Hironaka's example

Key takeaways

Hironaka's example belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Hironaka's example to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hironaka's example from memory before moving on to harder problems.

Reference excerpt

In algebraic geometry, Hironaka's example is a non-Kähler complex manifold that is a deformation of Kähler manifolds found by Heisuke Hironaka (1960, 1962). Hironaka's example can be used to show that several other plausible statements holding for smooth varieties of dimension at most 2 fail for smooth varieties of dimension at least 3.

Hironaka's example Take two smooth curves C and D in a smooth projective 3-fold P, intersecting in two points c and d that are nodes for the reducible curve C ∪ D {\displaystyle C\cup D} . For some applications these should be chosen so that there is a fixed-point-free automorphism exchanging the curves C and D and also exchanging the points c and d. Hironaka's example V is obtained by gluing two quasi-projective varieties V 1 {\displaystyle V_{1}} and V 2 {\displaystyle V_{2}} . Let V 1 {\displaystyle V_{1}} be the variety obtained by blowing up P ∖ c {\displaystyle P\setminus c} along C {\displaystyle C} and then along the strict transform of D {\displaystyle D} , and let V 2 {\displaystyle V_{2}} be the variety obtained by blowing up P ∖ d {\displaystyle P\setminus d} along D and then along the strict transform of C. Since these are isomorphic over P ∖ { c , d } {\displaystyle P\setminus \{c,d\}} , they can be glued, which results in a proper variety V. Then V has two smooth rational curves L and M lying over c and d such that L + M {\displaystyle L+M} is algebraically equivalent to 0, so V cannot be projective. For an explicit example of this configuration, take t to be a point of order 2 in an elliptic curve E, take P to be E × E / ( t ) × E / ( t ) {\displaystyle E\times E/(t)\times E/(t)} , take C and D to be the sets of points of the form ( x , x , 0 ) {\displaystyle (x,x,0)} and ( x , 0 , x ) {\displaystyle (x,0,x)} , so that c and d are the points (0,0,0) and ( t , 0 , 0 ) {\displaystyle (t,0,0)} , and take the involution σ to be the one taking ( x , y , z ) {\displaystyle (x,y,z)} to ( x + t , z , y ) {\displaystyle (x+t,z,y)} .

A complete abstract variety that is not projective Hironaka's variety is a smooth 3-dimensional complete variety but is not projective as it has a non-trivial curve algebraically equivalent to 0. Any 2-dimensional smooth complete variety is projective, so 3 is the smallest possible dimension for such an example. There are plenty of 2-dimensional complex manifolds that are not algebraic, such as Hopf surfaces (non Kähler) and non-algebraic tori (Kähler).

An effective cycle algebraically equivalent to 0 In a projective variety, a nonzero effective cycle has non-zero degree so cannot be algebraically equivalent to 0. In Hironaka's example the effective cycle consisting of the two exceptional curves is algebraically equivalent to 0.

A deformation of Kähler manifolds that is not a Kähler manifold If one of the curves D in Hironaka's construction is allowed to vary in a family such that most curves of the family do not intersect D, then one obtains a family of manifolds such that most are projective but one is not. Over the complex numbers this gives a deformation of smooth Kähler (in fact projective) varieties that is not Kähler. This family is trivial in the smooth category, so in particular there are Kähler and non-Kähler smooth compact 3-dimensional complex manifolds that are diffeomorphic.

A smooth algebraic space that is not a scheme Choose C and D so that P has an automorphism σ of order 2 acting freely on P and exchanging C and D, and also exchanging c and d. Then the quotient of V by the action of σ is a smooth 3-dimensional algebraic space with an irreducible curve algebraically equivalent to 0. This means that the quotient is a smooth 3-dimensional algebraic space that is not a scheme.

A Moishezon manifold that is not an abstract variety If the previous construction is done with complex manifolds rather than algebraic spaces, it gives an example of a smooth 3-dimensional compact Moishezon manifold that is not an abstract variety. A Moishezon manifold of dimension at most 2 is necessarily projective, so 3 is the minimum possible dimension for this example.

The quotient of a scheme by a free action of a finite group need not be a scheme This is essentially the same as the previous two examples. The quotient does exist as a scheme if every orbit is contained in an affine open subscheme; the counterexample above shows that this technical condition cannot be dropped.

A finite subset of a variety need not be contained in an open affine subvariety For quasi-projective varieties, it is obvious that any finite subset is contained in an open affine subvariety. This property fails for Hironaka's example: a two-points set consisting of a point in each of the exceptional curves is not contained in any open affine subvariety.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Hironaka's example

Start with the simplest possible case. Write down what Hironaka's example claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Hironaka's example before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Hironaka's example ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Hironaka's example

In research: Hironaka's example appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Hironaka's example in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Hironaka's example is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic geometry, so understanding it makes those chapters shorter.
In everyday life: Look for Hironaka's example outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Hironaka's example in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hironaka's example means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Hironaka's example out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hironaka's example in simple terms?

In algebraic geometry, Hironaka's example is a non-Kähler complex manifold that is a deformation of Kähler manifolds found by Heisuke Hironaka (1960, 1962). Hironaka's example can be used to show that several other plausible statements holding for smooth varieties of dimension at most 2 fail for sm…

Why does Hironaka's example matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Hironaka's example?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Hironaka's example.