Real Gromov-Witten theory in all genera and real enumerative geometry: Construction

Abstract

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.

Some Questions in the Theory of Pseudoholomorphic Curves

“…The counts of genus g degree d curves arising from the proofs of the mirror symmetry predictions for the projective CY complete intersections in genus 0 in [34,59] and in genus 1 in [106,83] via (2.14) have been shown to match the classical enumerative counts for g = 0, d ≤ 3 and for g = 1, d ≤ 4; see [12]. The genus 0 real GW-invariants of real symplectic fourfolds defined in [98] and of many higher-dimensional real symplectic manifolds defined in [26] directly provide lower bounds for counts of genus 0 real curves; the arbitrary genus real GW-invariants defined in [30] provide such bounds in arbitrary genera via the relation (2.16) proved in [68]. For local CY manifolds, Question 4 points to intriguing number-theoretic properties of GW-invariants; see G. Martin's conjecture in [74,Section 3.2].…”

“…So, the real analogues of (2.4) and (2.5) in this case also reduce to GW φ 0,A = E φ 0,A . Arbitrary genus real GW-invariants are constructed in [30] for many real symplectic manifolds, including the odddimensional projective spaces P 2n−1 and quintic threefolds X 5 ⊂ P 4 cut out by real equations. It is established in [68] that the analogue of (2.9) for the Fano classes A on a real symplectic sixfold…”

“…The approach of [46] to the integrality of the numbers n X g,A determined by (2.13) should be adaptable to other situations when the GW-invariants in question are expected to arise entirely from isolated J-holomorphic curves. These situations include the real genus 0 GW-invariants of many real symplectic manifolds and the real arbitrary genus GW-invariants of real symplectic CY sixfolds constructed in [26] and [30], respectively. In fact, the integrality of the numbers E X 0,A (µ) determined by (2.15) is already a (secondary) subject of [46].…”

Welschinger invariants of blow-ups of symplectic 4-manifolds

“…In 2005, Welschinger [36] first discovered such an invariant in dimensions 4 and 6, which was called the Welschinger invariant and revolutionized the real enumerative geometry. Recently it was partially extended to higher dimensions, higher genera and descendant type, see [8,31,32] and the references therein for the details. Itenberg-Kharlamov-Shustin [23] also extended the algebraic definition of Welschinger invariants to all del Pezzo surfaces and proved the invariance under deformation in algebraic setting.…”

Lower bounds and asymptotics of real double Hurwitz numbers

“…Such investigations have been carried out e.g. for real Schubert calculus [Sot97; MT16], counts of algebraic curves in surfaces passing through points [Wel05; IKS04] (see also [GZ15]) and counts of polynomials/simple rational functions with given critical levels [IZ16; ER17]. In most of these examples, a lower bound is constructed by defining a signed count of the real solutions (i.e., each real solution is counted with +1 or −1 according to some rule) and showing that this signed count is invariant under change of the conditions.…”