Real Algebraic and Analytic Geometry |

Spinor states of real rational curves in real algebraic convex $3$-manifolds and enumerative invariants.

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Submission: 2004, January 6.

*Abstract:
Let $X$ be a real algebraic convex $3$-manifold whose real part is equipped
with a $Pin^-$ structure. We show that every irreducible real rational curve
with non-empty real part has a canonical spinor state belonging to $\{\pm 1\}$.
The main result is then that the algebraic count of the number of
real irreducible rational curves in a given numerical equivalence class passing
through the appropriate number of points does not depend on the choice of the real
configuration of points, provided that these curves are counted with respect to their spinor states.
These invariants provide lower bounds for the total number of such real rational
curves independantly of the choice of the real configuration of points.*

Mathematics Subject Classification (2000): 14N35, 14P25.

Keywords and Phrases: Convex manifold, real algebraic manifold, stable map, enumerative geometry.

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