Some Examples of

Special Lagrangian Tori


Robert L. Bryant

Department of Mathematics

Duke University

Durham, NC 27708

12 February 1998

§0. Introduction

The purpose of this note is to point out some very elementary examples of special Lagrangian tori in certain Calabi-Yau manifolds that occur as hypersurfaces in complex projective space. All of these are constructed as real slices of smooth hypersurfaces defined over the reals. This method of constructing special Lagrangian submanifolds is, of course, well known. What does not appear to be in the current mirror symmetry literature is an explicit description of such examples in which the special Lagrangian submanifold is a 3-torus.

Some background. Let be a Ricci-flat Kähler -manifold, with Kähler form  and a parallel holomorphic volume form , normalized so that

The data  will be said to constitute a Calabi-Yau structure on .

Given any oriented -Lagrangian submanifold  with induced volume form , there exists a smooth map  so that

The submanifold  is said to be special Lagrangian Some authors require only that be constant. For simplicity, I will not consider this (slight) generalization, though it is important for some purposes. if  is identically .

As Harvey and Lawson [HL] show, a compact special Lagrangian submanifold  has minimal -volume in its homology class. In fact, the real part of  is a calibration on  and it calibrates the special Lagrangian submanifolds. Consequently, any smooth (or even ) special Lagrangian submanifold is real analytic in .

McLean [Mc] showed that for a smooth special Lagrangian submanifold , the moduli space of nearby special Lagrangian submanifolds is smooth and of dimension . He gave the following description: Let  denote the vector space of harmonic 1-forms on  (with respect to the induced metric). Then there exists an open neighborhood  of  and a smooth (in fact, real analytic) mapping  that vanishes to second order at  so that, in a -neighborhood of , any special Lagrangian normal graph is of the form  where

for a unique . (Here, denotes the vector in  dual to  and multiplication by  in  carries vectors tangent to  to vectors normal to .)

In particular, if  is an -torus and the induced metric on  is such that there is a basis  of  so that is nowhere vanishing, then these normal graphs foliate a neighborhood of  in . Such a special Lagrangian torus is said to be foliating. Every special Lagrangian -torus is foliating when  or , but this is not known to be the case in higher dimensions. This is trivial for . For  this follows from the fact a basis for the harmonic forms on  is formed by the real and imaginary parts of a nonzero (and hence nonvanishing) holomorphic 1-form on  (where  is regarded as a complex curve of genus  via the conformal structure and orientation induced on  by its inclusion into  as a special Lagrangian submanifold).

Conversely, if an open set  has a foliation  by compact smooth special Lagrangian manifolds, then each leaf  of  is diffeomorphic to the -torus and has a basis  of  so that is nowhere vanishing.

In the last few years, these special Lagrangian foliations have become interesting in the theory of mirror symmetry, particularly after the paper of Strominger, Yau, and Zaslow [SYZ]. Further work has been done on the geometry of moduli spaces of special Lagrangian tori and their augmentations by Hitchin [Hi1,2] and Lu [Lu], among others. For example, in [Hi2], on sees how special Lagrangian foliations can be constructed entirely in the holomorphic category in cases where  is even and  is actually hyperKähler.

In general, explicit examples of special Lagrangian tori seem to be difficult to construct. In the first place, the construction of Ricci-flat Kähler metrics itself relies on the celebrated theorem of Yau [Ya], which uses transcendental methods from nonlinear partial differential equations to construct a unique Ricci-flat Kähler form  in the cohomology class of any Kähler form on a compact complex  that is endowed with a holomorphic volume form . (By a Bochner argument, is then parallel with respect to .)

Many examples of compact complex Kähler manifolds with a holomorphic volume form are known, so Yau’s theorem provides a rich source of examples of Ricci-flat Kähler manifolds. Perhaps the simplest examples (other than complex tori) are the smooth hypersurfaces of degree  in .

Given the data  it is usually a highly nontrivial matter to construct a special Lagrangian submanifold in . One simple case where this can be done is when  possesses a real structure , i.e., an involution , that satisfies  and . If the locus  of fixed points of  is nonempty, then it is a smooth manifold of real dimension  that can be oriented so as to be a special Lagrangian submanifold of . Moreover, since  will necessarily preserve the underlying Kähler metric, is totally geodesic in  with respect to this metric.

Of particular interest is the case of hypersurfaces  of degree  that are invariant under the anti-automorphism induced by a real structure , i.e., a complex antilinear involution, with fixed subspace . In this case, the holomorphic volume form  on  can be chosen to be antiïnvariant under the induced antiholomorphism , i.e., so that . If one chooses a class  that is representable by a Kähler form that also is antiïnvariant under this conjugation, These always exist. For example, take  to be the class of the pullback to  of a -invariant Fubini-Study metric on . then the uniqueness part of Yau’s theorem implies that the unique Ricci-flat Kähler form  on  representing the class  must also be antiïnvariant under . Consequently  is a real structure on , implying that  is a special Lagrangian submanifold.

Thus, one method of exhibiting a special Lagrangian torus is to find a -invariant smooth hypersurface  whose real locus contains an -torus as one of its components. In the following sections, I will point out some simple examples of this for , and . Of course, the case  is well-known and completely classical. Examples with  are almost as classical. In fact, the possible topology of the real locus of a quartic surface in projective 3-space was determined by Kharlamov [Kh], and there are several cases where this real locus contains a 2-torus component. Despite their simplicity, examples when  seem not to have been pointed out in the mirror symmetry literature before. Of course it is this case that is of the most interest.

The construction. The basic idea is straightforward. For any polynomial function  that is homogeneous of degree , the equation defines a cone  and the equations  define a subcone . Assume that is a proper subset of . Let  be the usual projection. Then the image  is a hypersurface in  that is smooth away from the singular locus .

Let  denote the vector space consisting of polynomial functions  that are homogeneous of degree  and that satisfy where  is the usual conjugation fixing . This is a real vector space of dimension . For , the loci  and  are invariant under the induced conjugation on .

Let  be the dense open subset consisting of those  for which  is empty. For , endow  with the unique Ricci-flat Kähler form  whose Kähler class is dual to a hyperplane section. Then  is antiïnvariant under conjugation and the holomorphic volume form  can be chosen so that  and so that it satisfies the volume normalization needed to make its real part have comass one. This determines up to a sign.

Let  denote the open subset consisting of those  for which the real slice  contains an -torus  that is disjoint from . In the following sections, I will show that  is nonempty for , , and by producing explicit examples. I do not know whether or not  is nonempty for higher .

When  is nonempty, it follows that is nonempty (and open). For  , the smooth hypersurface  contains an -torus as a component of its real locus. Such an -torus is special Lagrangian in the Calabi-Yau structure .

§1. Cubic Curves in

This section is included for the sake of completeness and for comparison with the cases where and . Any smooth cubic curve defined over  is projectively equivalent to  where

for some real number . The curve  has two real components when  and one real component when . In either case, there is always exactly one odd component A simple closed curve in  is odd if it generates , otherwise it is even. and it contains the three real flexes , , and .

When , the curve  is singular, being the union of three lines, one of which is real, namely , where . Note that the singular locus  consists of three points, one of which is real, but which does not lie on .

Thus, lies in  for any finite . (When , the real locus consists of three real non-concurrent lines and hence has no smooth component.) Note also that has two components.

§2. Quartic Surfaces in

The topology of the real locus of a quartic surface in projective 3-space was determined in the 1970s. For a survey of these results, see Kharlamov [Kh]. However, it is easy to construct elements of  directly. Here are a few simple examples, generalizing both the ‘odd component’ and the ‘even component’ case of curves.

As a first example, consider

This is a nonsingular Fermat-type quartic. The real locus is

Since the curve  defined by  is diffeomorphic to the circle,  is diffeomorphic to  where

Note that is like the odd component of an elliptic curve: It is not contractible in  because its inverse image under the double cover  is a nontrivial double cover.

As a second example, take

where . In this case, the real locus consists of two disjoint 2-tori, neither of which is contractible in . The singular locus  consists of the four lines

and these have no real points.

The third example is more like the even component of a real elliptic curve. In , consider the circle  defined by the equations

This is the minimum locus of the quartic polynomial  defined by

The critical locus of  in  is the circle  plus the origin . Consequently, the regular values of  are all real numbers other than  and .

Let  be a real number satisfying  and consider the quartic polynomial . Its zero locus in  is smooth and is the boundary of the region , which retracts onto the circle . Consequently, the zero locus of  is diffeomorphic to a torus.

Now consider the homogeneous quartic  defined by

The singular locus  consists of two nonreal points .

Since  on  only at the origin, the real slice  is just

and hence is diffeomorphic to a torus. Thus  lies in .

Note that  misses a linear  in  and consequently is contractible in . Since  is not homotopic to , it follows that and lie in different components of .

By these three examples, it follows that  has at least three components.

§3. Quintic -folds in

In , consider the locus  defined by the equations

This is a 2-torus (it is the Clifford torus, up to scale). Because it is cut out by two independent equations, the normal bundle  of  in  is trivial, implying that its unit circle bundle  is diffeomorphic to the -torus.

Consider the quartic function

Then , with equality if and only if  lies in . The critical values of  are , , and .

Thus, for  satisfying , the hypersurface  defined by  is the boundary of a compact domain that has  as a deformation retract. In fact, by Morse theory,   is diffeomorphic to , which, as has been remarked, is a -torus.

Now consider the (reducible) quintic polynomial  defined by

The quintic hypersurface  is the union of the hyperplane  and the quartic hypersurface .

The singular locus  is a union of four lines

None of these lines have real points.

The intersection  is a union of two quadric surfaces

Neither of these surfaces has any real points. Note that consists of the four lines . Consequently, .

Thus, the real slice  is smooth and is the disjoint union of  and , which is diffeomorphic to the -torus.

Thus,  lies in , which is thereby shown to be nonempty. Consequently,  is nonempty, thus proving that there are smooth (and hence irreducible) quintic hypersurfaces in  whose real locus contains a torus as a component.

§4. References

[Hi1] N. Hitchin, The moduli space of special Lagrangian submanifolds, dg-ga/9711002.

[Hi2] N. Hitchin, The moduli space of complex Lagrangian submanifolds, dg-ga/9901069.

[HL] F. Harvey and H. Lawson, Calibrated Geometries, Acta. Math. 148 (1982), 47–157. MR 85i:53058.

[Kh] V. Kharlamov, On the classification of nonsingular surfaces of degree  in with respect to rigid isotopies. (Russian) Funktsional. Anal. i Prilozhen. 18 (1984), 49–56. (English translation in Functional Anal. Appl. 18 (1984), 39–45.) MR 85m:14034.

[Lu] P. Lu, Special Lagrangian Tori on a Borcea-Voisin Threefold, dg-ga/9902063.

[Mc] R. McLean, Deformations of calibrated submanifolds, Comm. Anal. Geom. 6 (1998), 705–747.

[SYZ] A. Strominger, S.T. Yau, and E. Zaslow, Mirror Symmetry is T-Duality, Nucl. Phys. B479 (1996), 243–259. MR 97j:32022.

[Ya] S.T. Yau, On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampére equations. I, Comm. Pure Appl. Math.  31 (1978), 339–411. MR 81d:53045.



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