus: Y 5 contains 2,875 (isolated) lines S. Katz 86 (via Schubert): Y 5 contains 609,250 conics For each line L Y 5 and conic C Y 5, { [f : (Σ, j) Y5 ] M 0 (Y 5, 2): f (Σ) L } M 0 (CP 1, 2) { [f : (Σ, j) Y5 ] M 0 (Y 5, 2) : f (Σ) C } M 0 (CP 1, 1) are connected components of M 0 (Y 5, 2) of dimensions 2 and 0

27 Expected dimension of M 0 (Y 5, d) Y 5 = Q 1 (0) CP 4 for a degree 5 polynomial Q = M 0 (Y 5, d) = { [f : (Σ, j) CP 4 ] M 0 (CP 4, d) : Q f = 0 } holomorphic degree d f : CP 1 CP 4 has the form f ([u, v]) = [ R 1 (u, v),..., R 5 (u, v) ] R 1,..., R 5 = homogeneous polynomials of degree d = dim M 0 (CP 4, d) = 5 (d +1) 1 3 Q f is homogen of degree 5d = Q f = 0 is 5d +1 conditions on R 1,..., R 5

28 Expected dimension of M 0 (Y 5, d) = dim vir M 0 (Y 5, d) = dim M 0 (CP 4, d) (5d + 1) = 0 A more elaborate computation gives dim vir M g (Y 5, d) = 0 g = want to define N g,d GW Y 5 g,d () M g (Y 5, d) vir

29 GW-Invariants of Y 5 CP 4 M g (Y 5, d) = { [f : (Σ, j) Y 5 ] g(f )=g, d(f )=d, j f = 0 } j f df + J Y5 df j N g,d Mg (Y 5, d) vir # { [f : (Σ, j) Y 5 ] g(f )=g, d(f )=d, j f = ν(f ) } ν = small generic deformation of -equation ν multi-valued = N g,d Q

30 What is special about Y 5? Y 5 is Calabi-Yau 3-fold: c 1 (TY 5 ) = 0 Y 5 is "flat on average": Ric Y5 =0 CY 3-folds are central to string theory

31 Physics Insight II: Mirror Symmetry A-Model partition function MIRROR for Calabi-Yau 3-fold Y principle B-Model partition function for mirror (family) of Y generating function for GWs of Y : Fg A (q) = d=1 N g,dq d? Fg B measures" geometry of moduli spaces of mirror CYs

32 B-Side Computations for Y = Y 5 Candelas-de la Ossa-Green-Parkes 91 construct mirror family, compute F0 B Bershadsky-Cecotti-Ooguri-Vafa 93 (BCOV) compute F1 B using physics arguments Fang-Z. Lu-Yoshikawa 03 compute F1 B mathematically Huang-Klemm-Quackenbush 06 compute Fg B, g 51 using physics

33 Mirror Symmetry Predictions and Verifications Predictions Fg A (q) N g,d q d d=1? = F B g (q). Theorem (Givental 96, Lian-Liu-Yau 97,... 00) g = 0 predict. of Candelas-de la Ossa-Green-Parkes 91 holds Theorem (Z. 07) g = 1 predict. of Bershadsky-Cecotti-Ooguri-Vafa 93 holds

34 General Approach to Verifying F A g = F B g (works for g = 0, 1) Need to compute each N g,d and all of them (for fixed g): Step 1: relate N g,d to GWs of CP 4 Y 5 Step 2: use (C ) 5 -action on CP 4 to compute each N g,d by localization Step 3: find some recursive feature(s) to compute N g,d Fg A d

35 From Y 5 CP 4 to CP 4 Q Y 5 Q 1 (0) CP 4 L O(5) V g,d M g (L, d) π Q M g (Y 5, d) Q 1 (0) M g (CP 4, d) π π ( [ξ : Σ L] ) = [π ξ : Σ CP 4 ] Q ( [f : Σ CP 4 ] ) = [Q f : Σ L]

36 From Y 5 CP 4 to CP 4 Q Y 5 Q 1 (0) CP 4 L O(5) V g,d M g (L, d) π Q M g (Y 5, d) Q 1 (0) M g (CP 4, d) π This suggests: Hyperplane Property N g,d M g (Y 5, d) vir Q 1 (0) vir? = e(v g,d ), [M g (CP 4, d)] vir

37 Genus 0 vs. Positive Genus g = 0 everything is as expected: M g (CP 4, d) is smooth [M g (CP 4, d)] vir = [M g (CP 4, d)] V g,d M g (CP 4, d) is vector bundle hyperplane prop. makes sense and holds g 1 none of these holds

38 Genus 1 Analogue Thm. A (J. Li Z. 04): HP holds for reduced genus 1 GWs M 0 1(Y 5, d) vir = e(v 1,d ) M 0 1(CP 4, d). This generalizes to complete intersections Y CP n. M 0 1(CP 4, d) M 1 (CP 4, d) main irred. component closure of { [f : Σ CP 4 ] M 1 (CP 4, d): Σ is smooth } V 1,d M 0 1(CP 4, d) not vector bundle, but e(v 1,d ) well-defined (0-set of generic section)

39 Standard vs. Reduced GWs Thm. A = N1,d 0 M 0 1(Y 5, d) vir = M 0 1 (CP4,d) M 0 1(Y 5, d) M 0 1(CP 4, d) M 1 (Y 5, d) e(v 1,d ) Thm. B (Z. 04, 07): N 1,d N 0 1,d = 1 12 N 0,d This generalizes to all symplectic manifolds: [standard] [reduced genus 1 GW] = F(genus 0 GW) to check BCOV, enough to compute M 0 1(CP 4,d) e(v 1,d)

40 Torus Actions (C ) 5 acts on CP 4 (with 5 fixed pts) = on M g (CP 4, d) (with simple fixed loci) and on V g,d M g (CP 4, d) M 0 g (CP4,d) e(v g,d) localizes to fixed loci g = 0: Atiyah-Bott Localization Thm reduces to graphs g = 1: M 0 g(cp 4, d), V g,d singular = AB does not apply

41 Genus 1 Bypass Thm. C (Vakil Z. 05): V 1,d M 0 1(CP 4, d) admit natural desingularizations: Ṽ 1,d V 1,d M 0 1 (CP4, d) M 0 1(CP 4, d) = M 0 1 (CP4,d) e(v 1,d ) = e(ṽ1,d) M 0 1 (CP4,d)

42 Computation of Genus 1 GWs of CIs Thm. C generalizes to all V 1,d M 0 1,k(CP n, d): L O(a) V 1,d M 1,k (L, d) π CP n M 1,k (CP n, d) π Thms A,B,C provide an algorithm for computing genus 1 GWs of complete intersections X CP n

43 Computation of N 1,d for all d split genus 1 graphs into many genus 0 graphs at special vertex make use of good properties of genus 0 numbers to eliminate infinite sums extract non-equivariant part of elements in H T (P4 )

44 Key Geometric Foundation A Sharp Gromov s Compactness Thm in Genus 1 (Z. 04) describes limits of sequences of pseudo-holomorphic maps describes limiting behavior for sequences of solutions of a -equation with limited perturbation allows use of topological techniques to study genus 1 GWs

45 Main Tool Analysis of Local Obstructions study obstructions to smoothing pseudo-holomorphic maps from singular domains not just potential existence of obstructions

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