Duplicate citations. The following articles are merged in Scholar. Their combined citations are counted only for the first article. Merged citations. This "Cited by" count includes citations to the following articles in Scholar. Add co-authors Co-authors. Upload PDF. Follow this author. New articles by this author. New citations to this author. New articles related to this author's research. Email address for updates. My profile My library Metrics Alerts. Sign in. Verified email at uu. Articles Cited by. Journal of Pure and Applied Algebra 10 3 , , In addition to the lecture notes submitted by its lecturers, this volume contains several research articles.

A group of fourty graduate students and young researchers attended the school. Some of them did so also for the accomodation. We are grateful to them. Sabine Buchmann is a French artist living in Istanbul, who likes to draw Ottoman-style miniatures of the boats serving across the bosphorus; these boats are an inseparable part of the city panorama. When asked, she liked the idea of a boat full of mathematicians and drew it for the conference poster — with the names of all the lecturers hidden inside, written in minute letters.

Her miniature helped us much in attracting the audience of the summer school. Finally we would like to thank warmly Prof. Daniel Allcock, James A. Problem Session. Carlson and Domingo Toledo. The moduli space of real 6-tuples in CP 1 is modeled on a quotient of hyperbolic 3-space by a nonarithmetic lattice in Isom H 3. These notes are an exposition of the key ideas behind our result that the moduli space Ms of stable real binary sextics is the quotient of real hyperbolic 3-space H 3 by a certain Coxeter group together with its diagram automorphism.

We hope they can serve as an aid in understanding our work [3] on moduli of real cubic surfaces, since exactly the same ideas are used, but the computations are easier and the results can be visualized. He is grateful to the organizers, fellow speakers and students for making the workshop very rewarding. The pictures are hand-drawn to encourage readers to draw their own.

Lecture 1 Hyperbolic space H 3 is a Riemannian manifold for which one can write down an explicit metric, but for us the following model will be more useful; it is called the upper half-space model. Its underlying set is the set of points in R3 with. Planes appear either as vertical half-planes, or as hemispheres resting on R2 :. If two planes meet then their intersection is a geodesic.

For now, ignore the col- ors of the nodes; they play no role until Theorem 2. After playing with it one discovers that it cannot appear as a vertical halfplane, so we look for a suitable hemisphere. It must be orthogonal to two of our three walls, so it is centered at the foot of one of the half- lines of intersection. We have drawn the picture so that the hemisphere is centered at the foot of the back edge. The dimensions we have drawn on the overhead view refer to Euclidean distances, not hyperbolic ones.

Readers may enjoy trying their hands at this by drawing polyhedra for the diagrams. P1 3 P3. In the last two cases we describe the meaning by pictures: Parallelism means. That is, when two planes do not meet in H 3 , we call them parallel if they meet at the boundary of H 3 , and ultraparallel if they do not meet even there.

Diagrams like 1 and 3 are called Coxeter diagrams after H. The proof is a very pretty covering space argument; see [5] for this and for a nice introduction to Coxeter groups in general. We care about hyperbolic orbifolds because it turns out that moduli spaces arising in algebraic geometry are usually orbifolds, and it happens sometimes that such a moduli space happens to coincide with a quotient of hyperbolic space or complex hyperbolic space or one of the other symmetric spaces.

So we can some- times gain insight into the algebraic geometry by manipulating simple objects like tilings of hyperbolic space. Now we come to the case which concerns us. Let C be the set of binary sextics, i. The relation with hyperbolic geometry begins with the following theorem: Theorem 2. In the second lecture we will see that the faces of the Pj corresponding to blackened nodes and triple bonds are very interesting; we will glue the Pj together to obtain a real-hyperbolic description of the entire moduli space. The book is a highly polished treatment of a subset of the material in the notes, which inspired a great deal of supplementary material, e.

Lecture 2 We will not really provide a proof of Theorem 2; instead we will develop the ideas behind it just enough to motivate the main construction leading to Theorem 4 below. Although Theorem 2 concerns smooth sextics, it turns out to be better to consider mildly singular sextics as well. For those who have seen geometric invariant theory, Cs is the set of stable sextics, hence the subscript s. In the space of ordered 6-tuples in CP 1 this is clear; to get the picture in Cs one mods out by permutations.

More precisely, in a neighborhood of a point of Fs describing a sextic with k double points, the map to Cs is given locally by z1 ,. We call an element of Fs resp. F0 a framed stable resp. See the appendix for a sketch of the Hodge theory involved in the proof. Theorem 3 Deligne—Mostow [6].

Complex hyperbolic space is like ordinary hyperbolic space except that it has 3 complex dimensions, and hyperplanes have complex codimension 1. There is an upper-half space model analogous to the real case, but the most common model for it is the open complex ball. The idea 3. Together with the G-invariance of g, this proves that g R is invariant under the identity component of GR. A closer study of g R shows that it is actually invariant under all of GR. We write K0 for the set of pairs 4 in the image g R F0R.

Since H03 ,. It is not guaranteed to terminate, but if it does then it gives a fundamental domain. There is no big idea here; one just works out the answer and writes it down. Now, the deleted faces are very interesting, and the next step in our discus- sion is to add them back in. By Theorem 3 we know that points of H represent singular sextics, which occur along the boundary between two components of CR 0.

For example,. Varying the remaining four points gives a family of singular sextics which lie in the closures of both CR R 0,0 and C0,1. This suggests reinstating the deleted walls of P0 and P1 and gluing the reinstated wall of P0 to one of the reinstated walls of P1.

This gives a rule for identifying the points of P0 and P1 that lie in this H 2. We begin by assembling P1 and the copies of P0 and P2. This requires pictures of the polyhedra. P0 appears in 2 , and for the others we draw both 3-dimensional and an overhead views. As before, length markings refer to Euclidean, not hyperbolic, distances.

There is only one way to identify isometric faces in pairs, pictured in Figure 1. The result appears in Figures 2 and 3 in overhead and 3-dimensional views. It is time to attach the two copies of P3. Figure 1. Overhead view of instructions for gluing P1 to two copies of P0 and two copies of P2. Figure 2. Overhead view of the result of gluing P1 to two copies of P0 and two copies of P2.

Moduli of Real Binary Sextics Figure 3. Three-dimensional view of the result of gluing P1 to two copies of P0 and two copies of P2. Adjoining it, and another copy of P3 in the symmetrical way, completes the gluing described in 8. The result appears in Figures 4 and 5. Figure 4. Figure 5. Theorem 4. For the rest of the lecture we will focus on the perhaps-surprising subtlety regarding the orbifold structures of MR R s and Q.

In a neighborhood U of S, Cs is a real 6-manifold meeting the discriminant a complex 5-manifold along a real 5-manifold. Here are pictures of the relevant parts of CR s and FsR :. S and mod out by the stabilizer of S in GR , as explained in lecture 1. We choose a transversal to GR. This leads to iii in Theorem 4. Appendix We will give a sketch of the Hodge theory behind Theorem 3 and then make a few remarks.

Theorem 3 is due to Deligne and Mostow [6], building on ideas of Picard; our approach is more explicitly Hodge-theoretic, along the lines of our treatment of moduli of cubic surfaces in [1]. We have used the 6-fold cover because the residue calculus is less fussy in projective space than in weighted projective space. See remark 9 for a comparison of the approaches using the 3-fold and 6-fold covers. One can show see, e.

We close with some remarks relevant but not central to the lectures. Remark 1. In a similar way, one could consider the moduli space of ordered 6-tuples of distinct points in CP 1 such that say points 1 and 2 are conjugate and points 3,. It is only by considering unordered 6-tuples that one sees all four types of 6-tuples occurring together, leading to our gluing construction.

Remark 2. Remark 3. R This implies that Ms is not a good orbifold in the sense of Thurston [16]. Remark 4. For background see [9] and [11]. Remark 5. See [7]. Remark 6. The points of H43 correspond to 6-tuples in CP 1 invariant under the non-standard anti- involution of CP 1 , which can be visualized as the antipodal map on the sphere S 2. Remark 7.

When discussing the gluing patterns 7 and 8 we did not specify information such as which gluing wall of P2 is glued to the gluing wall of P3. It turns out that there is no ambiguity because the only isometries between walls of the Pj are the ones we used. The gluing wall of P0 is glued to one of the top gluing walls of P1 , the gluing wall of P3 is glued to the left gluing wall of P2 , and the other gluing wall of P2 is glued to one of the bottom gluing walls of P1.

Remark 8. In these notes we work projectively, while in [3] we do not. This means that our space C is analogous to the CP 19 of cubic surfaces in CP 3 , which is the projectivization of the space called C in [3], and similarly for the various versions of F. Remark 9. As mentioned above, our treatment of the Hodge theory uses the 6- fold cover when the 3-fold cover would do; in [2] we used just the 3-fold cover, and the translation between the approaches deserves some comment.

References [1] D. Allcock, J. Carlson, and D. Toledo, The complex hyperbolic geometry of the moduli space of cubic surfaces, J. Toledo, Non-arithmetic uniformization of some real moduli spaces, to appear in Geom. Carlson and D. Toledo, Hyperbolic geometry and moduli of real cubic surfaces, in preparation. Canary, D. Epstein and P. Lecture Note Ser. Press, Cambridge, Publishing, River Edge, NJ, Deligne and G.

Mostow, Monodromy of hypergeometric functions and non- lattice integral monodromy, Publ. IHES 63 , 5— Gromov and I. Piatetski-Shapiro, Nonarithmetic groups in Lobachevsky spaces, Publ. Kapovich and J. Kellerhals, On the volume of hyperbolic polyhedra, Math. Kojima, H. Nishi and Y. Milnor, How to compute volume in hyperbolic space, in Collected Works, Volume 1, — Publish or Perish, Houston, Nikulin, Involutions of integral quadratic forms and their applications to real algebraic geometry, Math. USSR Izvestiya 22 , 99— Berkeley , — Sebastiani and R. Thurston, Three-dimensional geometry and topology, Princeton Mathematical Series, Thurston, Shapes of polyhedra and triangulations of the sphere, Geom.

Monographs 1, — Vinberg, Some arithmetical discrete groups in Lobacevskii spaces. Oxford, The radius of convergence of 1 is 1 unless a or b is a non-positive integer, in which cases we have a polynomial. They are examples of orthogonal polynomials. The latter will be dis- cussed in later sections. These observations are part of the following theorem. Theorem 1. Thus we see that, under the assumptions of Theorem 1.

For many more identities and formulas we refer to [AS] and [E]. Most of it can be found in standard text books such as Poole, Ince, Hille. Lemma 2. Let f1 ,. There exists a C-linear relation between these function if and only if W f1 ,. Theorem 2. Suppose 0 is a regular point of 3. Then the vector space of solutions of 3 is spanned by n C-linear independent Taylor series solutions f1 ,.

Finally, the Wronskian determinant W f1 ,. In that case we call 0 an apparent singularity. However, we do have the following theorem which we shall repeatedly apply. Suppose there exists a basis of power series solutions f1 ,. Show that the local exponents at a regular point read 0, 1,. Example 2.

Let t be a local parameter around this point and rewrite the equation with respect to the variable t. The corresponding indicial equation will be called the indicial equation of 5 at P. The roots of the indicial equation at P are called the local exponents of 5 at P. As a shortcut to compute indicial equations we use the following lemma. Lemma 3. Then there exist n C-linear independent Taylor series solutions f1 ,. Moreover, any Taylor series solution of 5 is a C-linear combination of f1 ,. Corollary 3. Any analytic solution of 5 near a regular point can be continued analytically along any path in C not meeting any singularity.

Af- ter analytic continuation of f1 ,. The equation 5 is called Fuchsian if all points on P1 are regular or a regular singularity. Theorem 3. Suppose 5 is a Fuchsian equation. Since the local exponents at a regular point are always 0, 1,. The hypergeometric equation 1 is an example of a Fuchsian equation. The 24 solutions of Kummer can now be characterised very easily. Suppose we apply the above procedure to the hypergeometric equation itself. Example 3. Monodromy of the hypergeometric function Let us now turn to the monodromy of the hypergeometric equation.

Then there exists a common eigenvector of A, B if and only if A, B have a common eigenvalue. If these eigenvalues are equal, we are done. Suppose they are not equal. Then w, v span C2. So suppose v is not an eigenvector of A. Hence A, B have a common eigenvector. A hypergeometric equation is called reducible if its monodromy group is reducible. A hypergeometric equation is called abelian if its monodromy group is abelian. Here is a simple necessary condition for abelian equations, which has the pleasant property that it depends only on a, b, c mod Z : Lemma 3. Abelian monodromy implies reducibility of the monodromy, hence at least one of the four numbers is integral.

Clearly this is a contradiction. Since A, B have disjoint eigenvalue sets, v is not an eigenvector of A and B. Hence w, v form a basis of C2. With respect to this basis A, B automatically obtain the form given in our Lemma. Suppose that 2 is irreducible. Then, up to conjugation, the mon- odromy group depends only on the values of a, b, c modulo Z. The sets R, S are said to interlace if every segment on the unit circle, connecting two points of R, contains a point of S. Let G be the group generated by A, B. We call these three cases the euclidean, spherical and hyperbolic case respectively.

Of course, with respect to this basis A and B have the form given in the previous lemma. Since A and B do not have the same characteristic equation the solutionspace for F is one-dimensional. It now a straightforward excercise to see that these inequalities correspond to interlacing, coinciding or non-interlacing of the eigenvalues of A and B. We are left with the case when v is an eigenvector of A and B. It is now straightforward to verify that is the unique invari- 0 1 ant hermitean matrix.

Moreover it is degenerate, which it should be as A, B have a common eigenvector. With the assumptions as in the previous lemma let G be the group generated by A and B. Let F be the invariant hermitean form for the monodromy group. First a little geometry. The segments are called the edges of the triangles, the points are called the vertices. In particular it is non-zero in H. Hence D z maps 0, 1 to a segment of a circle or a straight line. For the exact determination of the image of the Schwarz map we need the following additional result.

Hopefully, the following picture illustrates how this works. The monodromy group modulo scalars arises as follows. In the following section we shall study precisely such groups. The edges are then automatically geodesics.

## Arithmetic and Geometry Around Hypergeometric Functions

Our condition is equivalent to saying that all angles are positive. The latter property is equivalent to positivity of all angles. First of all we note that there exists a positive d0 with the following prop- erty. This is a special case of the theorem of Coxeter—Tits on representations of Coxeter groups. In particular it follows from the previous theorem that triangle groups generated by elementary triangles act discretely.

The list of spherical cases was already found by H. Schwarz and F. Klein see [Kl]. Hence the quotient of any two solutions f, g of the corresponding hypergeometric is algebraic. Hence g and, a fortiori, f are algebraic. For a complete list of such dissections and the corresponding identities we refer to [V].

In fact: Lemma 3. We consider the following cases. In the spherical case the condition is not violated. Let M be the monodromy group of 2. References [AS] M. Abramowitz, I. Stegun, Handbook of Mathematical Functions, Dover, Vidunas, Transformations of Gauss hypergeometric functions, J. Computational and Applied Math. Box They provide an introduction to recent work on the complex ball uniformization of the moduli spaces of del Pezzo surfaces, K3 surfaces and algebraic curves of lower genus. For convenience to a non-expert reader we include an introduction to the theory of periods of integrals on algebraic varieties with emphasis on abelian varieties and K3 surfaces.

Primary 14J10; Secondary 14J28, 14H Hodge structure, periods, moduli, Abelian varieties, arrangements of hyperplanes, K3 surfaces, complex ball. The topic of the talks was an introduction to recent work of various people on the complex ball uniformization of the moduli spaces of del Pezzo surfaces and algebraic curves of lower genus [ACT], [Vo], [K1]—[K4], [HL]. Keeping in mind the diverse background of the audience we include a general introduction to the theory of Hodge structures and period domains with more emphasis on abelian varieties and K3 surfaces.

So, an expert may start reading the notes from Section 6. In fact, it represents an example of a Hermitian symmetric space of non-compact type, a Siegel half-plane of degree g. The fundamental fact is the Torelli Theorem which asserts that this map is an isomorphism onto its image. This gives a moduli theoretical interpretation of some points of the orbit space. All points can be interpreted as the period matrices of principally polarized abelian varieties, i. The development of the general theory of periods of integrals on algebraic varieties in the s due to P. For the spaces of classical types I—IV, this can be achieved by embedding any type of space into a Siegel half-plane, and introducing the moduli space of abelian varieties with some additional structure like a level, complex multiplication or some tensor form on cohomology.

All of this becomes a part of the fancy theory of Shimura varieties. However, a more explicit interpretation remained to be searched for. The fundamen- tal result of I. Piatetsky-Shapiro and I. Shafarevich in the s gives an analog of the Torelli Theorem for polarized algebraic K3 surfaces. Recently, some arithmetic quotients of type IV domain of dimension 20 have been realized as periods of holomorphic symplectic manifolds of dimension 4. A complex ball is an example of a Hermitian symmetric space of type I. In Section 6 we discuss in a more general setting the theory of what we call eigenperiods of algebraic varieties.

The periods of hypergeometric functions allows one to realize some complex ball quotients as the moduli space of weighted semi-stable ordered point sets in projective line modulo projective equivalence. In some cases these moduli spaces are isomorphic to moduli spaces of other structures. For example, via the period map of algebraic curves, the hypergeometric complex ball quotients are mapped onto a subvariety of Ag parametrizing principally polarized abelian varieties with a certain cyclic group action. Its Jacobian variety is an abelian variety of dimension 4 with a cyclic group of order 3 acting by automorphism of certain type.

The moduli space of such abelian varieties is a complex ball quotient. Another example is the moduli space of equally marked sets of 5 points which leads to the moduli space of marked del Pezzo surfaces of degree 4 [K3]. Other examples relating the arithmetic complex ball quotients arising in the Deligne—Mostow theory to moduli spaces of Del Pezzo surfaces were found in [MT], [HL].

It turns out that all of these examples are intimately related to the moduli space of K3 surfaces with special structure of its Picard group of algebraic cycles and an action of a cyclic group. In Section 10 we develop a general theory of such moduli spaces. Some of these examples arise from the Deligne—Mostow theory.

We conjecture that all Deligne—Mostow arithmetic complex ball quotients are moduli spaces of K3 surfaces. We would like to thank the organizers of the Summer School, and especially Professor Uludag for the hospitality and for providing a stimulating and pleasant audience for our lectures. Introduction to Hodge theory In this and the next four sections we give a brief introduction to the Hodge theory of periods of integrals on algebraic varieties. We refer for details to [GH] or [Vo]. Let M be a smooth compact oriented connected manifold of even dimension 2n.

It is also symmetric if n is even, and skew-symmetric if n is odd. Ak M with complex valued smooth k-forms. Then local coordinates t1 ,. We denote the holomorphic part by TX. Each cohomology class has a unique representative by a harmonic form, i. By a theorem of Kodaira this implies that X is isomorphic to a complex projective algebraic variety. In fact, this was shown in the 19th century for Riemann surfaces, i.

A polarization of AHS on V is a a non-degenerate bilinear form Q on V which is symmetric if k is even, and skew-symmetric otherwise. It is a closed algebraic subvariety of the product of the Grassmann varieties G fp , VC. Other conditions are open conditions in complex topology. Now, suppose we have a family of compact connected complex manifolds. One can prove and this is not trivial! This is not possible in general.

This image is called the monodromy group of f. Fix a smooth manifold M underlying some complex manifold. Consider the set of isomorphism classes of complex structures on M. If M exists it is called a coarse moduli space of complex structures on M. Fine moduli spaces rarely exist unless we put some additional data, for example a marking on cohomology as in 3. A Local Torelli Theorem is the assertion that this map is a local isomorphism which, together with Global Torelli Theorem, will assert that the map is an embedding of complex varieties.

Moduli of K3 Surfaces Consider a Hodge structure of weight 1 on V. This allows us to transfer the structure of a complex space on W to V. We will often identify V with W by means of this isomorphism. Now let us see the meaning of a polarized AHS of weight 1. The polarization form Q makes the pair V, Q a real symplectic vector space. It follows from the symmetry condition that the operator I is an isometry of the symplectic space V, Q.

This shows that the complex structure I is positive with respect to Q. Conversely, suppose I is a positive complex structure on V, Q. Let V 01 be the i-eigensubspace of VC of the operator I. This checks property ii of the Hodge structure. This checks property iii. Thus we have proved the following. Lemma 4. There is a natural bijection between the set of Hodge structures of weight 1 on V with polarization form Q and the set of positive complex structures on V with respect to Q. The restric- tion of H to the diagonal is a real-valued quadratic form on the real space E, and the signature of H is the signature of this quadratic form.

Obviously, formula 4. Let e1 ,. These vectors form a basis of VC. This shows that H is of signature g, g. We have proved the following. Let V, Q be a real symplectic space. There is a natural bijection between Hodge structures on V of weight 1 with polarization form Q and points in G g, VC H , where H is the associated hermitian form of Q. Let E be the real subspace of V spanned by the last g vectors of the standard symplectic basis.

The conditions 4. This checks that Sp 2g, R acts transitively on Zg. Thus a compact subgroup K could be taken to be a subgroup of Sp 2g, R isomorphic to the unitary group U g. The group of holomorphic isometries acts transitively on a hermitian symmetric space X and its connected component of the identity is a connected Lie group G X.

The isotropy subgroup of a point is a maximal compact subgroup K of G X which contains a central subgroup isomorphic to U 1. Any hermitian symmetric space is a symmetric space with respect to the canonical structure of a Riemannian manifold. It is called irreducible if the Lie group is a simple Lie group. For AHS of weight 1 this of course cor- responds to the operation of the direct sum of symplectic spaces and the direct sum of complex structures.

It follows from 5. As is easy to see the converse is true. Comparing this with 2. A pair T, Q is called a polarized complex torus. Recall that an abelian variety over complex numbers is a projective algebraic variety isomorphic, as a complex manifold, to a complex torus. It is called the period matrix of T. One can restate the condition 5. These are the so-called Riemann—Frobenius conditions.

Any one-dimensional torus is an abelian variety an elliptic curve. Thus it is not a projective algebraic variety. The vector d1 ,. We will always represent it by a primitive D i. A polarization of type 1,. Clearly, isomorphic po- larized varieties have the same type of polarization. Any polarized abelian variety of type d1 ,. The period matrix of this form is called a normalized period matrix. It is easy to see that any such matrix arises in this way from a matrix 0g D from Sp 2g, Z D. Then 5. Summarizing, we obtain the following. Theorem 5. It is a lattice in the complex space H 01 X.

This just replaces the complex structure of the torus to the conjugate one. Now let Q be a polarization of the Hodge structure on V with hermitian form H. It is called the dual polarization. It is also a duality of polarized Hodge structures. Let H be the associated hermitian form on VC and h be its restriction to W. In this case J n X is called the intermediate Jacobian of X. However, in one special case they are. This is the case when all summands except one in V 10 are equal to zero. In this case J is a polarized abelian variety. Let T be a connected complex variety.

The abelian variety X can be uniquely reconstructed from Pic0 X as the dual polarized abelian variety. This proves the Global Torelli Theorem for polarized abelian varieties. Let M be a compact smooth oriented 2-manifold. We choose a complex structure on M which makes it a compact Riemann surface X, or a projective nonsingular curve. It gives a principal polarization on Pic0 X.

### Categories

The Torelli Theorem for Riemann surfaces asserts that two marked Riemann surfaces are isomorphic if and only if their normalized period matrices with respect to the symplectic matrices are equal. Thus we see that two Riemann surfaces are isomorphic if and only if the corresponding Jacobian varieties are isomorphic as principal abelian varieties, i.

Thus the Torelli Theorem for Riemann surfaces is the statement that two Riemann surfaces are isomorphic if and only if their Jacobian varieties are isomorphic as principally polarized abelian varieties. Clearly, this also establishes the Global Torelli Theorem for Hodge structures on one-dimensional cohomology of Riemann surfaces. Note that similar to 2.

Note that in this case the decomposition 7. It is obviously injective since one can reconstruct the Hodge structure from the image using 7. Fix a real vector space V together with a non-degenerate bilinear form Q0 , symmetric resp. Conversely any complex auto- morphism of VC preserving the Hodge decomposition and Q arises from a R-linear automorphism of V by extension of scalars. Also W and W are orthogonal with respect to H. It is known that E admits a basis e1 ,. The condition on Z is Z. Any hermitian symmetric space of non-compact type I is isomorphic to such a domain. We construct the inverse of the map 7.

Also, using 7. Its restriction h to L is of signature p, q. Let E be a positive subspace of L of dimension p. This automorphism is of type p, q , i. After an obvious change of a basis this matrix becomes a matrix 7. Arrangements of hyperplanes Let H1 ,. We assume that the hyperplanes are in a general position. This is a nonsingular algebraic variety which can be explicitly described as follows.

The group Ad m acts naturally on X via multiplication of the coordinates xi by dth roots of unity. Lemma 8. One can show see [DM], p. Now consider the long exact sequence. We only sketch the proof. Another proof was given by P. Deligne [De]. One can compute explicitly the Hodge decomposition of a nonsingular complete inter- section n-dimensional subvariety X in projective space PN.

Let y1 ,. If G is a group of auto- morphisms of X induced by linear transformations of PN , then its action on the cohomology is compatible with the action on the ring R F the representation on this ring must be twisted by the one-dimensional determinant representation. In our case the equations fi are very simple, and the action of the group Ad m can be explicitly computed. Using the isomorphism from Lemma 8. Consider the family X parametrized by the set of all possible ordered sets of m hyperplanes in general linear position.

The set of equivalence classes is an algebraic variety Xn,m of dimension mn. Ge- ometrically this means a projective transformation sending a collection of hyper- planes to a projectively equivalent collection. The following result is a theorem of A. Varchenko [Va]. Theorem 8.

Then the eigenperiod map. Comparing with the formula from Lemma 8. By Theorem 8. Now recall some constructions from Geometric Invariant Theory. Let S be the projective coordinate ring of the image of the product in PN. Let S G be the subring of invariant elements. A standard construction realizes this ring as the projective coordinate ring of some projective variety. Let U s resp. U ss be the open Zariski subset of P1 m parametrizing collections of points p1 ,. This gives additional cases, with largest m equal to Recall that, by Lemma 8.

Thus the action of C3 on the Jacobian variety is of type 1, 3. As we see in Section 11, the locus of principally polarized abelian 4-folds admitting an auto- morphism of order 3 of type 1, 3 is isomorphic to a 3-dimensional ball quotient. This agrees with the theory of Deligne—Mostow. K3 surfaces This is our second example.

This time we consider compact orientable simply- connected 4-manifolds M. Recall that by a theorem of M. The corresponding integral quadratic form is called in such case a unimodular qua- dratic form. The group H2 M, Z equipped with the cup-product is an example of a lattice, a free abelian group equipped with a symmetric bilinear form, or, equivalently, with a Z-valued quadratic form. We will denote the values of the bilinear form by x, y and use x2 to denote x, x. We apply the terminology of real quadratic forms to a lattice whose quadratic form is obtained by extension of scalars to R.

The even type is when the quadratic form takes only even values, and the odd type when it takes some odd values. Its Sylvester signature is p, q. Note that changing the orientation changes the index I M to its negative. Theorem 9. It is enough to construct such a complex surface realizing the quadratic form LK3. Let X be a nonsingular quartic surface in P3 C. By the Index formula 2. Thus H 2 X, Z is an even lattice. Let X be an algebraic K3 surface e. The polarization form Q admits an integral structure with respect to the lattice H 2 X, Z.

Choose a marking. It is a sublattice of signature 2, Note that, in coordinates, the period space Dl is isomorphic to a subset of lines in P20 whose complex coordinates t0 ,. One can give another model of the period space as follows. It is easy to construct the inverse map. If v arises as the period of a K3 surface, this switching is achieved by the changing the complex structure of the surface to the conjugate one.

This description shows that the period space is not connected. In coordinate description of Dl the connected components are distinguished by the sign of Im tt It is known that the K3-lattice LK3 represents any even number 2d, i. This is easy to see, for example, considering the sublattice of LK3 isomorphic to U. As always in this case we assume that the polarization class h belongs to H 2 X, Z , i. We can always choose h to be a primitive element in H 2 X, Z. We call it a marking of a polarized K3 surface. Con- versely, suppose this happens. The following is a fundamental result due to I.

Shafarevich and I.

- Doppelte Zitate.
- Product | Arithmetic and Geometry Around Hypergeometric Functions?
- Homological Algebra II [Lecture notes].
- Sample Survey Methods and Theory, Volume II Theory;

In fact, it has a structure of a quasi-projective algebraic variety. Note that, although the period space Dl is not connected, the orbit space is an irreducible algebraic variety. This is truly one of the deepest results in mathematics generalizing the Torelli theorem for curves and polarized abelian varieties. However, it is isomorphic to a constant sheaf when the base T is simply connected. One can construct a coarse moduli space of polarized K3 surfaces.

We denote it by M2d.

A further result, the Local Torelli Theorem asserts that this map is a local isomorphism of complex varieties, and hence the period map is an isomorphism with its image. Before we state a result describing the image of the period map let us remind some terminology from algebraic geometry. Let Pic X be the Picard group of X. It is a subgroup of H 2 X, Z spanned by the fundamental classes of irreducible 1-dimensional complex subvarieties curves on X.

In this case the line bundle becomes isomorphic to the restriction of the line bundle corresponding to a hyperplane. The next result describes the image of the period map 9. It is a union of countably many closed hypersurfaces in Dl. We call it the dis- criminant locus of Dl. So we see that the image of the period map is contained in the complement of the discriminant lo- cus. In fact, it gives more. By weakening the condition on the polarization one can consider pseudo- polarized K3 surfaces as pairs X, h , where h is a primitive divisor class corre- sponding to a pseudo-ample line bundle.

Then one proves the following. The singular points are rational double points of types corresponding to the irreducible components of the lattice RX. The components are distinguished by the sign of x1. Take the Picard lattice SX as M. The corresponding con- nected component is denoted by KX 0 and is called ample cone. It turns out that the isomorphism classes of M -polarized K3 surfaces can be parametrized by a quasi-projective algebraic variety MM.

The construction is based on the period mapping. We shall identify M with its image. Then we consider a marking of X, j. The space DM is isomorphic to the period space of such abstract Hodge structures. Theorem Let X, j be a M -polarized K3 surface.

## Analytic Continuation of Hypergeometric Functions in the Resonant Case - INSPIRE-HEP

We refer for the proof of these as- sertions to [Do]. It is denoted by pX,j and is called the period of M -polarized surface X. Applying Theorem MM,K3 of M -polarized resp. The same is true and the same proof works if the lattice N contains a direct summand isomorphic to U [Do]. Eigenperiods of algebraic K3 surfaces First we recall the fundamental result due to Nikulin [N2].

Remark See Section By condition Consider the degeneracy lattice R X, j of X, j. Lemma Let R be as in Lemma By induction, we can prove the following result. Examples In this section, we shall give examples of eigenperiods of K3 surfaces.

For a lattice L and an integer m, we denote by L m the lattice over the same Z-module with the symmetric bilinear form multiplied by m. Using Theorem Choose a standard basis r1 , r2 of A2 corresponding to the vertices of the Dynkin diagram. It is also isomorphic to the moduli space of principally polarized abelian varieties of dimension 4 with action of a cyclic group of order 3 of type 1, 3. Del Pezzo surfaces of degree 2 Let R be a smooth del Pezzo surface of degree 2.