Sugaku Expositions
Volume 1, Number 2, December 1988

MODULIOFVECTOR

BUNDLESO N K 3 S U R FA C E S , A N D S Y M P L E C T I C M A N I F O L D S

SHIGERU

MUKAI

K3 surfaces have been studied from old times as quartic surfaces or as Kummer surfaces. The name 'K3' itself was introduced only a quarter of a century
ago. Since then remarkable progress has been made in its study. In the sixties,
the foundations were layed for the modem study on their position in the clas
sification of surfaces, on their moduli space, and on their period mapping, etc.
In the seventies, the Torelli type theorem was established, which is the main

source of further progress. Now a generalization to higher dimensions is triedand geometries (singularity, automorphism, degeneration, etc.) of K3 surfaces
are studied in detail by combining with theories in other fields.
The concept of moduli has long been known. For example, it has been wellknown that the number of moduli of Riemann surfaces of genus g is equal to3^ - 3. It has been widely understood that the automorphic function is noth

ing but the function on the moduli space of elliptic curves. In a broad sense,a moduli space is the set of equivalence classes (isomorphism classes in mostcases) of a certain type of geometric objects, endowed with a suitable structure.
Among geometric objects are manifolds, submanifolds in a fixed manifold, vec
tor bundles on a manifold, etc. Among stmctures are topology, differentiable

structure, complex structure, etc. For each type of geometric object and foreach structure, we can study the moduli problem. In this article, we restrict

ourselves to the moduli of vector bundles' . But, even under this restriction,
we meet various situations depending on which vector bundles we consider onwhich manifolds. As an example, let us consider complex topological vector
bundles on topological spaces (in the category of CW complexes). In this case,there exists a vector bundle on a topological space B with the following
This article originally appeared in Japanese in Sogaku 39 (3) (1987), 216-235.
1980 Mathematics Subject Classification (1985 Revision). Primary 14D22, 14F05, 14J28; Sec
ondary 14D15, 14J40,
All footnotes were added in translation.

' The reader may consult Seshadri's survey article [ 113] for a more rigorous introduction to thealgebro-geometric moduli problem. There he discusses the (local and global) moduli problem of(sub)varieties and of vector bundles and the construction of the moduli by means of the geometricinvariant theory.
©1989 American Mathematical Society
0898-9583/89 $1.00 + $.25 per page
139

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universal property: For every topological space X, the mapping{continuous mapping from X to 5}/homotopy equiv.
{vector bundle on A'j/isom.

[f:X-^B] ^[r^]
isbijective. B is called the classifying space and ^ the universal vector bundle.
In the moduli problem discussed in the sequel, we always fix a complex manifold
X and study the set of isomorphism classes of holomorphic vector bundles
on X. Though it seldom exists and we are forced to make a concession and amodification, the set with a structure of complex analytic space is a moduli
space in the most ideal sense if there exists a holomorphic vector bundle on
the product X with the following universal property: For every analytic
space S, the mapping{holom. mapping from S to V^} {holom. vector bundle on A' x 5}/equiv.

[/:5_KjH-.[(l^x/)X]is bijective. To a vector bundle F on the product XxS there is associated a set{F| of vector bundles on X. This is regarded as a family of vector bun
dles on X which vary holomorphically on the parameter 5. The holomorphic

mapping f:S—^V^ corresponding to F as above is called the classification
mapping of F . Thus, the moduli space controls how holomorphic vector
bundles on X vary holomorphically.

A moduli space parametrizes geometric objects of a certain type. Once it isconstructed, the moduli space itself becomes an interesting geometric object ofstudy. Absolute moduli spaces, such as the above classification space B andthe moduli spaces of abelian varieties and curves, have rich geometric structuresand plenty of symmetries. For relative moduli spaces, such as the above,
we are interested in how inherits various properties (cohomology group,
Riemannian metric, structure of algebraic (projective) variety and the field ofdefinition, etc.) from X. In this article, we study this problem in the case ofK3 surfaces. We are especially interested in how the moduli space of vector
bundles inherits the symplectic structure and the period from K3 surfaces. We

also discuss the relation with the theory of (holomorphic) symplectic manifolds,
higher dimensional analogues of K3 surfaces.
1. K3 surfaces
2 . Ve c t o r b u n d l e s o n K 3 s u r f a c e s

3. Symplectic structure of the moduli spaces
4. Higher dimensional symplectic manifolds
5. Period of the moduli space
6. Notes on references

2 Two vector bundles F and f on the product X x S are equivalent if there exists a line
bundle L on S such that . Equivalent vector bundles F and F' induce the samefamily {/='Uxj}j6S = {/^'l A-xihes of vector bundles on X.

MODULI OF VECTOR BUNDLES

141

Ve c t o r b u n d l e s
4-(period, metric and projectivity, etc.)
Symplectic
Moduli spacestructure

Notation. An (exterior) differential form of degree r is simply called an r-form ^. Let A' be a complex manifold. The sheaf of holomorphic r-forms on
X is denoted by . In the case r = dimAf, an r-form is called a canonicalform and the sheaf is called the canonical (line) bundle. Holomorphic
0-forms are simply holomorphic functions. The sheaf is denoted byand called the structure sheaf of X.

For a vector space or a vector bundle E, we denote its dual by .
1. K3 SURFACES

In a word, K3 surfaces are 2-dimensional analogues of elliptic curves. K3

surfaces and 2-dimensional complex tori have many common properties andtheir position in all complex surfaces is almost the same as that of elliptic curves

in all curves (i.e., compact Riemann surfaces). On one hand, every elliptic curve
E is expressed in the following way:
(A)

E

=

cir.

r~z©z

as a one dimensional complex torus. On the other hand, it has many projectivemodels. Among them the Weierstrass standard form is the most famous. By
using the p-function is expressed in the form

E.Y^ = 4X^-g^X-g,. X = p{z). Y = p'{z).

^. = 60 EA. ^3 = 140 E 7This shows that £ is a double cover of the (complex) projective line P' branch
ing at four points. This also shows that .E is a smooth cubic curve in the pro

jective plane . If we use the i3-functions ^, then we obtain Jacobi's standard^ Do not confuse the r-form with the following: If {A'o X„} is a system of homogeneous
coordinates of the projective space P" , then a homogeneous polynomial F{Xo X„) of degreed is called a form, of degree rf on P" .

^ For a suitable coordinate z of the universal covering C of £", the p-function is defined byp(z) = i/z2 + - y?- - l/y^).
^ Replacing by a suitable affine transformation, we may assume that F = Z+Zr and Im t > 0.
Put q = . Then fl-functions of E are defined byd3(z) = = I-h 2 f^q"' COS Innz .
nez

n=l

i?2(z) = ^'^V'-i?3(z + §) = cos(2/i + l);rz ,I?i(z) = ^2(2+5) and i?o(z) = i?3(z + i).

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(B.2)

MUKAI

f = kx,^^- k'x.^, X. = -dAz).

E:{
\
'
,
[ X^ = kX^ -k!x^ , i = 0,1,2,3.

k = 6,\0)/6,\0). A:' = V(0)/i?j'(0).
This shows that £ is a complete intersection ^ of two quadratic surfaces in the
projective space .

Among all the curves, the elliptic curves are characterized by the property
that they have nowhere zero holomorphic canonical forms. There are exactly
three types of surfaces with this property (Kodaira [46, I, §6]), One is the 2-

dimensional complex tori and another is the K3 surfaces ^. They inherit (A)and (B) from the elliptic curve, respectively.
Definition (1.1). A surface (i.e., 2-dimensional compact complex manifold) is
Si K3 surface if it satisfies(1) there exists a holomorphic 2-form w e H^{S, fl^) without zeroes, and(2) the first Betti number is equal to zero.
By (1), K3 surfaces are symplectic manifolds in the following sense.
Definition (1.2). A closed holomorphic 2-form cu on a complex manifold X isa {holomorphic) symplectic structure if co is nowhere degenerate, i.e., the skew-

symmetric bilinear form - h x ^ h x

C on the tangent space t.

is nondegenerate at every point x E X,
Every two K3 surfaces can be deformed to each other (ibid., §5). The isomorphism classes of all K3 surfaces are locally parametrized by a 20-dimensional

complex manifold. It is known that every K3 surface has a Kahler metric ® (Siu[86], cf. [8]). In this article, we do not treat nonalgebraic K3 surfaces. All alge

braic K3 surfaces^ are parametrized by a countable union of 19-dimensional
algebraic varieties. This relationship between all K3 surfaces and algebraic K3
surfaces is similar to that between 2-dimensional complex tori and abelian sur
faces.

Now we give some examples of (algebraic) K3 surfaces.

Example (1.3) (Quartic surface). Let {Xq,X^ ,X2,X^} be a system of homoge
neous coordinates of the projective space and / a homogeneous polynomial® An intersection J'=yin---ny„cA' of subvarieties Y\ Yn in X is a complete

intersection if the codimension of y in A' is equal to 52/= i codim;^ y, at every point of Y .

' The third type of surfaces with trivial canonical bundles is called Kodaira surfaces. They areneither algebraic nor Kahler. Their first Betti numbers are equal to 3.

® A Hermitian metric JZ" ,p=i of a complex manifold is a Kahler metric if its

fundamental form (v^/2) 5^,'J g^^jdz" A d'z^ is closed. A smooth projective variety always
has a Kahler metric.

' Every (smooth compact) algebraic surface has a projective embedding. In particular it has a

Kahler metric.

MODULIOFVECTOR

BUNDLES143

of degree 4 in the variables Xq.X^ , A'2 , and X^. The set of zeroes

S: f(Xg.X^.X^,X^) = 0 inP'
of / is a K3 surface if it is smooth at every point seS, that is, if the partial
derivatives df/dX. (/ = 0,1,2, and 3) have no common zeroes.
The quartic surface S satisfies (2) in Definition (1.1) by the theorem ofLefschetz. We show that S also satisfies (1). Let Uq be the open subset ofdefined by X^ ^ 0. Then Uq is an affine 3-space with the system of
coordinates XJXq , X^/Xq , and XJXq . We expand the 3-form= d{XJXQ) A diX^Xo) A d{XJXQ)
on Uq formally and obtain 4*^ = ^/Xq , where we put

T' = XQdX^ A dX^ A dX^ - X^ dXQ A dX^ A dX^^X^dXQ^dX^^dX^-X^dXQ^dX^ AdX^.
Hence the 3-form'¥/f{X„ .x,.x,.x,]=v„/ni. X, /X^, X,/X^. X,/X^)
has simple poles along the intersection 5" n C/q and is holomorphic on Uq\S .
This is the same for the other open subsets U-: X.^^ (/ = 1,2, and 3). Hence^//(^o '^x'Xj, ^3) is a meromorphic 3-form on with simple poles along
S. The residue Res^(4'//) of 4^// along S is defined as a meromorphic
2-form cu on 5. Since S is smooth, cu has no zeroes or poles. So we have
proved (1.3).

The above argument also works for higher dimensional projective spaces P" .
We putni = 0

for a system of homogeneous coordinates Xq, ... .X^ of P" . If /{Xq .XJ
is a homogeneous polynomial of degree n + I, then 4*// is an /?-form (orcanonical form) holomorphic on / ^ 0 and has simple poles along f = 0. By
this fact, we obtain another example of a K3 surface.

Example (1.4). Assume that in the projective 4-space P'', a quadratic hypersurface

Q:q{XQ.X,.X^,X^.X,) = 0
and a cubic hypersurface

D'.d{XQ,X,,X^,X^.X^) = Q
If K is a smooth ample divisor of a smooth projective algebraic variety X , then the natural
homomorphism H'{X. Z) -♦ H'{Y . Z) is an isomorphism for every 0 < / < dimX (cf. [97] and[98]).

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MUKAl

intersect transversaliy, that is, two vectors {dqldX^, ,dqfdX^ and{dd/dX^ , ... , ddjdX^) are linearly independent at every point of QdD. Then

S = QnD is a K3 surface.

In fact, taking residues of the 4-form ^/qd first along Q and next along
S, we obtain a holomorphic 2-form on 5. In a similar way, we also obtain thefollowing two examples.

Example (1.5). Assume that three quadratic hypersurfaces

Q.:q.(Xg.X^.X2.X^.X,,X^) = 0. i = 0.l, and 2intersect transversaliy in the projective 5-space . Then the intersection S =Qj n 02 ^ 03 is a K3 surface.
Example (1.6). Let C: yiX^ ,X^,X.^) = Q be a smooth sextic curve in the projective plane and let

Y^ = y{X„.X^,X^)
be the double covering which ramifies exactly along C. Then S is a K3 surface,{n*^/Y is a nowhere zero holomorphic 2-form on S.)
Each example above of a K3 surface carries a natural polarization (an equiv
alence class of finite morphisms to projective spaces). The next example is aclassical one but has no natural polarization.

Example (1.7) (Kummer surface). Let T = C^/T, T ^ , be a 2-dimensional complex torus and i the symmetry t —t of T with respect to theorigin. The fixed point set of i coincides with the set of 2-torsion points jT/T.

Hence the quotient space Tji has sixteen ordinary double points '*. Taking
the minimal desingularization of T/i ,v/o obtain a K3 surface. We call this K3
surface the Kummer surface associated to T.

2.

VectorbundlesonK3

surfaces

In this section, we give some examples of vector bundles on K3 surfaces andshow the existence of a symplectic structure on the moduli space in two concrete
examples.

Definition (2.1). A holomorphic mapping n: E ^ X between complex manifolds is a holomorphic vector bundle of rank r if there exist an open coveringof ^ and a family of biholomorphic mappings (pr. 7r~'(C/.) ^ C'^x C/.,'' An n-dimensional hypersurface singularity O € {/{Xq X„) = 0} is an ordinary double

point if the initial form of / is quadratic and nondegenerate. The singularity is resolved by a single
blowing-up. The exceptional divisor D is isomorphic to a smooth (n - 1 )-dimensional quadric
Q cP" and its normal bundle is isomorphic to . In particular, in the case n = 2, D is
isomorphic to P' and the normal bundle is of degree -2 .
A vector bundle of rank one is called a line bundle.

^

MODULIOFVECTOR

BUNDLES145/ e /, which satisfy(ti) for every pair of i and j € I, the difference of two mappings (p. and(Pj over the intersection U. n Uj is expressed by a holomorphic function

U.nUj^GL{r,C)C&
to GL(r. C), that is,

((Pil

°

(^1

■' )

=

•')

holds for every vector v eC and / e (7,. n Uj.
In the above definition, if we assume further that X is an algebraic variety,

C//s are Zariski open subsets, and are restrictions of rational functions

on X, then E is called an algebraic vector bundle on ^. If the base manifold
X is a complete (or compact) algebraic variety, then by the GAGA principle(Serre [84]), every holomorphic vector bundle on X is algebraic. In the sequel,
vector bundle (and its section) always means a holomorphic one unless otherwise
specified.

First we take a K3 surface S in Example (1.5). Let W be the vector space of

quadratic forms q{XQ, X^. X2. X^. X^ , X^) = 0 on which vanish identicallyon S. Then is a 3-dimensional vector space with basis q^y and q^
defining S. In other words, the set N of quadrics of containing 5 is a

projective plane spanned by Qq, Q^, and Q2. Let A. be the symmetric 6x6
matrix corresponding to the quadratic form q., for i = 1,2, and 3. A quadric

Q: Q!Q^QH-a,^j+a2^2 = ^ is smooth if and only if the matrix aQ^Q+a,^,+a2^2
is regular. Hence the set of singular members in N coincides with^o = {Q' "0^0 + "1^1 + ^2^2 = 0 in P^ I det(ao^o + a^A^ + a2^2) = 0}•
Since det(Q:Qy4Q + a^A^ + 02^12) = 0 is a homogeneous polynomial of degree 6

in the variables Qq , a,, and a2, is a. sextic curve in ~ P^.
Example (2.2) ([62]). Let S be a K3 surface in Example (1.5) and assume

that every quadric containing S is of rank >5. Let h e H^{S ,Z) be thecohomology class (i.e., the Poincare dual of the homology class) of hyperplanesections of 5 c P^. Then the moduli space of stable (with respect to S c P^)

A subset of an algebraic variety (resp. a compact complex manifold) is Zariski open if its
complement is a closed algebraic (resp. analytic) subset.

A quadratic hypersurface is simply called a (hyper)quadric. N is called a net of (hyper)
quadrics. See [115] for the general theory of nets of quadrics.
Let A" C be an n-dimensiona! projective algebraic variety. We denote the restrictionof the tautological line bundle by and its /cth power by <^x{k) • For a coherent sheaf F ,there exists a polynomial (/) such that Pfik) is equal to the dimension of the space of global

sections of F ®(^x{k) for ^ » 0. Ppit) is called the Hilbert polynomial of F . P/r(0 is apolynomial of degree < n and «! times the coefficient of /" is equal to the rank r{E) of £". A
torsion free coherent sheaf E is (semi-)stable (with respect to A C in the sense of Gieseker[29]) if Pf(k)/r{F) < PE{k)/r{E), /c » 0 (resp. < ) holds for every proper nonzero subsheaf F
of

E.

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MUKAI

rank 2 vector bundles with = h and Cj = 4 is a K3 surface described in(1.6). Moreover, it is canonically isomorphic to the double cover of iV ~with branch the sextic curve A'q. (Bhosle [12] generalizes this to complete
intersections of three quadrics in P" .)Now we explain the above relationship between the vector bundles on S andthe net of quadrics N. We recall that the Grassmann variety Grass(P' C P^)

of lines in the projective space P^ is a smooth quadric Pi2P3^ ~ /'i3/^24 +^14^23 = 0 in P' by the Pliicker coordinates . For a point p (resp. a planeP) in P^, let Lp (resp. Lp) be the subset of Grass(P' c P^) consistingof lines passing through p (resp. contained in P). Both and Lp areplanes contained in Grass(P' c P^) C P^. The family of planes L^'s areparametrized by P^ and Lp's by the dual projective space of P^. All smoothquadrics in P^ are isomorphic to each other. Hence we have proved that every
smooth 4-dimensional quadric Q contains two families of planes parametrized
by projective 3-spaces. Take a family of planes on Q and denote it by{P^cQ \ Pj is a plane in , A p\
For every point x of Q, the parameters t with xeP^ form a line in A, which
we denote by . Let V be the 4-dimensional vector space of linear forms on
A. Then we obtain the exact sequence

where is the space of linear forms that vanish on and is the space of

linear forms on . Both F^ and E^ are of dimension 2. So we define a rank
2 subbundle F^ and a rank 2 quotient bundle E^ of the trivial vector bundle
Let W be an r-dimensional subspace of a vector space V . Then the rth exterior prod
uct W is a 1-dimensional subspace of • The Pliicker coordinate of W is the pointof P.(A'^ corresponding to A*^ • By the Pliicker coordinates, the Grassmann variety of rdimensional subspaces of V (or (r - I)-dimensional subspaces of P.(K)) is a submanifold of■ This embedding Grass(P''~' C P^~') C P<^)~' is called the Plucker embedding. In
our case, = 4 and r = 2 , we put Pij = v, A Vj for a basis {u] .vj .vy , U4} of .

MODULI OF VECTOR BUNDLES

Kx5

147

by

£^ = U£,xM-f'x5.
ses

s€S

Under the assumption in (2.2), is a stable vector bundle satisfying thenumerical condition in (2.2). Moreover, the family is a complete set ofrepresentatives of all the isomorphism classes of such vector bundles, where

A runs over all families of planes in quadrics in N. Associating to each A
the quadric Q swept out by planes parametrized by it, we obtain a morphismfrom the moduli space to N. If a quadric Q degenerates and has a singularpoint, then the two families of planes on it become the same one. Hence thismorphism is generically 2 to 1 and ramifies along . This shows (2.2).

Next we consider the K3 surface 5 c P'* in Example (1.4). Let / be a linethat intersects S at exactly two points (counting with the multiplicities) and let

5 be a point of S. We denote by F) (resp. Fj ^) the space of linear forms onthat vanish on / (resp. on / and at s). Put

ses

Unless s lies on I, Ff ^ is of dimension 2. Hence F/ is a vector bundle over
S\{S n /). F'i extends a vector bundle on all of S by the following:
Proposition (2.3) ([34]). Let X be a 2-dimensional complex manifold and E avector bundle over X minus a point x e X. Then there exists a neighborhood
U of X such that E is trivial over U\{x}. Moreover, there exists a uniquevector bundle E on X whose restriction to A'\{jc} is isomorphic to E.

Example (2.4). Let 5 c P'* be a K3 surface in Example (1.4) and assume that S

contains no lines. Then F/ is a stable (with respect to 5 c P"*) rank 2 vector
bundle with Cj = -h and Cj = 4, where h e H^{S. Z) is the cohomology

class of hyperplane sections of S . Moreover, for every such stable vector

bundle F, there exists a unqiue line / with #(/n5') = 2 and such that F^ c- F.
We show the existence of a symplectic structure on the moduli space in theabove case. By our assumption, / is either a line that joins two distinct points
X and y on S OT a line tangent to 5 at a point x e S. The latter is thelimit of the former as y goes to x inside S {y becomes a 1-dimensional
subspace of the tangent space t^ ^ of S at x). Such a is called an infinitely
near point of x. The set of unordered pairs {x ,y}, where x and y are
distinct points on S or one is an infinitely near point of the other, is denotedby Hilb^5. For every point {.x.y} of Hilb^5, there exists a unique line /that joins x and y . Let 5" x 5 be the blow-up of the product S x S of two

copies of S along the diagonal. Then Hilb^ S is isomorphic to the quotient of
5 X 5" by the involution induced from the factor change. The natural mapping

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MUKAI

Hilb^ 5 —> Sym^ 5 is the minimal resolution of the second symmetric productof S. Moreover, by this description of Hilb^5, we have the following.

Proposition (2.5). If S is a K3 surface, then Hilb^ S has a natural symplectic
structure induced from that of S.
This proposition was first stated by Fujiki and established the existence of
higher dimensional simply connected symplectic manifolds, which had been
uncertain before. The isomorphism classes of stable vector bundles in (2.4)

are parametrized by the open subset(Hilb'S)° = {{x,y}|/,^nS = {x,y}}of Hilb^ S. Therefore, we conclude that, in both cases (2.2) and (2.4), themoduli space of stable vector bundles has a symplectic structure. In the next

section, we show that this always holds over K3 surfaces.
3.

Symplectic

structureofthemodulispacesLet A" be a complex manifold. By the moduli space of vector bundles on X

we mean the set of their isomorphism classes endowed with a natural complex
structure. But if we allow all the vector bundles, then we cannot obtain agood moduli space . We must choose a nice class of vector bundles carefully
according to the property we require of the moduli space. The following are
typical examples of nice classes of vector bundles.

(Aan) Simple vector bundles on a compact complex manifold. The moduli
space is an analytic space that may not be Hausdorff .

(Agig) Simple vector bundles on a complete algebraic variety. The moduli

space is an algebraic space that may not be separated (Altman-Kleiman [2]).
(Consult [48] for algebraic spaces.)(Baig) Stable vector bundles on a projective algebraic variety AT c . The
moduli space is quasiprojective (in particular Hausdorff^'). By adding the
Let w be a symplectic structure of S . Then \= n\o} + is a symplectic structureof SxS. Since w®^ is invariant under the factor change involution i, tu®^lsx5\A descends toa holomorphic 2-form S^(o on (S x S\A)/i C Hilb^5 . It is easy to see that S^(o extends to a
symplectic structure Hilb^w on Hilb^S.
Theorem 2 in [IS] is false. (2.5) is its counterexample.

"There exists a family of vector bundles {£/}/ec such that Ei is isomorphic to a vector
bundle E for every t ^0 but Eq is not. This is called a jumping phenomenon. For example,
let L be a line bundle such that H^{L) b a jtO and H^(L) = 0. By the canonical isomorphism
E x t' { < f.L ) ^ H ^ { L ) , e v e r y l a , t € C , d e te r m i n e s th e e x te n s i o n 0 - * L E , Th e n
the family jumps to Eq=(^ ® L at / = 0. If we allowed such E = E\ and Eq in our
moduli problem, then the point [F] would not be closed in the moduli space.
2° There exists a pair of families of simple vector bundles {£",} and {F,} such that £, F,
for every t 0 and Eq Fq. An example of such a pair is given over a curve of genus 3 inNarasimhan-Seshadri [72, Remark 12.3].

If a coherent sheaf E is stable, then PF{k)/r{F) > PEik)lr(E), /c » 0, holds for every
nonzero quotient sheaf F ^ E. Assume that both E and E' are stable and that E' is adeformation of E . Then every nonzero homomorphism from E to E' is an isomorphism. This
property of stable sheaves eliminates the jumping and non-Hausdorff phenomena.

MODULIOFVECTOR

BUNDLES149

set of certain equivalence classes of semistable sheaves, it is compactified

and becomes a projective scheme (Mumford [66], Narasimhan-Seshadri [72](dim A' = 1), Gieseker [29] (dim A' = 2), Maruyama [56]).

(Ban) Vector bundles with Einstein-Hermitian metrics on a compact Kahler

manifold {X ,g). The moduli space is a Hausdorff analytic space. The Kahler

metric g induces a natural Kahler metric^'' on the nonsingular part of themoduli space (Kobayashi [45]).
Differentiable vector bundles with anti-self-dual Yang-Mills con
nections on a compact Riemannian manifold {X, g) of real dimension 4. The
moduli space is Hausdorff. The Riemannian metric g induces a Riemannian
metric on the moduli space.

A vector bundle £ on Y is simple if every (holomorphic) endomorphismof E is the multiplication by a holomorphic function on A'. If Y is compact,

then every endomorphism of a simple vector bundle is a constant multiplication.
A Hermitian metric /i of a vector bundle E on {X, g) satisfies the Einsteincondition if the mean curvature g~^dd \ogh e C°°{^nd{E)) is a constant mul
tiplication. For a vector bundle on a projective variety, we have
/ / - s t a b l e = > s t a b l e = >■ s e m i s t a b l e = > / / - s e m i s t a b l e

indecomposable E-H simple => indecomposable Einstein-HermitianIn this section, we show that the moduli space of vector bundles on a K3

surface inherits the symplectic structure. We note that this is generalized in thefollowing form.

(C) Simple vector bundles on a compact symplectic manifold {X ,co). The
symplectic structure co induces a symplectic structure on the smooth part ofthe moduli space (Kobayashi [44]).
If E is a semistable sheaf, then there exists a filtration 0 = Eq C Ei C • • • C Ej-i C Es = E

such that each successive quotient F, := Ei/Ei-i is stable and satisfies PfJr{Fi) = PE/r{E).
This filtration is called a JHS-filtration of E. The isomorphism class of the direct sum Gr(F) :=0/=i does not depend on the choice of a JHS-filtration. Two semistable sheaves E and E'

are S-equimlent if Gr(F) ~ Gr(F').

(Bgig) is a beautiful application of the geometric invariant theory developed in Mumford[67] (cf. [77]).
The imaginary part of a Kahler metric induces a real symplectic structure. (Ban) can beviewed as a combination of two inheritances of Riemannian metrics and of real symplectic structures [5, p. 46].
If the Riemannian 4-fold is Kahlerian, then the vector bundles with anti-self-dual Yang-Mills
connections are holomorphic and essentially the same as the Einstein-Hermitian vector bundles in(Ban) ([43] and [103]).

2^ A vector bundle E on an n-dimensional projective variety X is //-stable or stable inthe sense of Mumford and Takemoto [88] (with respect to Y c ) if (C|(F) • h"~^)/ranV.F <(ci(F) •//""'j/rankF (resp. <) holds for every nonzero subsheaf F of F (or <^{E)) withrank F < rank E, where h is the cohomology class of hyperplane sections of X cP'^. Kobayashi[43, 105] proved that every Einstein-Hermitian vector bundle is a direct sum of //-stable bundles
with the same slope and conjectured that the converse holds on projective varieties. In the casedim A" = 1 , this conjecture is essentially the same as the equivalence of the stable vector bundle andthe unitary representation of the fundamental group, which had been proved by Narasimhan andSeshadri [72] (see also [19]). Donaldson [20] has proved this conjecture in the case dim A = 2 .

150

SHIGERU

MUKAl(Cym) Anti-self-dual Yang-Mills connections on a compact Riemannian 4fold with a covariantly constant quaternion structure. The moduli space also
has a covariantly constant quaternion structure (Itoh [39]).

The first (A^j,) is a consequence of the existence of the Kuranishi space forvector bundles ([22, 25, 87]).

Theorem (3.1). Let E be a simple vector bundle on a compact complex manifold
X. Then there exist an analytic space M{E) with a base point * and a vector

bundle ^ on the product X x M{E) which satisfy the following.

(1) The restriction ^\xx* ^ to X x* is isomorphic to E.
(2) Let T be an arbitrary analytic space with a base point *. If is a vector
bundle on X xT with ^ exists a holomorphic mapping(p from a neighborhood of the base point of T to M{E) such that (p{*) = * andr'~(ix^)'r.
(3) The above mapping (p is unique as a germ of holomorphic mapping from( T. * ) t o { M { E ) . * ) .

{{M{E), *) and ^ are called the Kuranishi space and the Kuranishi familyof E, respectively.)
Since simpleness is an open condition , we may assume that the restrictionof ^ to X X r is simple for every point t e M{E). We define topology andcomplex structure on the set of isomorphism classes of simple vector bundles

on X by those of M{E). We denote by SV^ the analytic space obtained in
t h i s m a n n e r.

In order to show some local properties of SV^ and an existence of holo
morphic 2-forms on it, we consider the infinitesimal deformation of vector

bundles. By Definition (2.1), to each vector bundle on X there are associateda pair of an open covering of X and a set of holomorphic mappings

g-^j: Uj n Uj GL{r, C). We denote by GL{r, df^) the sheaf of regular matri
ces of size r whose entries are holomorphic functions. Then g.j^s are sectionsof GL{r and satisfy gijgji^g^j = 1 for every i .j .k e I. Hence the set
is a (multiplicative) 1-cocycle with values in GL{r . Moreover,the set of isomorphism classes of rank r vector bundles is identified with the28

1

cohomology set H {X ,GL{r ,<^^)). Let e be the infinitely small number

such that = 0 and e ^ 0. We put g-j = g.j{ \ +ea,y), where a^j is anr X r matrix whose entries are holomorphic functions on X. The 1-cochain

is considered as a first order infinitesimal deformation of {g,y}, .
" For every family {E,} of vector bundles, the function t >-* dim EndCf/) is upper semicontinuous.

Consult, e.g., [101]. In particular, all the isomorphism classes of line bundles on X areparametrized by the cohomology group H\X . From the exact sequence 0 Z<fx-▶ 1, we have the exact sequence //'(A" ,Z) H^{X .(fx) -* H^{X .(f^) H^(X .Z). By
the Hodge theory, the neutral connected component Coke[A/'(A', Z) H^{X .(fx)] of H^{X .(fj^)
is a complex torus if X is Kahlerian.

MODULI OF VECTOR BUNDLESISl

It is a 1-cocycle if and only if( ^■2 ) S j k < ' l i g j k + < ' j k = O l kholds for every i ,j ,k g I. This is the same as saying that is an(additive) 1-cocycIe with values in the sheaf ^nd{E) of (local) endomorphismsof £■. By this correspondence we obtain the canonical isomorphism(3.3) {first order infinitesimal deformation of £}/isom. c^h\x , ^nd{E)).
If E is simple, by our construction of and Theorem (3.1), this isequivalent to saying(3.4) the Zariski tangent space of SV^ at the point [jE] is canonically isomorphic to the cohomology group h\x ,^nd{E)).

Here the Zariski tangent space at the point p is the dual vector space of thequotient m/m^, where m is the maximal ideal at p. In particular, we have theinequality(3.5) dimj^j < dim {X, ^nd{E)).
The equality holds if and only if SV^ is smooth at the point [£].
For an endomorphism of a vector bundle, its trace is a scalar. Hence we havethe trace homomorphism Tr: ^nd{E) . Associating Tr(/ o g) for eachpair {f ,g) of endomorphisms, we obtain the bihomomorphism^nd{E)x^nd{E)-*(^^.
Since this is symmetrical in / and g, the induced bilinear mapping(3.6) h\x. gnd(E)) >kH\x. ^nd(E)) ^ H^{X
is skew-symmetric. Combining with (3.4), we have (3.7).
(3.7) The Zariski tangent space of has a natural skew-symmetric bilinear

form with values in H^{X at each point.
Let F and G be vector bundles on a compact complex «-fold X and(3.8)
a bihomomorphism with values in the canonical line bundle . This induces
a bilinear mapping(3.9) H\X, F) X H^'^X. G) H"{X . Q^)
for every /. The duality theorem of Serre [82] claims that H"{X ,0."^) is 1dimensional and that (3.9) is nondegenerate if (3.8) is nondegenerate at every
point of X. Applying this fact to our situation {F = G = ^nd{E)), we havethat if X is a K3 surface, then (3.6) is nondegenerate. This and (3.7) are thereasons why the moduli space of vector bundles on a K3 surface has a symplectic
structure. To prove it we need to show the following.

152

SHIGERU

MUKAI(3.10) When is SV^ nonsingular?
(3.11) Does the bilinear mapping (3.6) vary holomorphically as [E] moves inSV^ ? Is the 2-form obtained in this way always closed?

First we consider (3.10). If SVj^ is smooth at the point [E], then the Kuran-ishi space M{E) of E is smooth at * and its tangent space is isomorphic toh\x ,i'nd{E)). Hence shrinking it if necessary, we may assume that M{E)is an open neighborhood of 0 in , N = dim//'{X ,^nd{E)). The Kuranishi family ^ (or = ^\xxt^ Theorem (3.1)) satisfies thefollowing;

The infinitesimal deformations = dE^/dtJ , u = \ ,2, ... ,N ,along at f = (/,,..., ^yy) = 0 form a basis of h\x . ^nd{E)) .
So, we try to construct a family of deformations of E with (0) fora neighborhood T of 0 in in search of a condition for the smoothness ofSV^. We assume that the vector bundle E is given by a 1-cocycle jg/
for a sufficiently fine open covering of X. We deform E by finding

a family of 1-cocycles {G,y(/)}, j^j parametrized by T such that G.y(O) = g^jfor every i J G I. We expand G^j{t) in a power series of t = , ... ,tf^).

G,(')= E C'"=tis)

Let beabasisof H\x,^nd{E)). Weassumethat ,are represented by 1-cocycles ••• • }/,;€/* «>),wemaypu«=satisfies

Gy(0 = Sij + Sij E "y 'nf ■y=lWhen yg/ are defined for |/i| < n so that {<J,y(0}/.yg/ satisfies the 1cocycle condition modulo (r, » we ask whether {\fi\ =n+\) can be chosen so that {<^/y(0}, .ye/ is a 1-cocycle modulo (r, ^An easy analysisleads us to define the 2-cocycles {objy^}. ^ with coef
ficients in the sheaf ^nd{E). Their cohomology classes are denoted by ob^^^
and are called obstructions. The above is possible if and only if its cohomol

ogy class ob^''^ G H^{X ,^nd{E)) vanishes for every fi with |/2| = n + 1. In

particular, SV^ is smooth at [E] if H^{X .^nd{E)) = 0.

If {G,y(0}/.y6/ is a 1-cocycle modulo (/[ . then there exists a family of matrices

ob'^jl whose entries are holomorphic functions on t/, n Uj n C/^ such that Gij{t)Gjf^{t)Gki{t) =1 + E|;i|=«+i mod (/| . For every n with \^\ = n + \ , 's a

2-cocycle with coefficients in ^nd{E).

Assume that all the cohomology classes ob^''^ G H^{X ,^nd{E)) vanish. We can choose"so that the power series G/y(/) = converges in a neighborhood of 0. {G,y(0}/.y6/
d e fi n e s t h e K u r a n i s h i f a m i l y o f E .

MODULI OF VECTOR BUNDLES

153

Now we look for a better sufficient condition for the smoothness of SV^.
Note

that

if

is

a

1-cocycle

modulo

,

then

so

is

{det(G^.(/))}. . The key observation is this. The trace {Tr(ob|J';J)}.
of an obstruction cocycle {ob|.^^}. j is an obstruction cocycle for{det(G^(0)},,,, to extend a 1-cocycle modulo ... . . We denote bydet£ the line bundle defined by the 1-cocycle {det^.^.}. . Then the trace
Tr(ob^''^) is an obstruction for the moduli space (of line bundles) to be smoothat the point [det E]. In other words, the following diagram is commutative.

{obstruction for deformation of E} (obstruction for deformation of det^"}
n

h\x

n

.i'nd{E))^h\x.^^)

But every infinitesimal deformation a e h\x = h\x .^nd{L)) of aline bundle L can be integrated by exp(a) e h\x ,^*) on a complex man
ifold. Hence every obstruction vanishes for deformation of det£. It follows

that Tr(ob^''^) vanishes. So we have
Proposition (3.12). Let E be a vector bundle on a compact complex manifold

X. Then every obstruction for the moduli space SV^ to be smooth at [£:] liesin the kernel of the natural linear mapping

H^(X ,^nd(E)) - H^{X
In particular, SV^ is smooth at [£] if is injective.
Since is a vector bundle, the sheaf ^nd{E) is the direct sum of a structure

sheaf and the sheaf ^nd^{E) of trace zero endomorphisms of £". So thekernel of //^(Tr) is isomorphic to H^{X ,^nd^{E)). Proposition (3.12) andits proof have the advantage of being easily generalized to the case that £ is asheaf that may not be locally free (cf. [62]).

Corollary (3.13). If X is a K3 surface, then SV^ is smooth.

In fact, since the canonical line bundle is trivial, the injectivity of //^(Tr)is equivalent to the surjectivity of the linear mapping C ~ H^{X ,(^y)
H {X ,^nd[E)) by virtue of the Serre duality. The latter holds since E issimple.

Next we discuss (3.11). Let {C/,y(0}, je/ ^ family of 1-cocycles parame

trized by an open subset T of . {G.ft)}^ defines a family of vector
bundles on X. We denote it by . Let (p be its classification mappingfrom T to the moduli space. For every aeT, {d(p)^\ tj ^ h\x ,^nd{Efj)maps the tangent vector dldtf[ to the cohomology class of the (additive) 1-

cocycle {Gij{a)~\dG.j{t)ldt^^)\. Let be the bilinear mapping inSHIGERU

154

MUKAI

(3.7) and put u> = {&>,£,}(£,gspi, • Then we have=r f -idG.-dG..
I T t I' jG : d t ^ ^dGt gJ I
" p / J i J

,

.kei

H\X

By the cocycle condition G^jGjj^G/^i = 1, the pull-back of w by ^9 is expressed as follows:(3.14)
a.p

\

\

"

P

/

f

i.j

.kel

Hence co is holomorphic. The pull-back of its exterior derivative is equal todt^ A dtp A dt^
a.fi .y K \ ^ P ^ y f i.j .k€l

By the cocycle condition of j^/. we have
^

y

y

y

y

Hence it follows thatd(<p<' o)=a\. fi£. yTrL|g,'' 'KI " j V 9'. "I
11 " K J «'/> V " ^'ydt^AdtpAdty = 0.

Thus we have proved the following

Proposition (3.15). Let SVj^ be the moduli space of simple vector bundles on acompact complex manifold. Then the smooth part ('S'F^)reg of SV^ has a closed

holomorphic 2-form (o with coefficients in H^{X such that coincides
with (3.6) for every [E] e (■S'K^)reg •If A" is a K3 surface, then (3.6) is nondegenerate. Combining with (3.13),
we have

Theorem (3.16). If X is a K3 surface, then the moduli space SV^ is smoothand has a natural symplectic structure.

As is easily seen from its proof, the theorem also holds for 2-dimensional
complex tori.

• X uo s3AB3i|s dtduiis JO s3SSBp lusiqdiouiosi JO PS am uo pduyap si ajmotujs

xafduiOD B 'sajpunq jopsA sb Xbm auiBS aqj ui '3du3h "[z,8] ubuipbjx pus njs Xq

pdAOJd SI (ssinpoui--*^ jo) ssABaqs (juajaqoo) joj ssands iqsiuBjn">i jo dauspixs
aqX saABoqs ludjsqoo oj (9r£) uiajoaqxjo uopBziiBJauaS b uiBidxs 3m 3J3h

■ Z + (3^1) uoisuauiip 3jnd jo si {a)^AS "(ZZ'i) (ere) ^9 ' ^ASui pasop puB u3do si {a)^AS 'uoijBuuojap japun iububaui ajB ssBp iuaq3
aq; puB quBJ aqi aauis 'a = {3)a q;iM 3 safpunq jopaA ajduiis jo sassBp

uisiqdiouiosi JO las aqj {a)^AS M 3louap aM '{z' X)H ^ JopaA b joj• Z + (jCjT)^) = {{3)pus 'X)^H^]P = ^AS iZZ'i)aABq aAv '(^re) P"® (l^'e) 3uiuiquiOD 'aouaH• I = {{3)PUS' X)^H uiip = {{3)pus' X)^H tuiPaABq aM '((6*e) '[^g]) uiajoaqj /(iqBnp ajjas sq; /iq uaqi 'aiduiis si 3 ji

:uuoj ajduiis SuiMonoj aq; sauioaaq (£re)■ (Z' X)H B ((• {3] '3 ■ (5)j) = (;?)rt (OZT)
Xq 3 o; pa;BpossB {3) a Jo;oaA aq; augap aM ' uo 3 ajpunq
jo;aaA b joj 'jaAoajopj -aDijjvi e"M p^puajxa aq; (♦) q;iM (z' x)H 11®^ pu®

-(,/•/) + /■'- = ((/ ■,/' Z) • (r' /' J)) (61 •£)Xq (.) ;Dnpojd jaoui aq; pua;xa 9/^'7.®{T X)^H®T. = {T X)H;nd * X 33BJJns uBipqB jo £y[ b joj uuoj JBauqiq iBj3a;ui ub q;iAv ajiipoiu-7 aajj B 'p-i 'aaijjvi b aDnpoj;ui aM '(gre) JO 3uipuB;sjapun ja;;aq b joj■ (Z' X)^H uo SuuiBd uoi;Dasja;ui aq; si (•) puB 'uBipqB jo o adX; jo si x sb

SuipjoDDB 0 JO I o; iBuba si a ' {3)h - (j(^)'a)f + {3)J3 = {3)5 ;nd aM ajaqM' U)-I(3)s + (U) • (?) 'j) - (3)s{3)-i = {J'3)X (81 ■£)aABq aM uaq; 'aoBjjns uBipqB ub joe">I ® SI X JI 'uiajoaq; adX; qao^-uuBuiai^ aq; jo anyiA Xq {3)0 puB ' {3)0'{3)-^ '{3)^ JO suua; ui passajdxa si {3' 3)X oi;sua;oBJBqo ajBOuioj-jainj'3 JO SSBP luaqo aq; puB jiubj aq; aq {3)'^'^ = {3p pu^ {3)j ;ai■ {(j ■ 3)tuo^' x),H mip,(I-)2 = (?'?)' (iI■£)!

Xq {3' 3) JiBd aq; jo oi;sua;oBJBqa ajBauioj-jainj aq; augap 9/^ {' 3^3 oipunq jo;aaA aq; jo suopaas (ibooi) jo jBaqs b si
{3' 3)^0^) ' 3 o\ 3 uiojj suisiqdjouiouioq (iBDOf) jo jBaqs aq; {3' 3)1110^
Xq a;ouap aM ' 3 puB 3 sajpunq jo;aaA jo jiBd b joj suoisuauiip jiaq; a;Bi-naiBO aM ajajj *s;uauoduioD pa;aauuoD jo jaquinu a;iugui ub SBq ^aS"
S H i a N fi a B O J O H A J O n n a o w

156

SHIGERU

MUKAI

We denote by Spl^ the analytic space obtained in this way. For a vector bundle
E, we denote the sheaf of its sections by ^x^E). ^x^^) locally free andthe mapping E >-* (^xi^) gives an open immersion^' of SVx into Spl^ . So
we identify SVx with its image in Spl^^.. All assertions so far for SVx remaintrue and are proved by improving the above arguments if we replace SVx with

Spl;^. and H\X ,^nd{E)) with Ext'(£,£).
Theorem (3.23) ([62]). If X is a K3 or an abelian surface, then the moduli space

Spl;j. of simple sheaves on X is smooth and has a natural symplectic structure.

Moreover, for every simple sheaf E on X, the dimension of Spl^j. at the point[£■] is equal to {v{Ef) + 2.
This honest generalization of (3.16) yields two important corollaries. We

denote by Hilb" X the set of 0-dimensional subschemes N of length n of
X. Hilb" X has a natural complex structure as a connected component of theHilbert scheme Hilb^ of X (Grothendieck [100]). Hilb"A^ is compact if X
is compact. By forgetting the scheme structure of N, v/e obtain the 0-cycle[A^I = Hp mp{N){p) of length n , where p runs the support of N and m^iN)is the dimension (or multiplicity) of at p. [A^] is regarded as a point of the«th symmetric product Sym" X of X. The mapping (p; Hilb" X Sym" X,
fiN !-▶ [A^] is holomorphic. If dim X <1, Hilb X is smooth and connected

32

Hence the mapping ^9 is a desingularization of Sym" X. So we call Hilb" X
the wth Hilbert product of X in the case dimX = 2. For a 0-dimensional

subscheme N of .Y, let be the sheaf of ideals defining N. The sheaf® L is a simple sheaf of rank 1 for every line bundle L on X. Every small

deformation of is also of the form ® L! . Hence the isomorphismclasses of all L with length N = n form an open subset U in Spl^.
If dimX > 2, (8) L ~ ig)L' implies N = N' and L L' . Hence U
fi

33

is isomorphic to the product of Hilb X and the Picard variety PicX of
X. If X is a K3 surface, then Pic A' is discrete. If Jif is a complex torus,
then every connected component of Pic X is isomorphic to the dual torus X

of X. Hence (3.23) implies the following generalization of (2.5), which was
first proved by a different method in Beauville [9].
Corollary (3.24). If X is a K3 [resp. an abelian) surface, then the Hilbert product

Hilb" X {resp. the product X x Hilb" X) has a natural symplectic structure.
Thus we have obtained compact symplectic manifolds as open subsets of themoduli of rank 1 simple sheaves, ^ow we consider the case of rank > 2.
By Definition 2.1, the sheaf (^x{E) is isomorphic to (fx®'' on each open subset Uj. A
sheaf with such an open covering {tZ/l/g/ is called a locally free sheaf (of if^-modules) of rankr . A family of vector bundles E is recovered from a family of locally free sheaves (fx{E) •See Fogarty [26]. In contrast to this fact, Hilb" X can be reducible if dim X > 3 (see [102]).
The moduli space of line bundles on X is denoted by Pic A" and called the Picard variety of
X . Since the tensor product ® induces a group structure. Pic X is also called the Picard group.
If X is projective, then Pic A' is an abelian variety (cf. [112] and footnote 28).

MODULI OF VECTOR BUNDLES

157

The moduli space -SF^(v) (see (3.22)) very rarely contains a compact open

subset In contrast with this, the moduli space Spl^(t;) of simple sheaves
E on X with v{E) = v often contains a compact open subset by virtue of(Bgig). We fix a projective embedding A' c . For a vector v = {r ,l ,s) of
the extended lattice H{X. Z), let M^{v) be the set of isomorphism classesof stable (with respect to AT c P^) sheaves on Af. A stable sheaf is simple andsemistable. Since stability is an open condition (Maruyama [107]), M^{v) is an
open subset of Spl;j.(u). By (Bg,g), M^{v) is naturally compactified by adding
the equivalence classes of nonstable, semistable sheaves E with v{E) = v.
Therefore, if it happens that every semistable sheaf E with v{E) = v is stable,

then M^{v) is compact. This happens, for example, if the greatest common
divisor of the three integers r, {I-h), and s is equal to one , where h is the^ cohomology class of hyperplane sections of X .
Corollary (3.25). Let X be a K3 or abelian surface and v = {r ,l ,s) a vectorof the extended lattice H[X • Z). If GCD(r ,{l - h) ,s) = 1, then every con

nected component of the moduli space M^{v) is a smooth projective variety ofdimension (v^) + 2 with a natural symplectic structure.
Corollary (3.24) is the special case of (3.25) with v = { \ .0 ,e - n). The

moduli space M^(v) is connected in many cases, e.g., if r <2.

Conjecture. The moduli space M^{v) is connected for every u if AT is a K3
or an abelian surface.
4. Higher dimensional symplectic manifolds

In this section, we recall the general theory of compact Kahler manifolds with(holomorphic) symplectic structures. We give some examples of them and posesome problems concerning them.
If cu is a symplectic structure of 2n-dimensional complex manifolds, thenits PfaffianA • • • A w
n limes

is a (holomorphic) canonical form without zeroes. Hence the canonical line
bundle of a symplectic manifold is trivial. Conversely, let AT be a compact
complex manifold with trivial canonical line bundle. We further assume thatAs an example we consider the moduli space of vector bundles in Example 2.4. The K3
surface S is contained in a quadric Q and every line in Q meets S at three points. Hence themoduli space ~ (Hilb^S*)® of stable vector bundles is not compact. Take a curve {x, in

Hilb^5 such that {xt.yt} € (Hilb^5)® for every 0 / e A c C and {xo.yo} ^ (Hilb^5)®,then Iim,_o £/, is not a vector bundle at the third point of /q n 5 , where A is a line joining Xt
and yt.

See [29], [55], or [57, Part I] for a more precise definition of Mx .
If E is semistable and not stable, then there exists a subsheaf F of £ with(l/r(F))(r(F).(c,(F)-/j).^(F)) = (l/r(F)){f(F),(c,(F).A).5(F)) and 0 < r{F)< r(F). Hencer{E), (ci (F) • h), and 5(F) have a common divisor greater than one.

158

SHIGERU

MUKAl

X has a Kahler metric. (This is the case if A' is a projective algebraic variety.)
By virtue of Yau's [95, 96] solution of Calabi's conjecture, X has a Kahler

metric g = {g.^ whose Ricci curvature {R.j) is identically zero. Let X be theuniversal covering of X. The decomposition of the holonomy representation(with respect to g) into irreducible ones induces a decomposition of X into the
product of a complex Euclidean space and Kahler manifolds with irreducible
holonomy representations. (This is called the de Rham decomposition.)
Decomposition Theorem (4.1) (Bogomolov [14], Kobayashi [42], Beauville[9]) . Let X be a compact Kahler manifold and assume that the first Chernclass c, (Y) ^H^{X, Z) is torsion. Then there exists a finite unramified covering
X' of X which is isomorphic to the productr x n. t / . x H ' O .

.

where

(1) T is a complex torus,(2) each C/. is a simply connected projective variety such that H^{U^, Of) =0 for every 0 < p < dim [/., and(3) each Vj is a simply connected symplectic manifold such that

dimH°(K^.,Jl^) = 1.
Remark (4.2). The holonomy group is a special unitary group SU{*) for eachand a symplectic group Sp(*) for V.. The hypersurfaces of degree n + 1

in the projective spaces P" are examples of UjS. Algebraic K3 surfaces satisfy
both (2) and (3).
The manifold V satisfying (3) in the theorem is called an irreducible sym
plectic manifold. The symplectic structure w of K is unique up to constantmultiplications. Moreover, the algebra .OF) of holomorphic forms
on V is generated by o).

For a K3 surface S, its Hilbert product Hilb^S is an irreducible sym
plectic manifold (Corollary (3.24)). For a 2-dimensional complex torus T,

the fibers of the Albanese mapping Hilb""^' T ^ T are irreducible symplec
tic manifolds. We denote their isomorphism class by Kum" T and call it thenth Kummer product of T. Kum" T is a desingularization of the subvariety{{/q . • • • - I Z), ^ = 0} 0^^ the (n + l)st symmetric product Sym""*"' T of T.

The first Kummer product is nothing but the Kummer surface (1.7) associatedto T. Kum" T appears as a decomposition factor when we apply (4.1) to thesymplectic manifold f x Hilb""^' T (see (3.24)). In fact, the mapping

T X Kum" r - Hilb"^' T. (;, ,J) + ,is an unramified Galois covering of degree (n + 1)'*.
The Decomposition Theorem is also proved by Michelsohn [108]. But Theorem 7.18 in [108]is not correct because an incorrect Theorem 2 in [ 1S] is applied.

_^MODULI OF VECTOR BUNDLES

159

Example (4.3) (Donagi-Beauville [11]). Let K be a smooth cubic hypersurface

in and Grass(P' c P^) the Grassmann variety of lines in P^. Let F(V)be the subvariety of Grass(P' c P^) consisting of the lines contained in F.
Then F(F) is an irreducible symplectic manifold.

It is proved in [ 1 ] that F(V) has a trivial canonical line bundle and that F (V)is a 4-dimensionaI subvariety of degree 108 in p"* by the Pliicker coordinates.
If V is deformed to another cubic hypersurface V', then F(V) is deformedto F(V'). Hence, in view of (4.1), for the proof of (4.3), it suffices to show(4,3) for one cubic hypersurface. In [11], this is shown by using a K3 surface ofdegree 14. Here we prove it by using a K3 surface of degree 6. Let Fq be a cubic

hypersurface in P^ which has an ordinary double point at p = (0:0:0:0:0: 1)
and is smooth elsewhere. The defining equation of Fq is of the form^'o:9(-Vo,X, ,Ar2,Jr3,Jf4)^5 + </(Ar„,^i,^2.Ar3,A^,) = 0 inp'
for quadratic and cubic forms ^ and flf. Let S be the surface in P"* defined as

the common zero locus of q and d. By our assumption on Fq , the intersectionof ^ = 0 and = 0 is transversal. Hence 5 is a K3 surface by Example (1.4).
It is easy to see that for every pair ofpoints {a,b}e Hilb^ S of S, there exists
a unique line ^ in Fq that meets the two lines 'pa and ^. The mapping(p- Hilb^5 F(Fq), {a,b} »-▶ ^ is holomorphic and birational. F(Fq)has ordinary double points along a subvariety isomorphic to S and (p is its

minimal resolution. Since Hilb^5 is a symplectic manifold and since F{V)is a deformation of F{Vf), F{V) is also a symplectic manifold^® for every
smoothF.

As another example, we explain a way to obtain a new symplectic manifoldfrom an old one. Let A' be a 2/2-dimensional complex manifold with a sym
plectic structure w and T a complex submanifold of X. There exists a natural
exact sequence

0 Ty Y —* Nyj^ —^ 0
^ andsubmanifold(o induces Ya isskew-symmetricbilinear form on Ty. An n-dimensional
called Lagrangian (with respect to cu) if the restriction of(0 to Ty is identically zero. Since the restriction of w to r^| j, is nondegenerate, the normal bundle bfj^^y and the tangent bundles Ty of a Lagrangiansubmanifold are each other's dual. Since a global (holomorphic) 2-form on arational variety is always zero, an /i-dimensional rational submanifold of X isalways Lagrangian. Let us coi^ider the special case T ~ P" . We blow up X
along Y. The inverse image 7 of T is isomorphic to the projectivization of

the normal bundle Ny^^^ ~ . Hence Y is isomorphic to the (partial) flag
variety{{P I P ^Y and H is a hyperplane passing through p} cY x Y*
If X has ordinary double points along a (smooth) subvariety of codimension 2, then theminimal resolution A' is a flat deformation of X .

160

SHIGERU

MUKAlof Y, where Y* is the dual projective space of 7. 7 is a P"~'-bundle not
only over 7 but also over Y*. The situation is symmetrical in 7 and Y*. 7
can be blown down in the direction Y —* Y* in X. We obtain a new complex

manifold X* which contains Y* and such that X*\Y* ~ X\Y. Moreover,
X* has a symplectic structure:

Theorem (4.4) ([62]). Let X be a In-dimensional symplectic manifold and Y

its submanifold isomorphic to P" . Then there exist a symplectic manifold X*,
its submanifold Y* canonically isomorphic to the dual projective space of Y,
and a birational mapping (p: X ^ X* that satisfy the following-.

(1) (p {resp. (p~^) is not defined on Y {resp. on 7") but an isomorphism
outside it, and(2) the indeterminacy of (p {resp. (p~^) is resolved by the blowing up along
Y {resp. Y*).

This theorem can be easily generalized to the case in which 7 is a subman

ifold of codimension r and is a P^-bundle over a manifold. The mapping(p (resp. the symplectic manifold X*) is called the elementary transformation(resp. elementary transform) of X along 7. The elementary transformation isan example of a birational mapping that is not an isomorphism but an isomorphism in codimension 1. This phenomenon does not occur for manifolds ofdimension < 2: Every birational mapping (p between surfaces X and 7 is an

isomorphism if both (p and (p~^ are defined in codimension one. Concerning
the elementary transformation, the following problems are interesting.

Problem (4.5). Classify the birational mappings between symplectic manifolds,
especially in the 4-dimensional case .
The birational mappings between two 3-folds with trivial canonical bundles, more generally
between two minimal models of 3-folds, are classified by Kawamata [104] and KolMr [106].

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162

SHIGERU

MUKAI

We expect that higher dimensional symplectic manifolds have a similar prop
erty. In this section we formulate a Torelli type problem for them and study theperiods of symplectic manifolds that are obtained as moduli spaces of vector
bundles (Corollary (3.25)).
In a naive sense, a period is the integral of a (holomorphic) r-form on a
topological r-cycle or a set of such integrals. Let C be a curve of genus g. We

take a basis cu, , ... of the space //°(C,Q^) of holomorphic 1-forms on

C and a basis Oj, ... of the first homology group H^{C ,Z) of C. We

obtain 2g^ integrals X, (1 < / < ^, \ < j < 2g). The g x2g matrix\<j<2g is called the period matrix of C. The Torelli theorem

asserts that the isomorphism class of C is uniquely determined by a certain
equivalence class of its period matrix. To a l-form co we associate a homo-morphism (C, Z) C, J^co. We identify coe H^{C. Q) with
4 G Hom(//,(C.Z),C) and //°(C.a) with a subspace of h\C.C). The
natural orientation of C and the cup product induce the intersection pairing(5.1) (•): H\C ,Z)xH\C .Z)^Z
on the first cohomology group h\c ,Z). The following form of the Torelli
theorem seems to be most natural in the geometric point of view.

Torelli Theorem in Strong Form (5.2) (Matsusaka [58]). Let C and C' be
curves of the same genus and let (p: h\c' ,Z) ^ {C ,Z) bean isomorphism
between their first cohomology groups. Assume that (p is compatible with theintersection pairings (5.1) and that (p®C maps H^{C' ,0) onto
Then their exists an isomorphism f.C^C' from C onto C' that induces (por -(p on the first cohomology group.
Torelli Theorem (5.3) (Weil [94], Andreotti [3]). A curve C is isomorphic to

C' if and only if there exists an isomorphism (p: h\c' ,Z) ^ h\C .Z) that iscompatible with the intersection pairings and such that (p®C maps H^(C', Q)o n t oIn the case of K3 surfaces, we use 2-forms and 2-cycles instead of 1-forms

and 1-cycles. As in the case of curves, we identify the space /f°(5',f2^) ofholomorphic 2-forms on S with a subspace of the second cohomology group

H^{S, C). The natural orientation of S and the cup product induce the intersection pairing( 5 . 4 ) {■) • . H ^ { S , Z ) x H ^ { S . Z ) ^ Z
on the second cohomology group H^{S ,Z). Though the situation is different
in that the intersection pairing (5.4) is symmetric while (5.1) is skew-symmetric,an analogue of (5.3) holds for K3 surfaces.
The mapping //®(C, Q) -▶ //' (C. C) is injective by the theorems of de Rham and Dolbeault(see e.g..[102]).

The ideas of its several proofs are illustrated in [69, Lecture IV].

MODULI OF VECTOR BUNDLES

163

TorelH Type Theorem (5.5) (Pijateckii-Sapiro and Safarevic [81, 16, 52]), A
K3 surface S is isomorphic to S' if and only if there exists an isomorphism(p: H^{S', Z) ^ H^{S, Z) that is compatible with the intersection pairing (5.4)

and such that (p®C maps H^{S' onto

A Torelli type theorem in strong form, which is an analogue of (5.2), is
also proved for K3 surfaces (Looijenga and Peters [52], Bums and Rapoport[16], cf. [7]). We call the pair of the lattice H^{S ,Z) c- Z®^^ and a subspace
H^{S, fl^) ~ C of H^{S, Z) ® C the period of S. The Torelli type theorem

claims that the isomorphism class of S is uniquely determined by its period.

Remark (5.6). The period of a 2-dimensional complex toms T is also definedin the same way by using 2-forms and 2-cycles. But (5.5) does not hold forcomplex tori. In fact, T and its dual torus f have the same period withrespect to 2-forms. Shioda [85] has proved that every 2-dimensional complex

toms with the same period as T is isomorphic to T or f.
Let A' be a 2rt-dimensional irreducible compact symplectic Kahler mani

fold. We identify the space H^{X, Q^) of holomorphic 2-forms on X witha subspace of H^{X ,C) as in the case of a K3 surface. For a cohomology
class a e H^{X ,Z), we denote by (a") its self-intersection number, that is,^2n ^ ,Z) measured by the natural orientation. The self-intersection
form H^{X .Z) Z, a {a^") is not quadratic if n >2. But the second

cohomology group H^{X ,Z) still has a natural inner product (•) (Beauville[10], Fujiki [27]). The following is also interesting from the view point of the
topology of symplectic manifolds.
Theorem (5.7) ([27]). Let X be as above. Then the self-intersection form on

H^{X .Z) is an nth power of a quadratic form. To be precise, there exist an
integral bilinear form{'): H^iX ,Z) X H\X ,Z) ^ Z
and a rational number r such that(a^") = r{a • a)"holds for every a e H^{X. Z).
We normalize the inner product so that (cu • w) is positive and the greatest

common divisor of the (a'/?)'sisequalto 1, where a and p mnover H^{X ,Z)
and CO is a symplectic stmcture of X. Then the rational number r is equal to{ln)\/n\2" (resp. (2/2)!(m-H l)/«!2'*) if X is the wth Hilbert (resp. Kummer)
product of a K3 surface (resp. a 2-dimensional complex toms). By virtue ofthe inner product ( • ), we can define periods of similar type as K3 surfaces
for symplectic manifolds. It is important that the period thus defined be abirational invariant of X.

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Proposition (5.8). Let X and Y be irreducible compact symplectic Kdhler man
ifolds. Assume that X and Y are birationally equivalent, that is, there exist
a complex manifold Z and proper holomorphic mappings f'.Z^X andg: Z ^ Y that are isomorphisms over nonempty Zariski open subsets. Thenthe homomorphism

induced by the rational mapping 0 = f o g-': Y ■ ■ ■ -> X is an isomorphism.
Moreover, <t>* is compatible with the {normalized) inner products ( • ) andmaps H^{X,a^) onto H^{Y ,d^).
Since the canonical line bundles of X and Y are trivial, the exceptional
divisors of / and g are the same. This is a key to the proof of the proposition.

We denote by NS{X) the subgroup of H^{X, Z) consisting of the integral cohomology classes that are perpendicular to the symplectic structure w eH^{X ,^) with respect to the inner product ( • ). We call NS{X) with therestriction of (•) the Neron-Severi lattice of X. NS{X) is identified with theset of Chem classes of all line bundles on X.

Corollary (5.9). If two irreducible compact symplectic Kdhler manifolds are bi

rationally equivalent, then their Neron-Severi lattices NS{X) and NS{Y) are
isomorphic to each other.
Two symplectic manifolds with the same period are not necessarily isomor
phic to each other [17]. By Proposition (5.8), we formulate the Torelli typeproblem for symplectic manifolds as follows.
Torelli Type Problem (5.10). Let X and Y be compact irreducible symplectic
Kdhler manifolds of the same deformation type. Assume that there exists an isomorphism (p\ H^{X ,Z) ^ H^{Y ,Z) that is compatible with the [normalized]
inner products (•) and such that (p^C maps h'^{X ,^) onto H^{Y,Q?).
Is X birationally equivalent to Y ?
Now we calculate the period of the moduli space of stable sheaves on aK3 surface S. We fix a projective embedding 5 c P'^ of 5 and a vector
V = [r ,l ,s) of the extended K3 lattice H{S ,Z). Let M^{v) be the moduli
space of stable sheaves E on S with v{E) = v (§3). By Theorem (3.16),

M^{v) has a symplectic structure. By Corollary (3.25), Mg{v) is projective
if GCD(r ,{h'l),s)= 1 . We first consider the case of dim Afy(i;) = 2. By

Corollary (3.25), this happens if and only if v is isotropic, i.e., (v^) = 0. In
this case, a period is constructed from the extended K3 lattice (3.19) in thefollowing way.

(1) Let be the orthogonal complement of v in H{X,Z). Since v isisotropic, the inner product on H{X, Z) induces an inner product on

the quotient group v^jZv .

MODULIOFVECTOR

BUNDLES165(2) Since the Chera class of E is algebraic, v = v{E) is perpendicular toin H{X,Z). Hence H^{S ,C1^) is contained in <S>Cand determines a one-dimensional subspace of (v^/Zv) <S> C.

Theorem (5.11) ([63]). Let S be an algebraic K.3 surface and assume that M^{v)
is compact and of dimension 2. Then M^{v) is a K3 surface and its period isequal to V^/Zv constructed above. {In particular, M^{v) is connected.) To
be precise, there exists an isomorphism cp: V^/Zv ^ H^{Mg{v) ,Z) that iscompatible with the inner products and such that (p ®C maps H^{S ,d!') to

H\M,{v),a^).
Define the linear mapping ij/-. H^{S ,Q) ^ H{S, Q) by on-* {0 ,a ,{l •a)/r).
Then y/ induces an isometry between H^{S ,Q) and {v^lZv)®Q, Since y/®Cmaps H^{S, D^) onto itself, we have the following.
Corollary (5.12). Under the same conditions as in Theorem (5.11), the two K3

surfaces S and M^{v) have the same period over Q.
Remark (5.13). (1) Theorem (5.11) also holds for abelian surfaces T. This

follows, e.g., from an explicit description of Mj.{v). If Mj{v) is of dimension 2, then it is an abelian surface isogeneous to T. Every member of Mj{v)is a vector bundle and is said to be semihomogeneous or projectively flat, inthe case r > 0. A detailed analysis of semihomogeneous vector bundles showsthat Mj{v) is isomorphic to TI(pf{Tf) [59], where T is the dual abelian sur
face of r, T^ is the group of r-torsion points of T and (Pf-. T ^ T is ahomomorphism associated to a line bundle L with c^{L) = / (cf. [68]).
(2) Applying Theorem (5.11) to Example (2.2), we obtain a relation between
periods of two types of K3 surfaces (1.5) and (1.6). A generalization of thisrelation is studied by O'Grady [79].

A sheaf ^ on the product SxMg{v) is called a universal sheaf if it satisfies
the following three conditions:(a) ^ is flat over M^{v) (this is automatically satisfied if f is locallyfree),(b) the restriction of ^ to 5 x [E] is isomorphic to E for every member

E of M^{v), and(c) the restriction of ^ to 5 x is isomorphic to the Kuranishi familyof E (cf. (3.1) and [87]) for every member E of M^{v) and for a
sufficiently small neighborhood of [£].
The homomorphism ^ in Theorem (5.11) can be constructed from the Chemclass of a universal sheaf. But a universal sheaf does not always exist. Hence weneed its substitute. A sheaf ^ on 5 x M^{v) is called a quasiuniversal sheaf ofsimilitude s if it satisfies (a), (b') the restriction of f to 5x[£] is isomorphic

to E®^, and (c') the restriction of ^ to 5 x is isomorphic to the direct
sum of s copies of the Kuranishi family of E, for every member E of M^{v).

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MUKAI

A quasiuniversal sheaf of some similitude always exists. If both W and are

quasiuniversal sheaves on SxM^iv), then there exist vector bundles V and V'

on Mg{v) such that , where ^ x M^(v) -y M^(v)
is the projection to the second factor. We construct an algebraic cycle Z

on S X M^{v) from the Chem class of a quasiuniversal sheaf and define thehomomorphism

by a H-f • 7[*^a). We consider the restriction of its H^-part to u"*" OQ:(5->4)

By the above uniqueness property, q is independent of the choice of a quasi-

universal sheaf. Since is algebraic, 0,, q maps H^{S, to H^{M^{v). Cl^).
In the case dimAfy(u) = 2, 6^ is surjective and its kernel is Qv, which is ^the essential part of Theorem (5.11). In the case dimAfj(v) > 2, we have thefollowing:

Theorem (5.15). Let S be an algebraic YJi surface and fix a projective embedding

S . Let V be a vector of rank 1 or 2 in H{X. Z) and M^{v) the moduli
space of stable sheaves E on S with v{E) = v. Assume that M^{v) is compact

and of dimension > 4. Then the homomorphism (5.14) induces an isomorphism=>h\M,(v).Z)
between the orthogonal complement of V in H{S .Z) and the second cohomology

group H^{M^{v).Z) of Mg{v). Moreover, 6^ is compatible with the innerproducts {') and (•) and maps H^{S .Q^) onto .Q^).
Corollary (5.16). The Niron-Severi lattice NS{M^{v)) of the moduli space
Mg{v) is isomorphic to the intersection of and Z © NS{S) © Z c H{S. Z).
In the rank one case, say u = (1,0.1 - /i), every stable sheaf with v{E) = v

is an ideal (sheaf) defining a 0-dimensional subscheme of length n. Hence ^M^{v) is isomorphic to Hilb''5. (This does not depend on the choice ofthe embedding S c .) The orthogonal complement u"*" is generated by

J = (1,0,« - 1) and H^{S, Z). Hence the period of Hilb" S is isomorphic to
the pair of the lattice Zd ± H^{S, Z) and H^{S, f2^). In particular, the NeronSeveri lattice of Hilb" S is isomorphic to ZS ± NS{S) with {d^) = -2{n - 1).
This can be proved also by direct computation (see Beauville [9]).

Example (5.17). Let Grass(P' c P'*) be the Grassmann variety of lines in the
projective A-space P"*. Let

Grass(P' c P"*) c P^
See [63] for the explicit construction.

Since Ms{v) is connected, H^(Ms(v) .Z) has the inner product (•) by (5.7).

MODULI OF VECTOR BUNDLES

167

be the Pliicker embedding and cut it three times by a general hyperplane and onceby a general hyperquadric. Then we obtain a {polarized) K3 surface^^, 5 c .
We denote by h the cohomology class of hyperplane sections of S . We
assume that the three hyperplanes and the hyperquadric are sufficiently general

so that NS{S) is generated by h. Then we can describe all members of MJv),
v = {2,h.2).

Let Q be a singular hyperquadric of that contains S. The projection

from a singular point of Q induces a morphism from S to . The image of
S is contained in the image Q of Q. Q is a hyperquadric in P^. Hence, as
in ^ample (2.2), we obtain a rank 2 vector bundle from a family A of planes
in Q. Moreover, we can show that all locally free members of Mg{2 ,h ,2) are
obtained in this way.

Since Grass(P' c P'*) is a projective variety of degree 5 in P^, we have {h^) =10. Hence the moduli space M^{2 ,h,2) is a 4-fold by Corollary (3.25). This
4-fold is explicitly described as follows: Let N be the set of all hyperquadrics of
P that contain S and the subset consisting of singular ones as in Example(2.2). Then N is isomorphic to P^ and Nq is a hypersurface of degree 1 in N.

But Nq is reducible. Let Nq be the set of restrictions of all hyperquadrics of P^

that contain Grass(P' c P'*). is a hyperplaneof N contained in Nq.
Hence Nq is the union of Nq and a sextic hypersurface Nq of N. The moduli
space M{2 ,h ,2) is a double cover of the {singular) sextic hypersurface Nq in
A^=:;P^In this example, the intersection of the orthogonal complement and Ze

NS{S) © Z is generated by m = (1,0, - 1) and w = {2, h,Z) since NS{S) =Zh. Hence, by Corollary (5.16), the Neron-Severi lattice NS of M^{v) isisomorphic to the lattice

Zu®Zw, with (m^) = 2. (m-ly) =-1 and {w^) = -2.
This is an even integral quadratic form in two variables of discriminant 5. In

particular, NS is indecomposable. Hence M^{v) is not birationally equivalent

to Hilb^S for any K3 surface S by Corollary (5.9).'*® Moreover, M^{v) is not

birationally equivalent to any symplectic 4-fold obtained as in Example (4.3),
This kind of construction of (polarized) K3 surfaces from Grassmann varieties is extensively
generalized in [110] and [111].
Moishezon [109] proved the following generalization of Noether's theorem: Let A" cbe a projective 3-fold. If dim //°( Yq , Q^) > dim H^{X, for a smooth hyperplane section Yq
of X C , then there exists a smooth hyperplane section Y such that the restriction mapping
Pic X Pic Y is an isomorphism.

Let {u| be a basis of and put p/y = VjAvj . The Grassmann variety G{2. V) c

P»(A^ is defined by (^) quadratic forms qi = PijPi^, - p,^py/ + PuPjk = 0, where I runs

over all 4-element subsets {i < j <k< I) of {1 A^} . In particular, the set of quadratic forms

identically zero on Grass(P' c P*) C P' is a S-dimensional vector space spanned by these Pliicker
relations.

This will dispell the doubt expressed in [9, p. 781].

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MUKAl

since NS has no length 6 vectors. Thus we have obtained a new irreducible
symplectic manifold as a moduli space of stable sheaves on S.
Though we can construct many new symplectic manifolds in a similar way,

every component of the moduli space of rank 2 stable sheaves on a K3 surface
becomes birationally equivalent to a Hilbert product of a K3 surface under asuitable deformation of complex structure. (It is expected that this will also
hold in the higher rank case.) Hence the following is still open.
Problem. Is there any irreducible symplectic manifold that is not equivalent toHilb" S or Kum" T modulo deformation of complex structures and modulo
b i r a t i o n a l m o d i fi c a t i o n ?
6. Notes on references

The following is a short guide to references related to the topics on vector
bundles, discussed in this article.

The projective line P' has the unusual property that every vector bundle
over it decomposes into a direct sum of line bundles [30]. Atiyah [4] classified
the vector bundles on an elliptic curve on C. Later Oda [78] classified them overan arbitrary algebraically closed field. By their classification, the moduli space ofindecomposable vector bundles with fixed rank and degree is always isomorphic
to the base elliptic curve. This is also an unusual property. The moduli spaceof vector bundles on a smooth projective variety X contains arbitrarily higher
dimensional subvarieties if X is not a projective line or an elliptic curve. The
rank 2 vector bundles on a curve of genus 2 and their moduli space are explicitly

described by [71, 74], etc. Desale and Ramanan [18] generalize this result torank 2 vector bundles over a hyperelliptic curve of an arbitrary genus. For themoduli space of rank 2 vector bundles over an arbitrary curve, its topology

and rationality are studied by Newstead [73, 75, 76], Harder [31], Harder andNarasimhan [32]. Mumford and Newstead [70] study the relationship between
the periods of a curve and the moduli space of vector bundles on it.

The global property of the moduli space of vector bundles over an abeliansurface was first studied by Umemura [91, 92]. The author [61] has found a
Fourier transformation for vector bundles on an abelian variety. The Fouriertransformation is very useful for the study of vector bundles on abelian varieties
For every cubic 4-fold V C , the Neron-Severi lattice of F{V) contains an (integral)

vector of length 6. In fact, the cohomology class h e H^(F{V) ,Z) of hyperplane sections of
F{V) c Grass(P' C P^) c P"* satisfies (A") = 108 [1] and (h^) = s/{h*)IZ = 6 . Let be theorthogonal complement of h in H^(F{V) .Z) with respect to (•)• Then the Hodge structure /i-*is isomorphic to the orthogonal complement of C2( V) in //"*( V . Z) with respect to the intersection
pairing.
This is also true for the symplectic manifolds in (4.3) as we saw in §4.

5' Cf. [53, Corollary 3.4.1].

As a corollary to its main theorem, it is proved there that the isomorphism class of a curve

C is uniquely determined by that of the moduli space of rank 2 stable vector bundles with a fixed
determinant line bundle of an odd degree on C. This Torelli type theorem, together with its

generalization to higher rank vector bundles, is also proved in [89] and [90].

MODULI OF VECTOR BUNDLES

169([60, 64, 65]). The vector bundles on the projective plane are studied by many

authors. In the rank 2 case, the moduli space is explicitly described by Barth [6](cf. [57, Part II]), Ellingsrud and Stromme [24], and Hulek [35]. The moduli ofvector bundles of an arbitrary rank on is studied by Drezet-Le Potier [23].
Le Potier [50, 51] studies the Picard group of the moduli space. Among varietiesof higher dimension, the vector bundles over P" are well studied. Consult [33,80] and their references on this subject.
The moduli space of (/t-stable) vector bundles can be regarded as the moduli space of anti-self-dual Yang-Mills connections or as the moduli space ofEinstein-Hermitian metrics. Many problems on the moduli space of vector bun

dles, e.g., topology, complex structure, Kahler metric, compactification, symplectic structure, etc., can be approached by the differential geometric technique.
(See Atiyah and Bott [5], Kobayashi [45], Itoh [36], Donaldson [19], Kirwan[40].) Recently Donaldson [21, 99] constructed an example of two compact
complex surfaces that are homeomorphic but not diffeomorphic to each other.
He defined new invariants for differentiable manifolds by using the moduli of

Yang-Mills connections. The moduli space of stable vector bundles plays animportant role in the calculation of his invariants (cf. [28, 93]).
In this article, we have restricted ourselves to the moduli space. Last wenote that the geometry of vector bundles themselves on a K3 surface is alsointeresting. It has applications to special divisors on a curve [49] and to theclassification of Fano 3-folds [110, 111].
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Translated by S. MUKAI
D e p a r t m e n t o f M a t h e m a t i c s , F a c u l t y o f S c i e n c e , N a g o v a U n i v e r s i t y, C h i k u s a - K u ,Nagoya 464, Japan

Current address: Department of Mathematics, University of California, 405 Hilgard Avenue,
Los Angeles, California 90024

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