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Invariants of real symplectic 4-manifolds and lower bounds in real enumerative geometry. (English) Zbl 1082.14052
This outstanding work opens a new direction in the enumerative geometry: the author discovers an enumerative invariant of real rational pseudo-holomorphic curves on real rational symplectic four-folds, which is a real analogue of the Gromov-Witten invariants. Namely, given a real rational symplectic four-fold $X$ with a generic tame real almost complex structure $J$, an ample class $d\in H_2(X)$, and a generic equivariant configuration $x$ of $c_1(X)\cdot d-1$ points in $X$, the set of real rational pseudo-holomorphic curves in $X$, realizing the class $d$ and passing through the configuration $x$, is finite; counting these curves $C$ with weights $(-1)^{s(C)}$, where $s(C)$ is the number of real isolated nodes of $C$ (i.e., locally given by $x^2+y^2=0$), one obtains the number $\chi$. The main theorem states that $\chi$ does not depend neither on the choice of $J$, nor on the choice of $x$, but only on $X$, $d$, the number of real points in $x$, and their distribution among the connected components of $X({\Bbb R})$. In particular, this theorem covers the case of real generic del Pezzo surfaces and rational algebraic curves in them. The absolute value of $\chi$ provides a uniform lower bound for the number of real rational pseudo-holomorphic curves in $X$ homologous to $d$ and passing through any generic equivariant configuration of $c_1(X)\cdot d-1$ points, whereas an upper bound is given by the corresponding Gromov-Witten invariant. The author shows that it is non-trivial, i.e., does not vanish identically, and establishes relations between invariants, corresponding to configurations, containing different amounts of real points. Reviewer’s remark. {\it G. Mikhalkin} [C. R., Math., Acad. Sci. Paris 336, No. 8, 629--634 (2003; Zbl 1027.14026)], {\it I. Itenberg, V. Kharlamov}, and the reviewer [Int. Math. Res. Not. 2003, No.49, 2639--2653 (2003; Zbl 1083.14523)] have shown that the new invariant is always positive for totally real configurations in real toric del Pezzo surfaces, which, in particular, implies that for any such surface $X$, and ample class $d\in H_2(X)$, through any generic configuration of $c_1(X)\cdot d-1$ generic real points in $X$ one can trace a real rational curve.

##### MSC:
 14N10 Enumerative problems (algebraic geometry) 14N35 Gromov-Witten invariants, quantum cohomology, etc. 53D05 Symplectic manifolds, general 53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds 14P05 Real algebraic sets
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##### References:
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