Wednesday, November 29, 2023

Some facts about birational geometry

Birational geometry has as its morphisms birational maps: both the map and its inverse are expressible in terms of rational functions. The coefficients of polynomials appearing in rational functions are in the TGD framework rational. They map rationals to rationals and also numbers of given extension E of rationals to themselves (one can assign to each space-time region an extension defined by a polynomial).

Therefore birational maps map cognitive representations, defined as discretizations of the space-time surface such that the points have physically/number theoretically preferred coordinates in E, to cognitive representations. They therefore respect cognitive representations and are morphisms of cognition. They are also number-theoretically universal, making sense for all p-adic number fields and their extensions induced by E. This makes birational maps extremely interesting from the TGD point of view.

The following lists basic facts about birational geometry as I have understood them on the basis of Wikipedia articles about birational geometry and Enriques-Kodaira classification. I have added physics inspired associations with TGD.

Birational geometries are one central approach to algebraic geometry.

  1. They provide classification of complex varieties to equivalence classes related by birational maps. The classification complex curves (real dimension 2) is best understood and reduces to the classification of projective curves of projective space CPn determined as zeros of a homogeneous polynomial. I had good luck since complex surfaces (real dimension 4) are of obvious interest in TGD: now however the notion of complex structure is generalized and one has Hamilton-Jacobi structure and Minkowski signature is allowed.
  2. In TGD, a generalization of complex surfaces of complex dimension 2 in the embedding space H=M4× CP2 of complex dimension 4 is considered. What is new is the presence of the Minkowski signature requiring a combination of hypercomplex and complex structures to the Hamilton-Jacobi structure. Note however the space-time surfaces also have counterparts in the Euclidean signature E4× CP2: whether this has a physical interpretation, remains an open question. Second representation is provided as 4-surfaces in the space M8c of complexified octonions and an attractive idea is that M8-H duality corresponds to a birational mapping of cognitive representations to cognitive representations.
  3. Every algebraic variety is birationally equivalent with a sub-variety of CPn so that their classification reduces to the classification of projective varieties of CPn defined in terms of homogeneous polynomials. n=2 (4 real dimensions) is of special relevance from the TGD point of view. A variety is said to be rational if it is birationally equivalent to some projective variety: for instance CP2 is rational.
  4. A concrete example of birational equivalence is provided by stereographic projections of quadric hypersurfaces in n+1-D linear space. Circle in plane is the simplest example. Let p be a point of quadric. The stereographic projection sends a point q of the quadric to the line going through p and q, that is a point of CPn in the complex case. One can select one point on the line as its representative. Another exammple is provided by Möbius transformations representing Lorentz group as transformations of complex plane.
The notion of a minimal model is important.
  1. The basic observation is that it is possible to eliminate or add singularities by using birational maps of the space in which the surface is defined to some other spaces, which can have a higher dimension. Peaks and self-intersections are examples of singularities. The zeros of a birational map can be used to eliminate singularities of the algebraic surface of dimension n by blowups replacing the singularity with CPn. Poles in turn create singularities.

    The idea is to apply birational maps to find a birationally equivalent surface representation, which has no singularities. There is a very counter-intuitive formal description for this. For instance, complex curves of CP2 have intersections since their sum of their real dimensions is 4. The same applies to 4-surfaces in H. My understanding is as follows: the blowup for CP2 makes it possible to get rid of an intersection with intersection number 1. One can formally say that the blow up by gluing a CP1 defines a curve which has negative intersection number -1.

  2. In the TGD framework, wormhole contacts are Euclidian regions of space-time surface, which have the same metric and Kähler structure as CP2 and light-like M4 projection (or even H projection). They appear as blowups of singularities of 4-surfaces along a light-like curve of M8. The union of the quaternionic/associative normal spaces along the curve is not a line of CP2 but CP2 itself with two holes corresponding to the ends of the light-like curve. The 3-D normal spaces at the points of the light-like curve are not unique and form a local slicing of CP2 by 3-D surfaces. This is a Minkowskian analog of a blow-up for a point and also an analog of cut of analytic function.
  3. The Italian school of algebraic geometry has developed a rather detailed classification of these surfaces. The main result is that every complex surface X is birational either to a product CP1× C for some curve C or to a minimal surface Y. Preferred extremals are indeed minimal surfaces so that space-time surfaces might define minimal models. The absence of singularities (typically peaks or self-intersections) characterizing minimal models is indeed very natural since physically the peaks do not look acceptable.
Mathematicians use invariants to characterize mathematical structures. In TGD birational invariants would be cognitive invariants. They would be extremely interesting physically if the 4-D generalization of holomorphy really to a fusion of complex and hypercomplex structrures make sense (see this and this).

There are several birationals invariants listed in the Wikipedia article. Many of them are rather technical in nature. The canonical bundle KX for a variety of complex dimension n corresponds to n:th exterior power of complex cotangent bundle that is holomorphic n-forms. For space-time surfaces one would have n=2 and holomorphic 2-forms.

  1. Plurigenera corresponds to the dimensions for the vector space of global sections H0(X,KXd) for smooth projective varieties and are birational invariants. The global sections define global coordinates, which define birational maps to a projective space of this dimension.
  2. Kodaira dimension measures the complexity of the variety and characterizes how fast the plurigenera increase. It has values -∞,0,1,..n and has 4 values for space-time surfaces. The value -∞ corresponds to the simplest situation and for n=2 characterizes CP2, which is rational and has vanishing plurigenera.
  3. The dimensions for the spaces of global sections of the tensor powers of complex cotangent bundle (holomorphic 1-forms) define birational invariants. In particular, holomorphic forms of type (p,0) are birational invariants unlike the more general forms having type (p,q). Betti numbers are not in general birational invariants.
  4. Fundamental group is birational invariant as is obvious from the blowup construction. Other homotopy groups are not birational invariants.
  5. Gromow-Witten invariants are birational invariants. They are defined for pseudo-holomorphic curves (real dimension 2) in a symplectic manifold X. These invariants give the number of curves with a fixed genus and 2-homology class going through n marked points. Gromow-Witten invariants have also an interpretation as symplectic invariants characterizing the symplectic manifold X.

    In TGD, the application would be to partonic 2-surfaces of given genus g and homology charge (Kähler magnetic charge) representatable as holomorphic surfaces in X=CP2 containing n marked points of CP2 identifiable as the loci of fermions at the partonic 2-surface. This number would be of genuine interest in the calculation of scattering amplitudes.

What birational classification could mean in the TGD framework?
  1. Holomorphic ansatz gives the space-time surfaces as Bohr orbits. Birational maps give new solutions from a given solution. It would be natural to organize the Bohr orbits to birational equivalence classes, which might be called cognitive equivalence classes. This should induce similar organization at the level of M8c.
  2. An interesting possibility is that for certain space-time surfaces CP2 coordinates can be expressed in terms of preferred M4 coordinates using birational functions and vice versa. Cognitive representation in M4 coordinates would be mapped to a cognitive representation in CP2 coordinates.
  3. The interpretation of M8-H duality as a generalization of momentum position duality suggests information theoretic interpretation and the possibility that it could be seen as a cognitive/birational correspondence. This is indeed the case M4 when one considers linear M4 coordinates at both sides.
  4. An intriguing question is whether the pair of hypercomplex and complex coordinates associated with the Hamilton-Jacobi structure could be regarded as cognitively acceptable coordinates. If Hamilton-Jacobi coordinates are cognitively acceptable, they should relate to linear M4 coordinates by a birational correspondence so that M8-H duality in its basic form could be replaced with its composition with a coordinate transformation from the linear M4 coordinates to particular Hamilton-Jacobi coordinates. The color rotations in CP2 in turn define birational correspondences between different choices of Eguchi-Hanson coordinates.

    If this picture makes sense, one could say that the entire holomorphic space-time surfaces, rather than only their intersections with mass shells H3 and partonic orbits, correspond to cognitive explosions. This interpretation might make sense since holomorphy has a huge potential for generating information: it would make TGD exactly solvable.

See the article Birational maps as morphisms of cognitive structures or the chapter New findings related to the number theoretical view of TGD.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.

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