Spinor states of real rational curves in real algebraic convex 3-manifolds and enumerative invariants
This paper makes a next step in the creation of the theory of enumerative invariants of real algebraic varieties, initiated by the author. In his pioneering work [Invent. Math. 162, No. 1, 195--234 (2005; Zbl 1082.14052)] the author has defined enumerative invariants of real rational symplectic four-folds, which count real rational pseudo-holomorphic curves, belonging to a given homology class and passing through a real configuration of points, with weights \(\pm 1\), and which depend only on the distribution of real fixed points among the connected components of the real part of the given symplectic variety. In the present paper, the author defines invariants of real convex smooth algebraic 3-folds equipped with a Pin\(^-\) structure. Namely, given such a variety \(X\) and a homology class \(d\in H_2(X)\) such that \(c_1(X)d\) is positive even, the set \(R(\omega)\) of real rational curves passing through a generic real configuration \(\omega\) of \(c_1(X)d/2\) points in \(X\) is finite. Each curve in \(R(\omega)\) gives a loop in the principal \(\text{ O}(3)\)-bundle on \({\mathbb R}X\), and the curve obtains weight \(+1\) or \(-1\) depending on whether the loop lifts to a loop in the respective Pin\(^-\)-bundle on \({\mathbb R}X\) or not. The main theorem states that the total sum of the weights of the curves counted does not depend on the chosen configuration \(\omega\), but only on the distribution of the fixed points among the connected components of \({\mathbb R}X\). As an immediate consequence, the absolute value of the invariant provides a lower bound for the number of real rational curves in the class \(d\) passing through any generic real configuration of \(c_1(X)d/2\) points with a given distribution among the components of \({\mathbb R}X\) (notice that the corresponding Gromov-Witten invariant serves as an upper bound). Another important result says that, in case \(X={\mathbb P}^3\), when a pair or real points in a configuration turns into a pair of imaginary conjugate points, the jump of the invariant is twice the invariant for the projective 3-space blown up at a real point. The proof consists in the study of the moduli spaces of real stable maps of rational pointed curves into \(X\) and of the bifurcations of the evaluation map. The crucial bifurcation is a passage of a configuration \(\omega\) through a critical value of the evaluation map, when a pair of real curves in \(R(\omega)\) disappear (they turn into a pair of imaginary conjugate curves), but the weights of these two curves cancel each other.
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