Odd-dimensional Projective Spaces Research Articles

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

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 recursion for counts of real curves in ℂℙ2n−1: Another proof

In a recent paper, we obtained a WDVV-type relation for real genus 0 Gromov–Witten invariants with conjugate pairs of insertions; it specializes to a complete recursion in the case of odd-dimensional projective spaces. This paper provides another, more complex-geometric, proof of the latter. The main part of this approach readily extends to real symplectic manifolds with empty real locus, but not to the general case.

$\mathbb{Z}_2^k$-actions fixing a disjoint union of odd dimensional projective spaces

Consider the real, complex and quaternionic $n$-dimensional projective spaces, $\mathbb{R}P^n$, $\mathbb{C}P^n$ and $\mathbb{H}P^n$; to unify notation, write $K_dP^n$ for the real ($d=1$), complex ($d=2$) and quaternionic ($d=4$) $n$-dimensional projective space. Consider a pair $(M,\Phi)$, where $M$ is a closed smooth manifold and $\Phi$ is a smooth action of the group $\mathbb{Z}_2^k$ on $M$; here, $\mathbb{Z}_2^k$ is considered as the group generated by $k$ commuting smooth involutions $T_1,T_2,...,T_k$. Write $F$ for the fixed-point set of $\Phi$. In this paper we prove the following two results: i) If $F$ is a disjoint union $F=\mathbb{R}P^{n_1} \sqcup \mathbb{R}P^{n_2} \sqcup ... \sqcup \mathbb{R}P^{n_j}$, where $j \ge 2$, each $n_i$ is odd and $n_i \not=n_t$ if $i \not= t$, then $(M,\Phi)$ bounds equivariantly. ii) If $F= K_dP^n \sqcup K_dP^m$, where $d=1,2$ and $4$ and $n$ and $m$ are odd, then $(M,\Phi)$ bounds equivariantly. These results are found in the literature for $k=1$.

Enumeration of real curves in CP2n−1 and a Witten–Dijkgraaf–Verlinde–Verlinde relation for real Gromov–Witten invariants

We establish a homology relation for the Deligne-Mumford moduli spaces of real curves which lifts to a WDVV-type relation for real Gromov-Witten invariants of real symplectic manifolds; we also obtain a vanishing theorem for these invariants. For many real symplectic manifolds, these results reduce all genus 0 real invariants with conjugate pairs of constraints to genus 0 invariants with a single conjugate pair of constraints. In particular, we give a complete recursion for counts of real rational curves in odd-dimensional projective spaces with conjugate pairs of constraints and specify all cases when they are nonzero and thus provide non-trivial lower bounds in high-dimensional real algebraic geometry. We also show that the real invariants of the three-dimensional projective space with conjugate point constraints are congruent to their complex analogues modulo 4.

Number of points of a nonsingular hypersurface in an odd-dimensional projective space

The numbers of Fq-points of nonsingular hypersurfaces of a fixed degree in an odd-dimensional projective space are investigated, and an upper bound for them is given. Also we give the complete list of nonsingular hypersurfaces each of which realizes the upper bound. This is a natural generalization of our previous study of surfaces in projective 3-space.

Real orientations, real Gromov-Witten theory, and real enumerative geometry

The present note overviews our recent construction of real Gromov-Witten theory in arbitrary genera for many real symplectic manifolds, including the odd-dimensional projective spaces and the renowned quintic threefold, its properties, and its connections with real enumerative geometry. Our construction introduces the principle of orienting the determinant of a differential operator relative to a suitable base operator and a real setting analogue of the (relative) spin structure of open Gromov-Witten theory. Orienting the relative determinant, which in the now-standard cases is canonically equivalent to orienting the usual determinant, is naturally related to the topology of vector bundles in the relevant category. This principle and its applications allow 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.

Open Gromov–Witten disk invariants in the presence of an anti-symplectic involution

For a symplectic manifold with an anti-symplectic involution having non-empty fixed locus, we construct a model of the moduli space of real sphere maps out of moduli spaces of decorated disk maps and give an explicit expression for its first Stiefel–Whitney class. As a corollary, we obtain a large number of examples, which include all odd-dimensional projective spaces and many complete intersections, for which many types of real moduli spaces are orientable. For these manifolds, we define open Gromov–Witten invariants with no restriction on the dimension of the manifolds or the type of the constraints if there are no boundary marked points; a WDVV-type recursion obtained in a sequel computes these invariants for many real symplectic manifolds. If there are boundary marked points, we define the invariants under some restrictions on the allowed boundary constraints, even though the moduli spaces are not orientable in these cases.

Rational families of instanton bundles on ℙ2n+1

This paper is devoted to the theory of symplectic instanton bundles on an odd dimensional projective space P 2n+1 with n > 2. We study the ’t Hooft instanton bundles introduced by Ottaviani and a new family of instanton bundles which generalizes one introduced on P 3 independently by Rao and Skiti. The main result is the determination of the birational types of the moduli spaces of ’t Hooft and of Rao{Skiti instanton bundles, respectively. Assuming a conjecture of Ottaviani, we show that the moduli space of all symplectic instanton bundles on P 2n+1 with n > 2 is reducible.

On the [formula omitted]-cohomology of some odd-dimensional projective spaces

Kitchloo and Wilson have used the homotopy fixed points spectrum ER(2) of the classical complex-oriented Johnson–Wilson spectrum E(2) to deduce certain non-immersion results for real projective spaces. ER(n) is a 2n+2(2n−1)-periodic spectrum. The key result to use is the existence of a stable cofibration Σλ(n)ER(n)→ER(n)→E(n) connecting the real Johnson–Wilson spectrum with the classical one. The value of λ(n) is 22n+1−2n+2+1. We extend Kitchloo–Wilsonʼs results on non-immersions of real projective spaces by computing the second real Johnson–Wilson cohomology ER(2) of the odd-dimensional real projective spaces RP16K+9. This enables us to solve certain non-immersion problems of projective spaces using obstructions in ER(2)-cohomology.

THE COMPACTIFIED JACOBIAN OF A REDUCIBLE HYPERELLIPTIC CURVE

There is a one-to-one correspondence between nonsingular pencils of quadrics in an odd-dimensional projective space and nonsingular hyperelliptic curves. There is also a close relation between the moduli spaces of vector bundles of ranks 1, 2 on the hyperelliptic curve and spaces associated to the intersection of quadrics of the corresponding pencil. This has been studied extensively by many authors [1, 5, 8, 9, 11]. An interesting application of this was in the proof of the Verlinde formulae by Szenes [12]. This relationship was generalised to a relation between spaces related to singular pencils of quadrics associated to irreducible nodal curves and moduli of torsion-free sheaves on the nodal curves [2, 4]. The projective models of the moduli of vector bundles on smooth curves (given by this relationship) led to explicit representations of integrable systems of [6, 7]. Similar results on singular curves would provide important classes of special solutions. The pencils of quadrics considered in previous works had associated Segre symbols of type [1 … 1] and [2 … 2 1 … 1]. Kleiman (and Emma Previato) had raised the question: What happens in the case of Segre symbol [2 2 2]? In this case the associated curve Y has two rational components intersecting transversely at three points. Unlike in the previous cases, the compactified Jacobian has more than one component. Moreover the stability of line bundles and the compactification of the Jacobian depends on the choice of a polarisation on Y, making the problem more interesting and difficult.

Involutions Fixing the Disjoint Union of Odd-Dimensional Projective Spaces

AbstractWe show that any differentiable involution on a closed manifold whose fixed point set is a disjoint union of odd-dimensional real projective spaces must be a bounding involution.

Bordism classes of vector bundles over real projective spaces

A basis is presented for the R P n {\mathbf {R}}{{\mathbf {P}}^n} classes in N n ( B O ) {\mathfrak {N}_n}(BO) . This basis is used to prove that every smooth involution ( M , T ) (M,T) on a closed manifold bounds if its fixed point set is a disjoint union of odd-dimensional projective spaces of constant dimension.

On Topological Properties of Some Coverings. An Addendum to a Paper of Lanteri and Struppa

AbstractLet π: X′ → X be a finite surjective morphism of complex projective manifolds which can be factored by an embedding of X′ into the total space of an ample line bundle 𝓛 over X. A theorem of Lazarsfeld asserts that Betti numbers of X and X′ are equal except, possibly, the middle ones. In the present paper it is proved that the middle numbers are actually non-equal if either 𝓛 is spanned and deg π ≥ dim X, or if X is either a hyperquadric or a projective space and π is not a double cover of an odd-dimensional projective space by a hyperquadric.

The hodge structure of the intersection of three quadrics in an odd dimensional projective space

The hodge structure of the intersection of three quadrics in an odd dimensional projective space

Orthogonal line packings of PG2 m − 1 (2)

Baker ( Discrete Math., 15 (1976), 205–211) has shown that there exists a packing of the lines of each odd dimensional projective space over the field of two elements as a corollary to a theorem asserting the existence of a 2-resolution of the Steiner quadruple system of planes in an even dimensional affine space over the field of two elements. Two packings are orthogonal if any two of their spreads have at most one line in common. A variation of the previous construction gives alternate packings so that, for example, the existence of orthogonal packings of PG 2 m − 1 (2) when three does not divide 2 m − 1 can be demonstrated.

Partitioning the planes of AG 2 m(2) into 2-designs

A t-design S λ ( t, k, v) is an arrangement of v elements in blocks of k elements each such that every t element subset is contained in exactly λ blocks. A t-design S λ ( t, k, v) is called t′-resolvable if the blocks can be partitioned into families such that each family is the block system of a S λ ( t′, k, v). It is shown that the S 1(3, 4, 2 2 m ) design of planes on an even dimensional affine space over the field of two elements is 2-resolvable. Each S 1(2, 4, 2 2 m ) given by the resolution is itself 1-resolvable. As a corollary it is shown that every odd dimensional projective space over the field of two elements admits a 1-packing of 1-spreads, i.e. a partition of its lines into families of mutually disjoint lines whose union covers the space. This 1-packing may be generated from any one of its spreads by repeated application of a fixed collineation.

Real Gromov–Witten theory in all genera and real enumerative geometry: Properties

The first part of this work constructs positive-genus real Gromov–Witten invariants of real-orientable symplectic manifolds of odd “complex” dimensions; the present part focuses on their properties that are essential for actually working with these invariants. We determine the compatibility of the orientations on the moduli spaces of real maps constructed in the first part with the standard node-identifying immersion of Gromov–Witten theory. We also compare these orientations with alternative ways of orienting the moduli spaces of real maps that are available in special cases. In a sequel, we use the properties established in this paper to compare real Gromov–Witten and enumerative invariants, to describe equivariant localization data that computes the real Gromov–Witten invariants of odd-dimensional projective spaces, and to establish vanishing results for these invariants in the spirit of Walcher’s predictions.