We construct positive-genus analogues of Welschinger's invariants for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold. In some cases, our invariants provide lower bounds for counts of real positive-genus curves in real algebraic varieties. Our approach to the orientability problem is based entirely on the topology of real bundle pairs over symmetric surfaces; the previous attempts involved direct computations for the determinant lines of Fredholm operators over bordered surfaces. We use the notion of real orientation introduced in this paper to obtain isomorphisms of real bundle pairs over families of symmetric surfaces and then apply the determinant functor to these isomorphisms. This allows us to endow the uncompactified moduli spaces of real maps from symmetric surfaces of all topological types with natural orientations and to verify that they extend across the codimension-one boundaries of these spaces, thus implementing a far-reaching proposal from C.-C. Liu's thesis for a fully fledged real Gromov-Witten theory. The second and third parts of this work concern applications: they describe important properties of our orientations on the moduli spaces, establish some connections with real enumerative geometry, provide the relevant equivariant localization data for projective spaces, and obtain vanishing results in the spirit of Walcher's predictions.

A desingularization of the main component of the moduli space of genus-one stable maps into P n RAVI VAKIL ALEKSEY ZINGERWe construct a natural smooth compactification of the space of smooth genus-one curves with k distinct points in a projective space. It can be viewed as an analogue of a well-known smooth compactification of the space of smooth genus-zero curves, that is, the space of stable genus-zero maps S M 0;k .P n ; d /. In fact, our compactification is obtained from the singular space of stable genus-one maps S M 1;k .P n ; d / through a natural sequence of blowups along "bad" subvarieties. While this construction is simple to describe, it requires more work to show that the end result is a smooth space.As a bonus, we obtain desingularizations of certain natural sheaves over the "main" irreducible component S M 0 1;k .P n ; d/ of S M 1;k .P n ; d/. A number of applications of these desingularizations in enumerative geometry and Gromov-Witten theory are described in the introduction, including the second author's proof of physicists' predictions for genus-one Gromov-Witten invariants of a quintic threefold. 14D20; 53D991 Introduction Background and applicationsThe space of degree-d genus-g curves with k distinct marked points in P n is generally not compact, but admits a number of natural compactifications 1 . Among the most prominent compactifications is the moduli space of stable genus-g maps, S M g;k .P n ; d/, constructed by Gromov [9] and Fulton-Pandharipande [6]. It has found numerous applications in classical enumerative geometry and is a central object in Gromov-Witten theory. However, most applications in enumerative geometry and some results in GWtheory have been restricted to the genus-zero case. The reason for this is essentially that the genus-zero moduli space has a particularly simple structure: it is smooth and contains the space of smooth genus-zero curves as a dense open subset. hand, the moduli spaces of positive-genus stable maps fail to satisfy either of these two properties. In fact, S M g;k .P n ; d / can be arbitrarily singular according to Vakil [19]. It is thus natural to ask whether these failings can be remedied by modifying S M g;k .P n ; d/, preferably in a way that leads to a range of applications. As announced in [20] and shown in this paper, the answer is yes if gD1.We denote by M 1;k .P n ; d/ the subset of S M 1;k .P n ; d/ consisting of the stable maps that have smooth domains. This space is smooth and contains the space of genusone curves with k distinct marked points in P n as a dense open subset, provided d 3. However, M 1;k .P n ; d / is not compact. Let S M 0 1;k .P n ; d/ be the closure of M 1;k .P n ; d/ in the compact space S M 1;k .P n ; d/. While S M 0 1;k .P n ; d/ is not smooth, it turns out that a natural sequence of blowups along loci disjoint from M 1;k .P n ; d/ leads to a desingularization of S M 0 1;k .P n ; d/, which will be denoted byThe situation is as good as one could possibly hope. A general strategy when attempting to desingularize some space i...

We compute the reduced genus 1 Gromov-Witten invariants of Calabi-Yau hypersurfaces. As a consequence, we confirm the 1993 Bershadsky-Cecotti-Ooguri-Vafa (BCOV) prediction for the standard genus 1 GW-invariants of a quintic threefold. We combine constructions from a series of previous papers with the classical localization theorem to relate the reduced genus 1 invariants of a CY-hypersurface to previously computed integrals on moduli spaces of stable genus 0 maps into projective space. The resulting, rather unwieldy, expressions for a genus 1 equivariant generating function simplify drastically, using a regularity property of a genus 0 equivariant generating function in half of the cases. Finally, by disregarding terms that cannot effect the non-equivariant part of the former, we relate the answer to an explicit hypergeometric series in a simple way. The approach described in this paper is systematic. It is directly applicable to computing reduced genus 1 GW-invariants of other complete intersections and should apply to higher-genus localization computations. * Partially supported by a Sloan fellowship and DMS Grant 0604874 1 It is equivalent to the associativity of the multiplication in quantum cohomology. 2 see Appendix B for a comparison of statements of mirror symmetry. A prediction for the genus 1 GW-invariants of X 5 was made in [BCOV], building up on [CaDGP]. Both of these predictions date back to the early days of the Gromov-Witten theory. More recently, predictions for highergenus GW-invariants of X 5 have been made; the approach of [HKlQ] generates mirror formulas for GW-invariants of X 5 up to genus 51.While the ODE condition on GW-invariants mentioned above is proved directly, the mathematical approach to the mirror principle has been to compute the relevant GW-invariants in each specific case. However, this is rarely a simple task. The prediction for genus 0 invariants was confirmed mathematically in the mid-1990s. The prediction for genus 1 invariants is verified in this paper.Theorem 1 If N 1,d denotes the degree d genus 1 Gromov-Witten invariant of a quintic threefold,This theorem is deduced from Theorem 2 in Subsection 0.3. An outline of this paper is contained in the next subsection.The author is very grateful to D. Zagier, for taking interest in the statements of and collaborating on the proof of Propositions 3.1 and 3.2, to Jun Li, for bringing [BCOV] to the author's attention and collaborating on related work concerning genus 1 GW-invariants, to R. Pandharipande, for generously sharing his expertise in the Gromov-Witten theory, and the referee, for suggestions on improving the exposition. The author would also like to thank One approach to computing GW-invariants of a projective hypersurface (and more generally, of a complete intersection) is to relate them to GW-invariants of the ambient projective space as follows. Whenever g, d, and k are nonnegative integers and X is a smooth subvariety of P n , denote by M g,k (X, d) the moduli space of stable degree-d maps into X from genus g curves wit...

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The main concrete result of this paper is enumeration of genus-two curves with complex structure fixed in P 2 and P 3 . Along the way, rational curves with certain simple singularities are counted as well. While the methods described can be used to count positive-genus curves in some other cases, the most powerful direct applications of the machinery developed are to enumeration of rational curves with a very large class of singularities in projective spaces.