The intersection algebra of a pseudoholomorphic cycle in a good class through generic points #

Let X be a closed symplectic four-manifold with canonical class κ and a generic compatible almost complex structure J. The virtual dimension of a class x ∈ H₂(X; ℤ) is d(x) = x · x - κ · x, and a class e is good if e · S ≥ -1 for every embedded J-holomorphic sphere S of square -1. Suppose that D = ∑ m_i C_i is a positive cycle of distinct simple J-holomorphic images C_i, with positive multiplicities m_i, representing a good class e with d(e) = 2k ≥ 0, and passing through k generic points. The geometry of J-curves supplies the following numerical facts about the classes [C_i] in the intersection lattice:

distinct images meet nonnegatively, C_i · C_j ≥ 0 for i ≠ j;
if C_i has singularity number δ_i and contains k_i of the generic points, then 2 k_i ≤ d(C_i) - 2 δ_i, the index of the simple curves in the class of C_i;
an image of negative square is an exceptional sphere, and as e is good it satisfies e · C_i ≥ -1;
the classes sum to e, and every generic point lies on some image.

This file records these facts as TauCeti.IsGoodCycle on an arbitrary integral bilinear form and derives their purely algebraic consequences, following Taubes's analysis of the curves arising from Seiberg–Witten solutions in a good class:

the good-curve component inequality ∑ m_i d(C_i) ≤ d(e), which holds for every combination e = ∑ m_i C_i with natural coefficients of classes meeting pairwise nonnegatively, provided e · C_i ≥ -1 whenever C_i · C_i < 0;
every image is embedded, δ_i = 0, and contains exactly d(C_i) / 2 of the points;
every point lies on exactly one image, and d(e) = ∑ d(C_i);
an image of negative square has square -1;
distinct images are homologically disjoint, and exceptional spheres occur with multiplicity one;
an image occurring with multiplicity at least two satisfies C_i · C_i = κ · C_i = 0 and contains none of the points, so by adjunction it is a square-zero torus orthogonal to κ.

The argument expands d(e) = ∑ m_i (d(C_i) + (e · C_i - C_i · C_i)): each excess m_i (e · C_i - C_i · C_i) is nonnegative, and comparing the result with the point count d(e) = 2 k ≤ ∑ 2 k_i ≤ ∑ d(C_i) forces equality in every step.

Main declarations #

TauCeti.sum_mul_apply_self_sub_apply_canonical_le: the good-curve component inequality.
TauCeti.IsGoodCycle: the numerical properties of a positive cycle of simple pseudoholomorphic images in a good class through generic points.
TauCeti.IsGoodCycle.singularity_eq_zero: every image is embedded.
TauCeti.IsGoodCycle.apply_self_sub_apply_canonical_eq: every image contains half its virtual dimension of the points.
TauCeti.IsGoodCycle.existsUnique_mem: every point lies on exactly one image.
TauCeti.IsGoodCycle.apply_eq_apply_self: e · C_i = C_i · C_i for every image.
TauCeti.IsGoodCycle.apply_self_eq_neg_one: images of negative square have square -1.
TauCeti.IsGoodCycle.apply_eq_zero: distinct images are homologically disjoint.
TauCeti.IsGoodCycle.mult_eq_one_of_apply_self_neg: exceptional spheres occur once.
TauCeti.IsGoodCycle.apply_self_sub_apply_canonical_eq_zero_of_one_lt_mult, TauCeti.IsGoodCycle.apply_self_eq_zero_of_one_lt_mult, TauCeti.IsGoodCycle.apply_canonical_eq_zero_of_one_lt_mult, TauCeti.IsGoodCycle.eq_empty_of_one_lt_mult: repeated images have virtual dimension zero, are square-zero and orthogonal to κ, and avoid the points.

References #

C. H. Taubes, SW ⇒ Gr: from the Seiberg–Witten equations to pseudo-holomorphic curves, J. Amer. Math. Soc. 9 (1996), 845--918.
C. H. Taubes, Seiberg–Witten and Gromov invariants for symplectic 4-manifolds, First International Press Lecture Series 2 (2000).