User jme - MathOverflow

Best strategy for small resolutions

2010-07-13T12:38:34Z

I would like to know if there is a standard technique to check if a singular variety admits a small resolution. What are the main references for these types of questions?

I am mostly interested in threefolds and fourfolds with singularities in codimension 2 or higher.

(By a small resolution, I mean a proper birational transformation $Y\rightarrow X$ such that Y is smooth and the exceptional locus does not contain any divisors.)

Canonical forms for elliptic fibrations with Mordell-Weil group of rank 1 and zero torsion

2012-10-22T23:13:43Z

Consider an elliptic fibration given by the following Weierstrass model: $$ E: y^2 + a_1 x y + a_3 y =x^3 + a_2 x^2 + a_4 x + a_6,\quad a_6=a_2 a_4. $$ ( I work with characteristic zero). With the choice $a_6=a_2 a_4$, we easily identify the existence of two sections at $x=-a_2$. More precisely, we get the following two points on each fiber: $$ P: (x,y)=(-a_2, 0) \quad \text{and}\quad -P:(x,y)=(-a_2, a_1 a_2-a_3). $$ These two points correspond to the same generator of the Mordell-Weil group since they are actually opposite of each other for the group law of the elliptic curve with the usual zero element as the point of infinity on each fiber.

I have two questions:

Is the example above a case of an elliptic fibration of rank 1 with zero torsion?
Is there a reference where I can find canonical forms for Weierstrass models with rank 1 or 2 (and no torsion)?

Thank you!

Answer by JME for Elliptic curves with Mordell-Weil group Z/2Z over Q

2011-11-06T14:57:28Z

Mazur's theorem ensures that there are exactly 15 possible cases for the torsion part of the Mordell-Weil group of an elliptic curve: the cyclic groups $\mathbb{Z}_n$ (with $1\leq n\leq 10$ or $n=12$) and the groups $\mathbb{Z}_2\times\mathbb{Z}_n$ for $n=2,4,6,8$.

In his paper Universal Bounds on The Torsion of Elliptic Curves, Proc. London. Math. Soc.(1976) 33, 193-237 , Daniel Sion Kubert (who was a student of Mazur) presents in table 3 (page 217) a list of parametrizations for the different possible cases.

In particular, curves with a $\mathbb{Z}_2$ torsion are parametrized by the following family: $$ \mathbb{Z}_2\ \text{torsion}:\quad y^2=x(x^2+a x+ b), \quad b^2(a^2-4b)\neq 0. $$

The example given in Francesco's answer is a special case with $a=0$. As another example, the case with torsion $\mathbb{Z}_2\times \mathbb{Z}_2$ is parametrized by the Legendre family:

$$ \mathbb{Z}_2\times\mathbb{Z}_2 \ \text{torsion}:\quad y^2=x(x+r)(x+s), \quad r\neq 0 \neq s \neq r. $$

A slight generalization of the Hesse family parametrizes the curves with torsion $\mathbb{Z}_3$:

$$ \mathbb{Z}_3 \ \text{torsion}:\quad y^2+a_1 x y +a_3 y =x^3, \quad a_3^3( a_1^3-27 a_3)\neq 0. $$ For the other groups you might have to use Tate's normal form $$ E(b,c): \quad y^2+(1-c)x y - b y =x^3- b x^2 $$ and the condition for a given torsion is expressed as an algebraic condition on $b$ and $c$.

For example for $\mathbb{Z}_4$, we have $c=0$ and $b^4(1+16b)\neq 0$, which gives: $$ \mathbb{Z}_4 \ \text{torsion}:\quad E(b,c=0): \quad y^2+x y - b y =x^3- b x^2, \quad b^4(1+16b)\neq 0. $$

For a review, you can read chapter 4 of the book of Husemoller . A friendly short review is also available in section 2 of this string theory paper by Aspinwall and Morrison ( they don't present all the 15 cases but for those they analyze, they express everything in Weierstrass form).

Answer by JME for A K3 over $P^1$ with six singular $A_1$- fibers?

2011-09-25T16:06:21Z

As explained by Noam Elkies, the fibers you want are of Kodaira type:

$I_2$ : two rational curves intersecting at two distinct points
$III$ : two rational curves meeting at a double point

Since these are the only fibers you want, I will recommand not using a Weierstrass equation but rather a Jacobi quartic form for your elliptic fibration. Indeed, a smooth Weierstrass model admits only singular fibers of type $I_1$ (nodal curves) and type $II$ (cuspidial curve). So if you want to have singular fibers of type $I_2$ and $III$, you will have to introduce singularities in the total space of your fibration and resolve them following Tate's algorithm.

You can avoid dealing with the resolution of singularities if you start with the Jacobi quartic form. Consider a weighted projective plane $\mathbb{P}^2_{1,2,1}$ bundle, a smooth quartic equation will define an elliptic curve as you can see by using the adjunction formula. The canonical form of such a curve is

$$ \text{Jacobi form}: \quad y^2=x^4+ e x^2 z^2+ f x z^3+ g z^4 $$

where $[x:y:z]$ are the projective coordinates of weight $1$, $2$ and $1$ respectively.

For a fibration $Y\rightarrow B$, the weighted projective plane is replaced by a weighted projective bundle $\mathbb{P}_{1,2,1}[\mathscr{O}\oplus\mathscr{L}\oplus\mathscr{L}^2]$ with weight $1,2,1$, where $\mathscr{L}\ $ is a line bundle over the base $B$ of your fibration. In that equation $e,f,g$ are now sections of $\mathscr{L}^2$, $\mathscr{L}^3$ and $\mathscr{L}^4$ respectively. The projective coordinates $x$, $z$, $y$ are respectively sections of $\mathscr{L}\otimes \mathscr{O}(1)$, $\mathscr{O}(1)$ and $\mathscr{L}^2\otimes\mathscr{O}(2)$, where $\mathscr{O}(1)$ is the tautological line bundle of the weighted projective bundle.

Using the adjuction, you can see that you have a $K3$ elliptic fibration iff $c_1(\mathscr{L})=c_1(B)$.

In case you actually want a Weierstrass model, the Jacobian fibration will give you a birationally equivalent Weierstrass model: $y^2 =x^3+ Fx + G$ with $F=-4 g +e^2/3$ and $G=f^2-8/3 eg +2e^3/27$. You can compute the birational transformation using Maple.

Going back to the Jacobi form, it is not difficult to see that a smooth elliptic fibration will have fibers of type $I_1$, $II$, $I_2$ and $III$.

In particular, the fibers are:

type $I_1$ at a general point of $4
F^3+27 G^2=0$
type $II$ at $F=G=0$ and $e\neq 0$
type $I_2$ iff $f= e^2-4 g=0$ and $g\neq 0$
type $III$ iff $f=e=g=0$.

With a $\mathbb{P}^1$ base and $\mathscr{L}=O(2H)$, you have a $K3$ surface with fibers $I_1,II, I_2, III$.

You can avoid the fiber $I_1$ and $II$ by manipulating the equation. For example if you take the equation with a $\mathbb{P}^1$ base and $\mathscr{L}=O(2H)$, you can write the following K3 elliptic surface

$$y^2=x^4+ g z^4\Longrightarrow \ \text{8 fibers of type}\ III $$

You need $g$ to be a smooth polynomial of degree 8 in $P^1$. For example $g=x_1^8+x_0^8$ with $[x_0:x_1]$ the projective coordinates of $\mathbb{P}^1$. You will have $8$ fibers of type $III$ at the zero of $g$.Note that with that choice, the singular fibers are at the vertices of an octagon on $\mathbb{P}^1$ defined by $x_1^8+x_0^8=0$.

These types of fibrations based on the Jacobi quartic form are known as $E_7$ elliptic fibration in the string theory literature. They have some nice topological properties and transitions. You can read more about them in this paper.

Answer by JME for minimal resolution of singularities

2011-03-11T13:25:48Z

Francesco explained beautifully the resolution. Since I had prepared a geometric description of the resolution, I thought I will still post it.

The singular surface $$ E: S^2 (X^3+Y^3+Z^3)-3 (S^2+T^2) X Y Z=0 $$ is an hypersurface of bidegree $(2,3)$ in $\mathbb{P}^1\times \mathbb{P}^2$. The rational curve $\mathbb{P}^1$ is parametrized by the projective coordinates $[S:T]$ and $[X:Y:Z]$ are projective coordinates of $\mathbb{P}^2$. For every point of $\mathbb{P}^1$, the equation defines a cubic in $\mathbb{P}^2$ which is in the form of Hesse pencil: $$ H: s (X^3+Y^3+Z^3)+ t XYZ=0, \quad [s:t]\in \mathbb{P}^1. $$ Hesse pencil is famous in number theory, in cryptography and also shows up examples of mirror symmetry in physics. It is related to the Hesse configuration of 9 points and 12 lines in $\mathbb{P}^2$. There is a nice review by Artebani and Dolgachev. Hesse pencil can be seen as an elliptic surface with base $\mathbb{P}^1$. It admits singular fibers of Kodaira type $I_3$ (three lines forming a triangle).

The fibration considered in the question is obtained from Hesse pencil with the following map: $$ [s:t]\mapsto [s^2:-3(s^2+t^2)]. $$ This map is two-to-one eveywhere except at $s=0$ and at $t=0$ where it is one-to-one. This is related to the $\mathbb{Z}_2$ singularities described by Francesco in his answer. The six singular points of the elliptic surface $E$ are the intersection points of the three lines that form the fibers $I_3$ above $[S:T]=[1:0]$ and $[S:T]=[0:1]$. After the resolution, the singular points are replaced by $(-2)$-curves. The resolution describes a topological transition where two singular fibers of type $I_3$ are replaced by fibers of Kodaira type $I_6$. The transition is realized by replacing on each $I_3$ fiber, each of the 3 intersections points of the three lines by a $\mathbb{P}^1$.

Answer by JME for Why are hypergeometric series important and do they have a geometric or heuristic motivation?

2011-03-10T16:47:02Z

Hypergeometric series are solutions of a large class of differential equations. A series $\sum_{k} a_k t^k$ is hypergeometric if $Q_{k}=\frac{a_{k+1}}{a_k}$ is a rational function. Many familiar functions (trigonometric functions, exponential,logarithm,Hermite polynomials, Laguerre polynomials, etc) are hypergeometric.

Hypergeometric functions show up as solutions of many important ordinary differential equations. In particular in physics, for example in the study of the hydrogene atom (Hermite polynomials) and in simple problems of classical mechanics.

Hypergeometric functions are also important in the study of elliptic elliptic curves where they can be used to compute the inverse of the $j$-invariant.

I guess you can read more about them in this wikipedia page or in these notes. Several examples of applications to number theory, physics and combinatorics can be read here .

Non-existence of small resolutions for the singularity $y^2=u^2+v^2+w^3$

2010-07-12T16:10:28Z

In his paper "Smooth models for elliptic threefolds" (In: The Birational Geometry of Degenerations, Progress in Mathematics, v. 29, Birkhauser, (1983), 85-133), Rick Miranda mentions in the example of section 8 (page 101-102) that it is an unfortunate fact of life that there are no small resolutions for the singularity of the threefold defined by the equation

$$y^2=u^2+v^2+w^3\quad \text{in}\quad \mathbb{C}^4.$$

He said that it is a theorem of Brieskorn but he does not give a reference. Anyone has a reference to this theorem and its proof?

Updated (following some of the answers, the question can be generalized right away):

More generally the same question can be asked for

$$y^2=u^2+v^2+w^{2k+1}\quad \text{in}\quad \mathbb{C}^4, \quad\text{where}\quad k\in \mathbb{N}_0.$$

Answer by JME for Algebraic varieties which are topological manifolds

2010-10-08T12:24:24Z

The simplest example of a singular algebraic variety which is a topological manifold is given by the cusp $$z_1^2-z_0^3=0.$$ The cusp is a topological manifold homeomorphic to a real plane $\mathbb{R}^2$ as can be seen by the parametrization $t\mapsto (z_1,z_0)= (t^2,t^3)$ where $t$ is a complex variable.
Mumford has proven that a two dimensional normal complex space which is a topological manifold is always nonsingular.
Mumford's result does not generalize to (odd) dimensions higher than 2 as proven by Brieskorn using the following counter examples which generalizes the case of the cusp:

$$z_1^2+ z_2^2+\cdots z_{2k+1}^2-z_0^3=0,\quad \text{where} \quad k\in \mathbb{N}_0.$$
More generally, given $a=(a_1, \cdots, a_n)\in \mathbb{N}^n_0$ with $a_j>1$ for all $j$, one can define the following variety $\Gamma(a)$ known as a Brieskorn-Pham variety: $$ \Gamma(a): \quad z_1^{a_1}+\cdots z_n^{a_n}=0. $$
Brieskorn has proved the following conjecture of Milnor:
$$\Gamma(a)\quad \text{is a topological manifold} \iff \prod_{1\leq k_l\leq
a_k-1}(1-\epsilon_1^{k_1}
\epsilon_1^{k_2}\cdots
\epsilon_n^{k_n} )=1,$$ where $\epsilon_k=\mathrm{exp}\Big({\frac{2\pi }{a_k}\mathrm{i} }\Big)$ for $k=1,\cdots, n$.

Geometric interpretation of exceptional Symmetric spaces

2010-08-19T13:20:51Z

Elie Cartan has classified all compact symmetric spaces admitting a compact simple Lie group as their group of motion.There are 7 infinite series and 12 exceptional cases. The exceptional cases are related to real forms of exceptional Lie algebra. Most of these symmetric spaces admit at least one geometric interpretation usually in terms of complex and real Grassmannians and their generalizations to quaternions and octonions($\mathbb{H}$), octonions($\mathbb{\mathbb{O}}$), bioctonions ($\mathbb{C}\otimes \mathbb{O}$), quateroctonions ($\mathbb{H}\otimes \mathbb{O}$) and octooctonions ($\mathbb{O}\otimes \mathbb{O}$). See for example the Wikipedia's entry for Symmetric Spaces.

Two of the exceptional symmetric spaces, don't seem to have such a geometric interpretation as far as I know. In Cartan notation, these two spaces are called $EI$ and $EV$ and correspond respectively to the exceptional symmetric spaces $\frac{E_7}{SU(8) / \mathbb{Z}_2 }$ and $\frac{E_6}{USp(4)/ \mathbb{Z}_2}$ of respective rank and dimension$(4,42)$ and $(7,70)$.

Now that the stage is set, here is my question:

What is the geometric description of the symmetric spaces $\frac{E_{7}}{SU(8)/ \mathbb{Z}_2}$ and $\frac{E_6}{Sp(4)/ \mathbb{Z}_2}$ ?

References on the subject are also welcome. This question is motivated by an answer to this MO question.

In order to give a more precise idea of the kind of answer I expect, let me give some examples: the symmetric space $\frac{F_4}{\mathrm{Spin}(9)}$ is geometrically described as the Cayley projective plane $\mathbb{O}P^2$, the space $\frac{E_6}{\mathrm{SO}(10) \mathrm{SO}(2)}$ is geometrically the Caylay bioctonion plane $(\mathbb{C}\otimes \mathbb{O}) \mathbb{P}^2$ and the symmetric space $\frac{E_6}{F_4}$ is the space of isometrically equivalent collineations of the Cayley plane $\mathbb{O}\mathbb{P}^2$.

NB:These two spaces also show up as scalar manifolds in maximal supergravity theories, this is for example review in this article of Boya. But for this question, I won't consider supergravity as a geometric interpretation.

Updates

Richard Borcherds has provided an answer thanks to a reference to the book "Einstein manifolds", but the book gives the answer without any proofs or explanation. So we now have an answer but we don't understand it. So if anyone could help with explaining how "antichains" enter the story, it will be highly appreciated. I have put some extra information in the comments.

Answer by JME for Is there a common genesis for ADE classifications?

2010-08-05T20:23:24Z

I will first address the string theory part of the question.

String theory provides examples of physical systems admitting several descriptions that provide natural bridges between Kleinian singularities (and therefore Platonic solids), ALE spaces, quiver diagrams, ADE diagrams and two dimensional Conformal Field Theories.

The scene is given by compactifications of string theory on Kleinian orbifolds $M_\Gamma=\mathbb{C}^2/\Gamma$ where $\Gamma$ is a discrete subgroup of $SU(2)$. The space $M_\Gamma$ admits a Kleinian singularity at the origin. After studying this physical system, one is less surprised to see that Kleinian singularities, quiver diagrams, ALE spaces, ADE diagrams and 2 dimensional Conformal Field Theories all admit the same ADE classifications since they provide different descriptions of the same underlying physical system.

Michael Douglas and Gregory Moore have studied the compactification of string theory on Kleinian orbifold $M_\Gamma$ using D-branes as probes of the geometry. D-branes are extended objects on which strings can end. D-branes provide a physical description of the geometry in terms of supersymmetric gauge theories. Such supersymmetric gauge theories are efficiently summarized by a quiver diagram with a very natural physical interpretation: the nodes correspond to D-branes with specific gauge groups on them and the links between the nodes are open strings ending on the branes.

The minimal energy configurations (the vacua) of these supersymmetric gauge theories are obtained finding the extrema of a potential whose construction is equivalent to the hyperkhäler quotient construction of Asymptotic Locally Euclidian Spaces (ALE spaces) first obtained by Kronheimer. ALE spaces are HyperKähler four dimensional real manifolds whose anti-self-dual metrics are asymptotic to a Kleinian orbifold $M_\Gamma=\mathbb{C}^4/ \Gamma$. Physically ALE spaces described gravitational instantons. ALE spaces provide small resolutions of the Kleinian singularities where the singular point is replaced by a system of spheres whose intersection matrix is equivalent to the Cartan matrix of an ADE Dynkin diagram. One can also consider Yang-Mills instantons on such spaces. The gauge group associated with the Yang-Mills instantons is given by the type of ADE diagram obtained by the resolution of the singularity. This was analyzed in the math literature by Kronheimer and Nakajima. Physically the ALE instantons moduli space is equivalent to the vacua of the gauge theory description of D-branes located at the singularities.

The link between D-branes on ALE spaces (or equivalently Kleinian singularities) and the ADE classification of two dimensional Conformal Field Theories (CFT) was studied by Lershe, Lutken and Schweigert. Although the geometry is singular, the CFT description is smooth. The 2 dimensional CFT is coming directly from the string description: as a string evolves it described a 2 dimensional surface called the string worldsheet. D-branes enter the CFT as boundary states. In the description of the CFT, one recovers Arnold's ADE list of simple isolated singularities.

Updates

I would like to comment on the non-stringy part of the question. This is motivated by the comments of Victor Protsak.

If one removes all the string theory interpretation in the discussion above. What is left is Kronheimer's description of ALE spaces. Kronheimer's construction provides a beautiful realization of McKay's correspondence between Kleinian singularities, their crepant resolutions and ADE diagrams. This is reviewed in chapter 7 of Dominic Joyce's book "Compact Manifolds with Special Holonomy". From that perspective, the string theory description provides a physical interpretation of Kronheimer's construction and adds a natural link with quiver diagrams and 2 dimensional Conformal Field Theories.

When is a surface in a threefold contractible to a curve?

2010-07-11T16:11:56Z

Given a threefold $Y$ containing a surface $S$. Under which conditions can I contract $S$ so that I still end up with a smooth variety? In other words

what are the conditions for the existence of a smooth variety $X$ and a morphism $Y\rightarrow X$ such that the image of $S$ under the morphism is a curve and the morphism is an isomorphism away from $S$?

What are the conditions when $Y$ is a fourfold and $S$ is still a surface?

Answer by JME for How do you define the Euler Characteristic of a scheme?

2010-08-12T04:36:41Z

If $X$ is a smooth variety over the complex numbers, you can use the topological Euler characteristic of the support $X(\mathbb{C})$ of the scheme. The most efficient way to compute it is to use Chern classes and the Poincaré-Hopf theorem: the Euler characteristic is the degree of the top Chern class. Chern classes can be defined purely algebraically (in homology) as it is done for example in Chapter 3 of Fulton's book "Intersection Theory".

When $X$ is singular, there are many nonequivalent generalizations of the Euler characteristic. One efficient approach is again to work in the spirit of the Poincaré-Hopf theorem by first defining Chern classes for singular varieties. This provides a purely algebraic treatment. I will give a tour of the 4 most famous examples:

-The Chern-MacPherson class (also known as the Chern-Schwartz-MacPherson class ) is probably the most important. It can be defined on the Chow group of a variety using constructible functions. It enjoys beautiful functorial properties under proper maps and it is compatible with specializations. The existence of the Chern-MacPherson class realizes a conjecture of Deligne and Grothendieck of 1969 as the unique natural transformation between the covariant functor of constructible functions and the usual covariant functor of $\mathbb{Z}$-homology such that it maps the characteristic function of a manifold to its (homological) Chern class. The Chern-MacPherson class computes the topological Euler characteristic of a (possibly singular) variety.

-The Chern-Mather-class enters the definition of the Chern-MacPherson class. It is relevant to compute stringy invariants. It has nice properties under proper birational maps and can be simply defined using a Nash-blow-up.

These two classes (MacPherson and Mather) can be generalized to scheme by considering the support of the scheme. This makes calculations much more easy, but obviously one looses a lot of the information carried by the scheme. The next two definitions are much more "scheme friendly" but they have less understood functorial properties.

-The Chern-Fulton and the Chern-Fulton-Johnson classes are both extremely sensitive to the scheme structure. They coincide in the case of local proper intersections. In general they don't satisfy the "inclusion-exclusion principle" which is satisfies by the topological Euler characteristic.

References

For a friendly introduction with references to the appropriate literature, I would recommend this short lecture notes by Paolo Aluffi. The general treatment is discussed in Fulton's book Intersection Theory (specially section s 4.2.6, 4.2.9 and 19.1.7). For applications to motivic integration and stringy invariants see for example this review or this one.

Are there any (interesting) consequences of the irrationality of π?

2010-08-12T11:59:34Z

I am not sure how appropriate this question is for MO. If it is not, I apologize in advance but I could not resist asking it and if by any chance I get some interesting answers, it will for sure be very useful to keep my students excited about mathematics and physics as September arrives.

We all know very well that $\pi$ (the ratio of the circumference of a circle to its diameter in Euclidean space) is irrational and even transcendental. These are some of the famous results in all mathematics.

So I was wondering what will go wrong if $\pi$ was just an integer number?

Are there important theorems that are based on the fact that it is actually irrational and/or transcendental?

Answer by JME for Exceptional Lie algebras

2010-08-05T13:11:54Z

The appearance of exceptional Lie algebra, Kac-Moody algebras and Borcherds algebra in gravitational theories is a very elegant and exciting corner of current research in supergravity and string theory. I would like to discuss how exceptional Lie algebras naturally appear in the context of maximal supergravity theories and how this is connected to $E_{11}$ and Borcherds superalgebras. I will also discuss how Kac-Moody algebra occurs naturally in the context of gravitational singularities.

11 dimensional supergravity and exceptional Lie algebra

11 dimensional supergravity was constructed in 1978 by Cremmer, Julia and Scherk. Nowadays it is considered as the low energy limit of M-theory. The field content of 11-dimensional supergravity is simply given by a metric $g$ and a 3-form $A_{(3)}$. We can construct all the (massless) maximal supergravity theory in $D$-dimension with $2 < D <11$, by considering (Kaluza-Klein) reduction of 11 dimensional supergravity on a $(11-D)$-torus. This process produces a lot of additional fields (2-forms, 1-forms and 0-forms) coming from the reduction of the metric and the 3-form. In general for a compactification of 11 dimensional supergravity on a torus $T^{11-D}$ to a D-dimensional spacetime, we produce a scalar manifold $$ \frac{E_{11-D}}{K(E_{11-D})},$$ where $K(G)$ is the maximal compact subalgebra of $G$. In particular, we have in dimension 5, 4 and 3

$$ 5D\rightarrow \frac{E_6}{USp(8)}, \quad 4D\rightarrow \frac{E_7}{SU(8)}, \quad 3D\rightarrow \frac{E_8}{SO(16)}. $$

$E_{11}$ conjecture and Borcherds algebras

We recall that $E_9=E_8^+$ is understood as the extended Dynkin diagram of $E_8$. In the same way $E_{10}=E_8^{++}$ and $E_{11}=E_8^{+++}$ are the over-extended and the very-extended Dynkin diagram of $E_8$. There is a conjecture introduced by Peter West in 2001 and supported by several facts that the Kac-Moody algebra $E_{11}$ is related to a non-linear realization of M-theory and that $E_{11}$ can provide an 11 dimensional origin not only of all massless maximal supergravity theories (including type IIB) but also of the massive ones.

A beautiful duality was discovered by Iqbal, Neitzke and Vafa between compactifications of M-theory on tori and the second cohomology of some associated del Pezzo surfaces. Now the full cohomology of theses surfaces spans the root lattice of a Borcherds superalgebra.
Henry-Labordere Julia and Paulot have shown that some truncations of these Borchers algebras provide a classification of $p$-forms coming from tori reduction of (massive) maximal supergravity. This classification matches the one of the $E_{11}$ conjecture of Peter West. The Borcherds description was recently proven to be systematically derived from the split real form of $E_{11}$ by Henneaux, Julia and Levie.

Space-time singularities, Kac-Moody algebra and Cosmic billiards

A fascinating and non-speculative occurrence of $E_9$ and $E_{10}$ in a theory of gravity occurs when studying the behavior of gravity near a spacetime singularity. Belinskii, Khalatnikov and Lifchitz (BKL) have studied in details the general solution of Einstein equations near a spacetime singularity. As one reaches the singularity, the Einstein equations admit a chaotic behavior in time. Chitre and Misner has reformulated the BKL analysis in terms of a billiard motion in a 2 dimensional hyperbolic space.

In higher dimension, the chaotic behavior disappear in spacetime dimensions greater than 10. In particular, in 11 dimensions, there is no chaos at all. But if one add a 3-form (like the one of 11 dimensional supergravity), chaos comes back. In higher dimension one can also describe the chaotic behavior by a billiard in a higher dimensional hyperboloic space.

When a theory admits a compactification to three dimensions on a higher dimensional torus such that in the reduced 3 dimensional theory, the Lagrangian is given by Einstein-Hilbert action and a sigma model with target space a $G/H$ such that $G$ is a simple Lie group and $H$ its maximal compact subgroup, the billiard table is a Coxeter polyhedron and the billiard group is a Coxeter group. The table billiard can be described by the over-extended Kac-Moody algebra $G^{++}$ associated with the group $G$. In particular $$ \text{The billiard associated with eleven supergravity is } E_8^{++}=E_{10}.$$

One can formulate the billiard dynamics as a motion in the Cartan subalgebra of the Kac-Moody algebra.

Answer by JME for Algebraic geometry examples

2010-08-02T14:46:23Z

If you discuss the Whitney umbrella, I guess it is to show that blowing-up the singular point does not resolve the singularity whereas blowing-up the double line does. Another interesting properties of blow-up can be discussed by considering the 3 different resolutions of the conifold $x_1 x_2- x_3 x_4 =0$ in $\mathbb{C}^4$, namely the two small resolutions related by a flop and the blow-up of the isolated singularity with exceptional locus $\mathbb{P}^1\times\mathbb{P}^1$.
A quadric surface $Q$ in $\mathbb{P}^3$ is isomorphic to $\mathbb{P}^1\times \mathbb{P}^1$ via the Segre embedding. However, the Zariski topology of $Q$ is not homeomorphic to the product topology of $\mathbb{P}^1\times \mathbb{P}^1$ when each $\mathbb{P}^1$ is considered with the Zariski topology.
The orbifolds $\mathbb{C}^2/ \Gamma$ where $\Gamma$ is a discrete group of $SU(2)$. These orbifolds can be expressed as the simple isolated singularities of a surface and their resolution gives all the ADE Dynkin diagrams.

Answer by JME for ADE type Dynkin diagrams

2010-07-15T08:41:51Z

Extended Dynkin diagrams appear naturally in Kodaira classification of singular fibers of an elliptic surface.

What are the possible singular fibers of an elliptic fibration over a higher dimensional base?

2010-07-17T14:18:47Z

An elliptic fibration is a proper morphism $Y\rightarrow B$ between varieties such that the fiber over a general point of the base $B$ is a smooth curve of genus one. It is often required for the morphism to be flat, but I am open minded about it. In order to save your valuable time, I will first summarize my understanding of the current status and will then ask my question.

If the elliptic fibration admits a section, it is birationally equivalent to a (singular) Weierstrass model.
The singular fibers of an elliptic surface have been classified by Kodaira and it is closely related to ADE extended Dynkin diagrams. There are 9 types of singular fibers including two infinite series. Kodaira classification also determines the monodromy around each singular fiber. Kodaira classification is also reproduced by Néron and there is a famous algorithm by Tate (called Tate's algorithm) which helps determining the type of singular fibers for a Weierstrass model.
In the case of an elliptic threefold, Rick Miranda considers flat resolutions of Weierstrass models. He has a classification of singular fibers that results from his analysis. He looks at 'collisions of singularities' at the intersection of two divisors of the discriminant locus. He assumes several strong conditions like normal crossing of the discriminant locus and a well defined $j$-invariant. He often blow-up the base to get rid of "bad collisions''. He classifies the fibers that appears over the remained "good collisions''. There are 7 non-trivial types of collisions and 5 of them lead to non-Kodaira fibers including an infinite family.
Miranda's analysis has been generalized by Michael Szydlo in his Phd thesis at Harvard under the supervision of Barry Mazur. He considers collisions for higher dimensional elliptic fibrations. However, this part of thesis has not been published (but he has a published paper on the part of his thesis that deals with elliptic fibrations over non-perfect residue fields). He assumes the same conditions as Miranda and shows that in higher codimension, there are a finite number of types of non-Kodaira fibers resulting from collisions of singular fibers. Actually his list his similar to the one of Miranda except for one case where they do their blow-ups in different order and (therefore) end up with different fibers.

So here is my question:

What else is known about the possible fibers of a higher dimensional elliptic fibrations? Is there a hope to get a classification which does not assume normal crossing? If not, why? Non-trivial examples of non-Kodaira and non-Miranda fibers are also welcome and examples of non-flat fibration as well.

The reason why I won't ask for normal crossing is because geometrically it is not a natural thing to ask. After all, the discriminant of a Weierstrass equation is of the form $\Delta=4 f^3+27 g^2$. This is not really an invitation to ask for normal crossing... Moreover, in many cases relevant for applications to string theory, the normal crossing condition will not hold.

Answer by JME for What is the Batalin-Vilkovisky formalism, and what are its uses in mathematics?

2010-07-19T05:30:38Z

The BV formalism provides a (co)homological reformulation of several important questions of quantum field theory. The kind of problems that are usually addressed by the BV formalism are:

the determination of gauge invariant operators,
the determination of conserved currents,
the problem of consistent deformation of a theory,
the determination of possible quantum anomalies (the violation of the gauge invariance due to quantum effects).

The BV formalism is specially attractive because it does not require one to make a choice of a gauge fixing and it maintains a manifest spacetime covariance. It can also deals with situation that the traditional BRST formalism can not handle. This is for example the case of gauge theories admitting an open gauge algebra ( a gauge algebra that is closed only modulo the equations of motion). The typical example are supergravity theories. The BV formalism also allows an elegant and powerful mathematical reformulation of certain questions of quantum field theories in the language of homomological algebra.

Mathematically, the BV formalism is simply a clever application of homological perturbation theory. In order to understand the relation, I will first review the geometry of a physical model described by a Lagrangian $\mathcal{L}$ depending of fields $\phi^I$ and a finite number of their derivatives and admitting a gauge symmetry $G$. The starting point is the space $\mathcal{M}$ of all possible configurations of fields and their derivatives. This can be formalized using the language of jet-spaces. The Euler-Lagrange equations give the equations of motion of the theory and together with their derivatives, they define a sub-space $\Sigma$ of $\mathcal{M}$ called the stationary space. The on-shell functions are the functions relevant for the dynamic of the theory, they are defined on the stationary space $\Sigma$, they can be described alegebraically as $\mathbb{C}^\infty(\Sigma)=\mathbb{C}^\infty(\mathcal{M})/ \mathcal{N}$ where $\mathcal{N}$ is the ideal of functions that vanish on $\Sigma$. Because of the gauge invariance, the Euler-Lagrange equations are not independent but they satisfy some non-trivial relations called Noether identities. One has to identify different configurations related by a gauge transformation. Indeed, a gauge symmetry is not a real symmetry of the theory but a redundancy of the description.

The two steps that we have just described (restriction to the stationary surface and taking the quotient by the gauge transformations) are respectively realized in the BV formalism by the homology of the Koszul-Tate differential $\delta$ and the cohomology of the longitudinal operator $\gamma$. The Koszul-Tate operator defines a resolution of the equations of motion in homology. This is done by introducing one antifield $\phi^*_I$ for each field $\phi^I$ of the Lagrangian. The antifields are introduced to ensure that the equations of motion are trivial in the homology of the Koszul-Tate operator. The gauge invariance of the theory is taking care of by the cohomology of the longitudinal differential $\gamma$. In the case of Yang-Mills theories, the cohomology of $\gamma$ is equivalent to the Lie algebra cohomology.

The full BV operator is then given by

$$s=\delta + \gamma+\cdots,$$

where the dots are for possible additional terms required to ensure that the BV operator $s$ is nilpotent ( $s^2=0$). The construction of $s$ from $\delta$ and $\gamma$ follows a recursive pattern borrowed from homological perturbation theory. One can trace the need for the antifields and the Koszul-Tate differential to this recursive pattern. For simple theories like Yang-Mills, we just have $s=\delta+\gamma$ because the gauge algebra closes as a group without using the equations of motion. In more complicate situation when the algebra is open there are additional terms in the definition of $s$. One can generates $s$ using the BV bracket $(\cdot ,\cdot)$ (under which a field and its associated antifields are dual) and a source $S$ such that the BV operator can be expressed as $$ s F= (S,F). $$ The classifical master equation is
$$(S,S)=0,$$ and it is just equivalent to $s^2=0$.

At the quantum level, the action $S$ is replaced by a quantum action $W=S+\sum_ i \hbar^i M_i$ where the terms $M_i$ are contribution due to the path integral measure. The gauge invariant of quantum expectation values of operators is equivalent to the quantum master equation : $$ \frac{1}{2}(W,W)=i\hbar \Delta W, $$ where $\Delta$ is an operator similar to the Laplacian but defined in the space of fields and their antifields. This operator naturally appears when one considers the invariance of the measure of the path integral under an infinitesimal BRST transformation. When $\Delta S=0$, we can take $W=S$.

We will now review the BV (co)homological interpretation of some important questions in quantum field theory:

The observables of the theory are gauge invariant operators, they are described by the cohomology group $H(s)$ in ghost number zero.
Non-trivial conserved currents of the theory are equivalent to the so-called characteristic cohomology $H^{n-1}_0(\delta |d)$ which is the cohomology of the Koszul-Tate operator $\delta$ (in antifield number zero) modulo total derivatives for forms of degree $n-1$, where $n$ is the dimension of spacetime.
The equivalent class of global symmetries is equivalent to $H^n_1(\delta| d)$.
The gauge anomalies are controlled by the group $H^{1,n}(s|d)$ (that is $H(s)$ in antifield number 1 and in the space of $n$-form modulo total derivative). The conditions that define the cohomology $H^{1,n}(s|d)$ are generalization of the famous Wess-Zumino consistency condition.
The group $H^{0,n}(s|d)$ controls the renormalization of the theory and all the possible counter terms.
The groups $H^{0,n}(\gamma,d)$ and $H^{1,n}(\gamma, d)$ control the consistent deformations of the theory.

References:

For a short review, I recommend the preprint by Fuster, Henneaux and Maas: hep-th/0506098.
The classical reference is the book of Marc Henneaux and Claudio Teitelboim (Quantization of Gauge Systems).
For applications there is also a standard review by Barnich, Brandt and Henneaux: ``Local BRST cohomology in gauge theories,'' Phys. Rept.338, 439 (2000) [arXiv:hep-th/0002245].

Answer by JME for Does anyone recognize this surface?

2010-07-15T14:10:29Z

Your surface is actually a Whitney umbrella.

To see that, just perform the following substitution (which is just a translation):

$$x =1+\frac{z^2}{4} -t.$$

After this, your surface is defined by the simpler equation:

$$z^2 t =y^2.$$

This is exactly the canonical form of the Whitney umbrella. The Whitney umbrella is a singular surface in $\mathbb{C}^3$ which looks like a self-intersecting plane. It has a line of double points at $z=y=0$ and the singularity worsen to a pinch point at the origin $z=y=t=0$. You can resolve it by blowing-up the double line.

The Whitney umbrella is studied in many books of algebraic geometry as an example of a surface with a pinch point and a first tricky example of blow-up: it shows that blowing-up the worse singularity is not the best way to smooth a space (if you blow-up the pinch points you will again get the same equation). In classical singular theory, it is an example of a singular surface which does not have a regular Whitney stratification. It also appears very naturally in the study of elliptic fibration in the context of string theory (more precisely F-theory) when one consider "Sen's weak coupling limit" which gives the orientifold limit of F-theory at weak coupling.

How can I get a small resolution for the binomial fourfold $x_1 x_2 x_3- y_1 y_2=0$ in $\mathbb{C}^5$?

2010-07-13T12:35:28Z

I consider the singular fourfold $X$ defined as follows:

$$X: \quad x_1 x_2 x_3 -y_1 y_2=0\quad \text{in}\quad \mathbb{C}^5.$$

Its singular locus is a bouquet of three planes meeting at the origin:

$$Sing(X):\quad y_1=y_2=x_1 x_2=x_2 x_3=x_1 x_3=0.$$

How can I described the small resolution of this space (if any)?

Comment by JME

2012-06-12T15:07:27Z

@David, do you have a reference on this?

Comment by JME

2011-11-07T22:14:32Z

One thing that is not clear to me is what are the conditions for this to be true? There are many fibrations for which $\chi(E)\neq \chi(B)\chi(F)$.Take for example an elliptic fibration.

Comment by JME

2011-09-25T17:37:55Z

Thanks, typo corrected.

Comment by JME

2011-07-30T20:24:05Z

@Karl When you blow up divisors that are not Cartier, under which conditions one can be sure that the resulting small resolution is projective?

Comment by JME

2011-07-30T10:53:50Z

Nice example! How do you see that it is non-Kahler? Do you have a reference?

Comment by JME

2011-03-27T18:59:45Z

@Pete I really like these examples. Can you give a reference?

Comment by JME

2011-03-27T18:54:26Z

Where can we read about this?

Comment by JME

2011-03-12T11:20:58Z

@Alfonz Thanks a lot! I just corrected it.

Comment by JME

2011-03-11T23:07:12Z

There must be something wrong with the argument using the smoothness of $\sigma$ to detect the singularities of the surface since it fails for the change base $[s:t]\mapsto [s^2:t^2]$...singular fibers are necessary related to singular points of the total space. Also what are exactly $\xi,\zeta$ and $\eta$ ? can you express them in terms of the original variables $x,y,z,u,v$? I am asking to see how the base change $[s:t]\mapsto [s^2:s^2+t^2]$ will turn $\zeta$ into $\zeta^2$.

Comment by JME

2011-03-11T20:51:57Z

Maybe I am missing something, but it seems to me that you are considering a different surface than the one in the question. Shouldn't your base change be $[s:t]\mapsto [s^2:s^2+t^2]$? Otherwise you end up with the surface The surface $s^2 (x^3+y^3+z^3)-3 t^2 x y z=0$ which has only 3 singular points and over $[1:0]$, the fiber is just $x^3+y^3+z^3=0$ which is a smooth elliptic fiber.

Comment by JME

2011-03-11T11:45:47Z

@unknown I notice that your space is actually an elliptic surface. Why are you interested in it? This can help to identify the kind of resolutions you want.

Comment by JME

2011-03-11T11:05:21Z

@unknown I think that your calculation for the singular locus is wrong. The singular locus is made of only six isolated points. There is no line of singularities. To avoid missing points, you might actually want to work with the projective coordinates, it is also more easy.For example, you are missing all the singular points with $S=0$.

Comment by JME

2011-01-21T09:11:22Z

Andrei, I think we will not agree on this but this is your question and obviously that is not the answer you were looking for. I will remove it.

Comment by JME

2011-01-20T19:12:56Z

@Andei If the existence is automatic, how do you explain that Joyce has produced examples of structure $GL(n,\mathbb{H})Gl(1,\mathbb{H})$ which are not reducible to $Sp(n)Sp(1)$? This is done in his paper "Compact hypercomplex and quaternionic manifolds".

Comment by JME

2011-01-20T18:30:43Z

@Andrei I know that they are called quaternionic-kahler, I wrote that as a comment on your question. You have written in your question that one could consider a metric compatible with Q. Without a metric the structure is $GL(n, \mathbb{H})Sp(1)$ and not $Sp(n)Sp(1)$, but even in your own answer, you refer to $Sp(n)Sp(1)$ which implies the existence of a compatible metric!