Complete Intersections In Projective Space Research Articles

Conjecture 𝒪 for Projective Complete Intersections

Abstract Conjecture ${{\mathcal{O}}}$ and Gamma conjectures for quantum cohomology of Fano manifolds were proposed by Galkin, Golyshev, and Iritani [ 16]. We prove Conjecture ${{\mathcal{O}}}$ for Fano complete intersections in projective spaces. The main tools to compute relevant genus-zero Gromov–Witten invariants with primitive insertions are the genus-one topological recursion relation and Zinger’s standard-versus-reduced formula. We also prove a related conjecture of Galkin [ 15] for projective Fano complete intersections.

Euler-symmetric complete intersections in projective space

Euler-symmetric complete intersections in projective space

Gromov–Witten theory of complete intersections via nodal invariants

AbstractWe provide an inductive algorithm computing Gromov–Witten invariants in all genera with arbitrary insertions of all smooth complete intersections in projective space. We also prove that all Gromov–Witten classes of all smooth complete intersections in projective space belong to the tautological ring of the moduli space of stable curves. The main idea is to show that invariants with insertions of primitive cohomology classes are controlled by their monodromy and by invariants defined without primitive insertions but with imposed nodes in the domain curve. To compute these nodal Gromov–Witten invariants, we introduce the new notion of nodal relative Gromov–Witten invariants. We then prove a nodal degeneration formula and a relative splitting formula. These results for nodal relative Gromov–Witten theory are stated in complete generality and are of independent interest.

Algebraic reduced genus one Gromov–Witten invariants for complete intersections in projective spaces, Part 2

In Lee and Oh (Preprint, arXiv:1809.10995, 2018), we provided an algebraic proof of Zinger's comparison formula (Geom. Topol. 12 (2008), no. 2, 1203–1241; J. Differential Geom. 83 (2009), no. 2, 407–460) between genus one Gromov–Witten invariants and reduced invariants when the target space is a complete intersection of dimension 2 or 3 in a projective space. In this paper, we extend the proof in Lee and Oh (Preprint, arXiv:1809.10995, 2018) to all dimensions and to descendant invariants.

Moduli Space of Quasi-Polarized K3 Surfaces of Degree 6 and 8

In this paper, the authors study the moduli space of quasi-polarized complex K3 surfaces of degree 6 and 8 via geometric invariant theory. The general members in such moduli spaces are complete intersections in projective spaces and they have natural GIT constructions for the corresponding moduli spaces and they show that the K3 surfaces with at worst ADE singularities are GIT stable. They give a concrete description of boundary of the compactification of the degree 6 case via the Hilbert-Mumford criterion. They compute the Picard group via Noether-Lefschetz theory and discuss the connection to the Looijenga’s compactifications from arithmetic perspective. One of the main ingredients is the study of the projective models of K3 surfaces in terms of Noether-Lefschetz divisors.

A note on a Griffiths-type ring for complete intersections in Grassmannians

We calculate a Griffiths-type ring for smooth complete intersections in Grassmannians. This is the analogue of the classical Jacobian ring for complete intersections in projective space and allows us to explicitly compute their Hodge groups.

Grothendieck�Lefschetz for vector bundles

According to the Grothendieck-Lefschetz theorem from SGA 2, there are no nontrivial line bundles on the punctured spectrum $U_R$ of a local ring $R$ that is a complete intersection of dimension $\ge 4$. Dao conjectured a generalization for vector bundles $\mathscr{V}$ of arbitrary rank on $U_R$: such a $\mathscr{V}$ is free if and only if $\mathrm{depth}_R(\mathrm{End}_R(\Gamma(U_R, \mathscr{V}))) \ge 4$. We use deformation theoretic techniques to settle Dao's conjecture. We also present examples showing that its assumptions are sharp and draw consequences for splitting of vector bundles on complete intersections in projective space.

Virtual complete intersections in [formula omitted

The minimal free resolution of the coordinate ring of a complete intersection in projective space is a Koszul complex on a regular sequence. In the product of projective spaces P1×P1, we investigate which sets of points have a virtual resolution that is a Koszul complex on a regular sequence. This paper provides conditions on sets of points; some of which guarantee the points have this property, and some of which guarantee the points do not have this property.

Quasimap wall-crossings and mirror symmetry

We state a wall-crossing formula for the virtual classes of epsilon-stable quasimaps to GIT quotients and prove it for complete intersections in projective space, with no positivity restrictions on their first Chern class. As a consequence, the wall-crossing formula relating the genus g descendant Gromov-Witten potential and the genus g epsilon-quasimap descendant potential is established. For the quintic threefold, our results may be interpreted as giving a rigorous and geometric interpretation of the holomorphic limit of the BCOV B-model partition function of the mirror family.

Algebraic Reduced Genus One Gromov–Witten Invariants for Complete Intersections in Projective Spaces

Abstract In [ 17, 18], Zinger defined reduced Gromov–Witten (GW) invariants and proved a comparison theorem of standard and reduced genus one GW invariants for every symplectic manifold (with all dimension). In [ 3], Chang and Li provided a proof of the comparison theorem for quintic Calabi–Yau three-folds in algebraic geometry by taking a definition of reduced invariants as an Euler number of certain vector bundle. In [ 5], Coates and Manolache have defined reduced GW invariants in algebraic geometry following the idea by Vakil and Zinger in [ 14] and proved the comparison theorem for every Calabi–Yau threefold. In this paper, we prove the comparison theorem for every (not necessarily Calabi–Yau) complete intersection of Dimension 2 or 3 in projective spaces by taking a definition of reduced GW invariants in [ 5].

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

Gromov–Witten invariants of real-orientable symplectic manifolds of odd “complex” dimensions; the second part studies the orientations on the moduli spaces of real maps used in constructing these invariants. The present paper applies the results of the latter to obtain quantitative and qualitative conclusions about the invariants defined in the former. After describing large collections of real-orientable symplectic manifolds, we show that the genus $1$ real Gromov–Witten invariants of sufficiently positive almost Kahler threefolds are signed counts of real genus 1 curves only and, thus, provide direct lower bounds for the counts of these curves in such targets. We specify real orientations on the real-orientable complete intersections in projective spaces; the real Gromov–Witten invariants they determine are in a sense canonically determined by the complete intersection itself, (at least) in most cases. We also obtain equivariant localization data that computes the real invariants of projective spaces and determines the contributions from many torus fixed loci for other complete intersections. Our results confirm Walcher’s predictions for the vanishing of these invariants in certain cases and for the localization data in other cases. The localization data is also used to demonstrate the non-triviality of our lower bounds for real curves of genus $1$ in the present paper and of higher genera in a separate paper.

Mirror Theorem for Elliptic Quasimap Invariants of Local Calabi–Yau Varieties

The elliptic quasi-map potential function is explicitly calculated for Calabi–Yau complete intersections in projective spaces in [13]. We extend this result to local Calabi–Yau varieties. Using this and the wall crossing formula in [5], we can calculate the elliptic Gromov–Witten potential function.

Mirror theorem for elliptic quasimap invariants

We propose and prove a mirror theorem for the elliptic quasimap invariants for smooth Calabi-Yau complete intersections in projective spaces. The theorem combined with the wall-crossing formula appeared in paper (arXiv:1308.6377) implies mirror theorems of Zinger and Popa for the elliptic Gromov-Witten invariants for those varieties. This paper and the wall-crossing formula provide a unified framework for the mirror theory of rational and elliptic Gromov-Witten invariants.

Systems of cubic forms in many variables

Abstract We consider a system of R cubic forms in n variables, with integer coefficients, which define a smooth complete intersection in projective space. Provided n ≥ 25 ⁢ R {n\geq 25R} , we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish. In particular, we obtain the Hasse principle for systems of cubic forms in 25 ⁢ R {25R} variables, previous work having required that n ≫ R 2 {n\gg R^{2}} . One conjectures that n ≥ 6 ⁢ R + 1 {n\geq 6R+1} should be sufficient. We reduce the problem to an upper bound for the number of solutions to a certain auxiliary inequality. To prove this bound we adapt a method of Davenport.

Rational curves on complete intersections in positive characteristic

We study properties of rational curves on complete intersections in positive characteristic. It has long been known that in characteristic 0, smooth Calabi–Yau and general type varieties are not uniruled. In positive characteristic, however, there are well-known counterexamples to this statement. We will show that nevertheless, a general Calabi–Yau or general type complete intersection in projective space is not uniruled. We will also show that the space of complete intersections of degree (d1,⋯,dk) containing a rational curve has codimension at least ∑i=1kdi−2n+2 and give similar results for hypersurfaces containing higher genus curves.

Counting rational points on hypersurfaces and higher order expansions

We study the number of representations of an integer n=F(x) by a homogeneous form in sufficiently many variables. This is a classical problem in number theory to which the circle method has been successfully applied to give an asymptotic for the number of such representations where the integer vector x is restricted to a box of side length P for P sufficiently large. In the special case of Waring's problem, Vaughan and Wooley have recently established for the first time a higher order expansion for the corresponding asymptotic formula. Via a different and much more general approach we derive a multi-term asymptotic for this problem for general forms F(x) and give an interpretation for the occurring lower order terms.As an application we derive higher order expansions for the number of rational points of bounded anticanonical height on the projective hypersurface F(x)=0 for forms F(x) in sufficiently many variables. The main term of this expansion is the one predicted by Manin's conjecture. Our new result gives some evidence for how the conjecture could be refined to cover lower order terms in the setting of high-dimensional complete intersections in projective space.

Complete intersections: moduli, Torelli, and good reduction

We study the arithmetic of complete intersections in projective space over number fields. Our main results include arithmetic Torelli theorems and versions of the Shafarevich conjecture, as proved for curves and abelian varieties by Faltings. For example, we prove an analogue of the Shafarevich conjecture for cubic and quartic threefolds and intersections of two quadrics.

ON THE MODIFIED FUTAKI INVARIANT OF COMPLETE INTERSECTIONS IN PROJECTIVE SPACES

Let $M$ be a Fano manifold. We call a Kähler metric ${\it\omega}\in c_{1}(M)$ a Kähler–Ricci soliton if it satisfies the equation $\text{Ric}({\it\omega})-{\it\omega}=L_{V}{\it\omega}$ for some holomorphic vector field $V$ on $M$. It is known that a necessary condition for the existence of Kähler–Ricci solitons is the vanishing of the modified Futaki invariant introduced by Tian and Zhu. In a recent work of Berman and Nyström, it was generalized for (possibly singular) Fano varieties, and the notion of algebrogeometric stability of the pair $(M,V)$ was introduced. In this paper, we propose a method of computing the modified Futaki invariant for Fano complete intersections in projective spaces.

Spectral sequences and vacua in N = 2 $$ \mathcal{N}=2 $$ gauged linear quantum mechanics with potentials

We study the behaviour of supersymmetric ground states in a class of one-dimensional N = 2 abelian gauged linear sigma models, including theories for which the target space is a complete intersection in projective space, and more generally, models with an interaction term introduced by Herbst, Hori and Page in which the vacua correspond to elements of hypercohomology groups of complexes of sheaves. Combining physical insights from recent work by Hori, Kim and Yi with the use of spectral sequences, we propose a way to reconcile the non-linear sigma model description, valid deep within a geometric phase, with the effective Coulomb branch description, valid near a phase boundary. This leads to a physical interpretation of the hypercohomology groups from the perspective of the Coulomb branch, as well as an interpretation for the spectral sequences used to compute them.

Holomorphic Yukawa couplings in heterotic string theory

We develop techniques, based on differential geometry, to compute holomorphic Yukawa couplings for heterotic line bundle models on Calabi-Yau manifolds defined as complete intersections in projective spaces. It is shown explicitly how these techniques relate to algebraic methods for computing holomorphic Yukawa couplings. We apply our methods to various examples and evaluate the holomorphic Yukawa couplings explicitly as functions of the complex structure moduli. It is shown that the rank of the Yukawa matrix can decrease at specific loci in complex structure moduli space. In particular, we compute the up Yukawa coupling and the singlet-Higgs-lepton trilinear coupling in the heterotic standard model described in arXiv:1404.2767