SYZ Conjecture Research Articles

Metric SYZ conjecture for certain toric Fano hypersurfaces

Metric SYZ conjecture for certain toric Fano hypersurfaces

Metric SYZ conjecture and non-Archimedean geometry

Metric SYZ conjecture and non-Archimedean geometry

Parabolic automorphisms of hyperkähler manifolds

A parabolic automorphism of a hyperkähler manifold M is a holomorphic automorphism acting on H2(M) by a non-semisimple quasi-unipotent linear map. We prove that a parabolic automorphism which preserves a Lagrangian fibration acts on almost all fibers ergodically. The existence of an invariant Lagrangian fibration is automatic for manifolds satisfying the hyperkähler SYZ conjecture; this includes all known examples of hyperkähler manifolds. When there are two parabolic automorphisms preserving two distinct Lagrangian fibrations, it follows that the group they generate acts on M ergodically. Our results generalize those obtained by S. Cantat for K3 surfaces.

Computing Riemann–Roch polynomials and classifying hyper-Kähler fourfolds

We prove that a hyper-Kähler fourfold satisfying a mild topological assumption is of K3 [ 2 ] ^{[2]} deformation type. This proves in particular a conjecture of O’Grady stating that hyper-Kähler fourfolds of K3 [ 2 ] ^{[2]} numerical type are of K3 [ 2 ] ^{[2]} deformation type. Our topological assumption concerns the existence of two integral degree-2 cohomology classes satisfying certain numerical intersection conditions. There are two main ingredients in the proof. We first prove a topological version of the statement, by showing that our topological assumption forces the Betti numbers, the Fujiki constant, and the Huybrechts–Riemann–Roch polynomial of the hyper-Kähler fourfold to be the same as those of K3 [ 2 ] ^{[2]} hyper-Kähler fourfolds. The key part of the article is then to prove the hyper-Kähler SYZ conjecture for hyper-Kähler fourfolds for divisor classes satisfying the numerical condition mentioned above.

Survey on the metric SYZ conjecture and non-Archimedean geometry

We survey the metric aspects of the Strominger–Yau–Zaslow conjecture on the existence of special Lagrangian fibrations on Calabi–Yau manifolds near the large complex structure limit. We will discuss the diverse motivations for the conjectural picture, what the best hopes are, and a number of subtleties. The bulk of the survey highlights the role of pluripotential theory, and non-Archimedean geometry in particular, with a list of open questions.

Effectivity of Semi-positive Line Bundles

We review work by Campana–Oguiso–Peternell (J Differ Geom 85(3):397–424, 2010) and Verbitsky (Geom Funct Anal 19(5):1481–1493, 2010) showing that a semi-positive line bundle on a hyperkähler manifold admits at least one non-trivial section. This is modest but tangible evidence towards the SYZ conjecture for hyperkähler manifolds.

On the Lefschetz standard conjecture for Lagrangian covered hyper-Kähler varieties

We investigate the Lefschetz standard conjecture for degree 2 cohomology of hyper-Kähler manifolds admitting a covering by Lagrangian subvarieties. In the case of a Lagrangian fibration, we show that the Lefschetz standard conjecture is implied by the SYZ conjecture characterizing classes of divisors associated with Lagrangian fibration. In dimension 4, we consider the more general case of a Lagrangian covered fourfold X, and prove the Lefschetz standard conjecture in degree 2, assuming ρ(X)=1 and X is general in moduli. Finally we discuss various links between Lefschetz cycles and the study of the rational equivalence of points and Bloch-Beilinson type filtrations, giving a general interpretation of a recent intriguing result of Marian and Zhao.

Moduli-dependent Calabi-Yau and SU(3)-structure metrics from machine learning

We use machine learning to approximate Calabi-Yau and SU(3)-structure metrics, including for the first time complex structure moduli dependence. Our new methods furthermore improve existing numerical approximations in terms of accuracy and speed. Knowing these metrics has numerous applications, ranging from computations of crucial aspects of the effective field theory of string compactifications such as the canonical normalizations for Yukawa couplings, and the massive string spectrum which plays a crucial role in swampland conjectures, to mirror symmetry and the SYZ conjecture. In the case of SU(3) structure, our machine learning approach allows us to engineer metrics with certain torsion properties. Our methods are demonstrated for Calabi-Yau and SU(3)-structure manifolds based on a one-parameter family of quintic hypersurfaces in ℙ4.

Motivic zeta functions of degenerating Calabi–Yau varieties

We study motivic zeta functions of degenerating families of Calabi–Yau varieties. Our main result says that they satisfy an analog of Igusa’s monodromy conjecture if the family has a so-called Galois equivariant Kulikov model; we provide several classes of examples where this condition is verified. We also establish a close relation between the zeta function and the skeleton that appeared in Kontsevich and Soibelman’s non-archimedean interpretation of the SYZ conjecture in mirror symmetry.

The SYZ mirror symmetry and the BKMP remodeling conjecture

The Remodeling Conjecture proposed by Bouchard-Klemm-Marino-Pasquetti (BKMP) relates the A-model open and closed topological string amplitudes (open and closed Gromov-Witten invariants) of a symplectic toric Calabi-Yau 3-fold to Eynard-Orantin invariants of its mirror curve. The Remodeling Conjecture can be viewed as a version of all genus open-closed mirror symmetry. The SYZ conjecture explains mirror symmetry as $T$-duality. After a brief review on SYZ mirror symmetry and mirrors of symplectic toric Calabi-Yau 3-orbifolds, we give a non-technical exposition of our results on the Remodeling Conjecture for symplectic toric Calabi-Yau 3-orbifolds. In the end, we apply SYZ mirror symmetry to obtain the descendent version of the all genus mirror symmetry for toric Calabi-Yau 3-orbifolds.

A glimpse of the SYZ conjecture and related developments

A glimpse of the SYZ conjecture and related developments

Kobayashi pseudometric on hyperkähler manifolds

The Kobayashi pseudometric on a complex manifold is the maximal pseudometric such that any holomorphic map from the Poincar\'e disk to the manifold is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this conjecture for any hyperk\"ahler manifold that admits a deformation with two Lagrangian fibrations and whose Picard rank is not maximal. The Strominger-Yau-Zaslow (SYZ) conjecture claims that parabolic nef line bundles on hyperk\"ahler manifolds are semi-ample. We prove that the Kobayashi pseudometric vanishes for any hyperk\"ahler manifold with $b_2\geq 13$ if the SYZ conjecture holds for all its deformations. This proves the Kobayashi conjecture for all K3 surfaces and their Hilbert schemes.

Mirror symmetry and the Strominger-Yau-Zaslow conjecture

This survey was written for the Current Developments in Mathematics conference, 2012, and is an updating of my article The Strominger-Yau-Zaslow conjecture: From torus fibrations to degenerations, in the Seattle 2005 proceedings. We trace progress and thinking about the SYZ conjecture since its introduction in 1996. We begin with the original differential geometric conjecture and its refinements, and explain how it led to the algebro-geometric program developed by myself and Siebert. After explaining the overall philosophy, I explain how recent results fit into this program.

SYZ mirror symmetry for toric Calabi-Yau manifolds

We investigate mirror symmetry for toric Calabi-Yau manifolds from the perspective of the SYZ conjecture. Starting with a non-toric special Lagrangian torus fibration on a toric Calabi-Yau manifold $X$, we construct a complex manifold $\check{X}$ using T-duality modified by quantum corrections. These corrections are encoded by Fourier transforms of generating functions of certain open Gromov-Witten invariants. We conjecture that this complex manifold $\check{X}$, which belongs to the Hori-Iqbal-Vafa mirror family, is inherently written in canonical flat coordinates. In particular, we obtain an enumerative meaning for the (inverse) mirror maps, and this gives a geometric reason for why their Taylor series expansions in terms of the K\"ahler parameters of $X$ have integral coefficients. Applying the results in \cite{Chan10} and \cite{LLW10}, we compute the open Gromov-Witten invariants in terms of local BPS invariants and give evidences of our conjecture for several 3-dimensional examples including $K_{\proj^2}$ and $K_{\proj^1\times\proj^1}$.

Hyperkähler Syz Conjecture and Semipositive Line Bundles

Let M be a compact, holomorphic symplectic Kahler manifold, and L a non-trivial line bundle admitting a metric of semipositive curvature. We show that some power of L is effective. This result is related to the hyperkahler SYZ conjecture, which states that such a manifold admits a holomorphic Lagrangian fibration, if L is not big.

Calabi–Yau varieties with semi-stable fibre structures

Motivated by the Strominger–Yau–Zaslow conjecture, we study Calabi–Yau varieties with semi-stable fibre structures. We use Hodge theory to study the higher direct images of wedge products of relative cotangent sheaves of certain semi-stable families over higher dimensional quasi-projective bases, and obtain some results on positivity. We then apply these results to study non-isotrivial Calabi–Yau varieties fibred by semi-stable Abelian varieties (or hyperkahler varieties).

Torus fibrations, gerbes, and duality

Introduction The Brauer group and the Tate-Shafarevich group Smooth genus one fibrations Surfaces Modified $T$-duality and the SYZ conjecture Appendix A. Duality for representations of $1$-motives, by Dmitry Arinkin Bibliography.

Singularities of special Lagrangian fibrations and the SYZ Conjecture

The SYZ Conjecture explains Mirror Symmetry between Calabi–Yau 3-folds M, ˆ M in terms of special Lagrangian fibrations f : M ! B and ˆ f : ˆ M ! B over the same base B, whose fibres are dual 3-tori, except for singular fibres. This paper studies the singularities of special Lagrangian fibrations. We construct many examples of special Lagrangian fibrations on open subsets of C 3 . The simplest are given explicitly, and the rest use analytic existence results for U(1)-invariant special Lagrangian 3-folds in C 3 . We then argue that some features of our examples should also hold for generic special Lagrangian fibrations of (almost) Calabi–Yau 3-folds, and draw some conclusions on the SYZ Conjecture.

Special Lagrangian Submanifolds with Isolated Conical Singularities. v. Survey and Applications

This is the last in a series of five papers studying compact special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x1, . . . , xn locally modelled on special Lagrangian cones C1, . . . , Cn in ℂm with isolated singularities at 0. Readers are advised to begin with this paper. We survey the major results of the previous four papers, giving brief explanations of the proofs. We apply the results to describe the boundary of a moduli space of compact, nonsingular SL m-folds N in M. We prove the existence of special Lagrangian connected sums N1#...#Nk of SL m-folds N1, . . . , Nk in M. We also study SL 3-folds with T2-cone singularities, proving results related to ideas of the author on invariants of Calabi-Yau 3-folds, and the SYZ Conjecture. Let X be a compact SL m-fold with isolated conical singularities xi and cones Ci for i = 1, . . . , n. The first paper studied the regularity of X near its singular points, and the the second the moduli space of deformations of X. The third and fourth papers construct desingularizations of X, realizing X as a limit of a family of compact, nonsingular SL m-folds Nt in M for small t > 0. Let Li be an asymptotically conical SL m-fold in ℂm asymptotic to Ci at infinity. We make Nt by gluing tLi into X at xi for i = 1, . . . , n.

Topological mirror symmetry

This paper focuses on a topological version on the Strominger-Yau-Zaslow mirror symmetry conjecture. Roughly put, the SYZ conjecture suggests that mirror pairs of Calabi-Yau manifolds are related by the existence of dual special Lagrangian torus fibrations. We explore this conjecture without reference to the special Lagrangian condition. In this setting, natural questions include: does there exist a nice class of T^3-fibrations for which the dual fibration can be constructed? Do such fibrations exist on a manifold such as the quintic threefold? If so, is the dual fibration the mirror? We answer these questions affirmatively. We introduce a class of topological T^3-fibrations for which duals can be constructed, including over the singular fibres. We then construct such a fibration on the quintic threefold, and show that by applying this general dualizing construction to this particular case, one obtains the mirror quintic. Thus we have constructed the mirror quintic topologically with no a priori knowledge of the mirror. This shows that in a non-degenerate (and representative) case, the Strominger-Yau-Zaslow conjecture correctly explains mirror symmetry.