VANISHING THEOREMS FOR TOPOLOGICALLY TRIVIAL LINE …mpopa/talks/robfest.pdf · Univ. of Illinois at Chicago ... Happy birthday! Mention that I ﬁnd it a little bit embarassing to

VANISHING THEOREMS FOR TOPOLOGICALLY TRIVIAL LINE BUNDLES AND APPLICATIONS

Mihnea Popa

Univ. of Illinois at Chicago

• The initial package.

• Some of the most striking applications over the years.

• Recent developments and extensions of the theory, and new applications they facilitated.

I will survey a very influential part of Rob’s work, namely Generic Vanishing theory. It was introduced by Rob together with Mark Green in the late 80’s and early 90’s, and has since become indispensable in studying the geometry of varieties with non-zero

1st Betti number (and beyond). Will describe:

It has been an important part of my life to be Rob’s student and it is a great honorto organize this event. I have learned many things from him, both in terms of mathand humanity. We met 17 years ago, and I find hard to believe that he has turned60, especially since he doesn’t look his years. This gives me great confidence that he’llbe around for many years to come. As we’ve heard he will start a new chapter in hiscareer in the Fall – I wish him best of luck! Happy birthday!

Mention that I find it a little bit embarassing to be an organizer speaking at theconference, and therefore I would like to use the opportunity to

I will survey a very influential part of Rob’s work, namely Generic Vanishing theory. Itwas introduced by Rob together with Mark Green in the late 80’s and early 90’s, andhas since become indispensable in studying the geometry of varieties with b1(X) ⌥= 0(and beyond). Will describe:

• The initial package.

• The most striking applications over the years.

• Recent developments and extensions of the theory, and the new applications thatthey facilitated.

For expanded versions of some of the things I will say here, a nice set of lecture notescan be found on Christian Schnell’s webpage:

http://www.math.sunysb.edu/˜cschnell/teaching.html

Will always denoteX = smooth projective variety overC (although much of the generaltheory works for compact Kahler manifolds). Say dimX = n.

As motivation, recall Kodaira Vanishing : if L ample on X, then

H i(X,�X ⇥ L) = 0, for all i > 0.

This is well known to have myriads of applications, and lots of useful generalizations.

Question. What can one say under positivity hypotheses that are milder than theampleness of L? For instance, how about �X itself? (Think �X = �X ⇥OX ...)

Examples. (1) Let X = C = smooth projective curve of genus g ⇤ 1. Then

H1(C,�C) ⇧ C ⌥= 0

but

H1(C,�C ⇥ L) = 0, for all L ⌃ Pic0(C)� {0}.

(Keep in mind C ⇥⌅ J(C).)

Examples. (1) Let X = C = smooth projective curve of genus g � 1. Then

H1(C,!C) ' C 6= 0

butH1(C,!C ⌦ L) = 0, for all L 2 Pic0(C)� {0}.

(Keep in mind C ,! J(C).)

(2) Let X = Sf�! C be an elliptic surface over a curve of genus g � 2. For

L 2 Pic0(S), as above

H2(S,!S ⌦ L) 6= 0 () L = OS.

However, the picture for H1 is interesting; for L 2 Pic0(C), one can use:

H1(S,!S ⌦ f ⇤L) ' H1(C, f⇤!S ⌦ L)�H0(C,!C ⌦ L) 6= 0.

• f non-isotrivial : Pic0(S) = f ⇤Pic0(C) = {M |H1(S,!S⌦M) 6= 0} (So Albanesemap contracts fibers of f , and no vanishing whatsoever.)

• f isotrivial : this time

f ⇤Pic0(C) ✓ {M | H1(S,!S ⌦M) 6= 0} ⇢6=Pic0(S).

(So Albanese map generically finite, and H1 vanishes generically.)

For any i define the algebraic subset

V i(!X) := {L 2 Pic0(X) | H i(X,!X ⌦ L) 6= 0} ⇢ Pic0(X),

called the i-th cohomological support locus of !X . Other examples of this sort, to-gether with some conjectures of Beauville and Catanese led Green and Lazarsfeldto study, dually, the deformation problem for cohomology groups of topologicallytrivial line bundles:

H i(X,L) with L 2 Pic0(X).

This in turn led to

Generic Vanishing package:

Dimension Theorem. (Green-Lazarsfeld, ’87)Consider the Albanese map a : X ! Alb(X). Then

codim V i(!X) � i� dimX + dim a(X) for all i.

Linearity Theorem. (Green-Lazarsfeld, ’91; Arapura, Simpson ’93)The irreducible components of each V i(!X) are torsion translates of abelian sub-varieties of Pic0(X).

Strong Linearity Theorem. (Green-Lazarsfeld, ’91)I’ll tell you later.

Examples. (1) Let X = C = smooth projective curve of genus g � 1. Then

H1(C,!C) ' C 6= 0

butH1(C,!C ⌦ ↵) = 0, for all ↵ 2 Pic0(C)� {0}.

(Keep in mind C ,! J(C).)

(2) Let X = Sf�! C be an elliptic surface over a curve of genus g � 2. For

↵ 2 Pic0(S), as above

H2(S,!S ⌦ ↵) 6= 0 () ↵ = OS.

However, the picture for H1 is interesting; for ↵ 2 Pic0(C), one can use:

H1(S,!S ⌦ f ⇤↵) ' H1(C, f⇤!S ⌦ ↵)�H0(C,!C ⌦ ↵).

• f non-isotrivial : Pic0(S) = f ⇤Pic0(C) = {� | H1(S,!S ⌦ �) 6= 0} (SoAlbanese map contracts fibers of f , and no vanishing whatsoever.)


f ⇤Pic0(C) ✓ {� | H1(S,!S ⌦ �) 6= 0} ⇢6=Pic0(S).




called the i-th cohomological support locus of !X . Other examples of thissort, together with some conjectures of Beauville and Catanese led Greenand Lazarsfeld to study, dually, the deformation problem for cohomologygroups of topologically trivial line bundles:


This in turn led to




Linearity Theorem. (Green-Lazarsfeld, ’91; Arapura, Simpson ’93)The irreducible components of each V i(!X) are torsion translates of abeliansubvarieties of Pic0(X).

Strong Linearity Theorem. (Green-Lazarsfeld, ’91)I’ll tell you later.

Generic Vanishing Package:

2(2) Let X = S

f�! C be an elliptic surface over a curve of genus g � 2. ForL 2 Pic0(S), as above

H2(S,!S ⌦ L) 6= 0 () L = OS.

However, the picture for H1 is interesting; for L 2 Pic0(C), one can use:

H1(S,!S ⌦ f ⇤L) ' H1(C, f⇤!S ⌦ L)�H0(C,!C ⌦ L) 6= 0.

• f non-isotrivial : Pic0(S) = f ⇤Pic0(C) = {M |H1(S,!S⌦M) 6= 0} (So Albanesemap contracts fibers of f , and no vanishing whatsoever.)


f ⇤Pic0(C) ✓ {M | H1(S,!S ⌦M) 6= 0} ⇢6=Pic0(S).




called the i-th cohomological support locus of !X . Other examples of this sort, to-gether with some conjectures of Beauville and Catanese led Green and Lazarsfeldto study, dually, the deformation problem for cohomology groups of topologicallytrivial line bundles:


This in turn led to




Linearity Theorem. (Green-Lazarsfeld, ’91; Arapura, Simpson ’93)The irreducible components of each V i(!X) are torsion translates of abelian sub-varieties of Pic0(X).

Strong Linearity Theorem. (Green-Lazarsfeld, ’91)I have to sacrifice something...

Fibrations. (Green-Lazarsfeld, ’91)Let W be a positive dimensional irreducible component of V i(!X) for some i.Then there exists a morphism f : X ! Y , with Y normal, of maximal Albanesedimension, and dimY n� i, such that

W ✓ ⌧ + f ⇤Pic0(Y ), ⌧ 2 Pic0(X).

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Basic example. A variety is called ofmaximal Albanese dimension if its Albanesemap is generically finite onto its image, i.e. dimX = dim a(X). In this case theDimension Theorem says that V i(!X) are proper subsets of Pic0(X) as soon asi > 0, so as a weaker statement we see the first instance of a “generic” analogueof Kodaira vanishing:

H i(X,!X ⌦ L) = 0, for all i > 0 and L 2 Pic0(X) general.

This means that sometimes Generic Vanishing can be applied similarly to standardvanishing theorems; for instance, if X is of maximal Albanese dimension, then

�(!X) = �(!X ⌦ L) = h0(X,!X ⌦ L) � 0

Recent developments. Generic Vanishing theory has been strengthened byhomological methods, which have opened the way to a wider class of applications.

(1) Derived category approach. Introduced by C. Hacon (2003); important stepforward, which eventually allowed for substantial extension of the theory. Haconshowed that using the Fourier-Mukai transform, duality and commutative algebra,one can reduce the Dimension Theorem to Kawamata-Viehweg vanishing.

X ⇥ bAp1

||✏✏

p2

""X

a

✏✏

A⇥ bAp1

||

p2

""

bA'✏✏

A bA

Consider P a Poincare line bundle on X ⇥ bA (NB: bA ' Pic0(X).) Define integralfunctor

�P : D(X) ! D( bA), F 7! Rp2⇤(p⇤1F ⌦ P ).

Hacon showed using vanishing that

(1) �P (OX) 2 D[n�k,n]( bA).

In particular, if X is of maximal Albanese dimension, it is a sheaf in degree n, i.e.

Rip2⇤P = 0, for all i < n,

a conjecture of Green-Lazarsfeld, and this implies Generic Vanishing.

Some of the first important applications:• paracanonical systems

• singularities of Theta divisors

• a characterization of abelian varieties

• pluricanonical systems on varieties of maximal Albanese dimension

• the list is by no means exhaustive...

Singularities of theta divisors. The initial result that really showed the typeof possible birational geometry applications:

Theorem.(Ein-Lazarsfeld, ’97) If (A,⇥) is an irreducible principally polarizedabelian variety, then ⇥ is normal and has rational singularities.

Idea: Use the adjoint ideal Adj(⇥), i.e. an ideal sheaf on A sitting in an exactsequence

0 �! OA �! OA(⇥)⌦ Adj(⇥) �! f⇤!X �! 0,

where f : X ! ⇥ is a resolution of singularities. It is a fact that ⇥ satisfies therequired conclusion if and only if Adj(⇥) ' OA. Method:

• Twist the sequence by general L 2 Pic0(A).

• Apply the Dimension Theorem to !X (X is of maximal Albanese dimension).

• Conclude Adj(⇥) ' OA since its zero locus has to satisfy Z ⇢ Ta2A(⇥+a) = ;.

Characterization of abelian varieties. A conjecture of Kollar improving pre-vious numerical characterizations of abelian varieties:

Theorem. (Chen-Hacon, 2001) A smooth projective variety is birational to anabelian variety if and only if q(X) = dimX and P1(X) = P2(X) = 1.

Notation: q(X) = h0(X,⌦1X); Pm(X) = h0(X,!⌦m

X ).

Idea: • Show that 0 is an isolated point in V 0(!X) by using GV methods tosee that otherwise there is a positive dimensional component 0 2 Z such that�Z ⇢ V 0(!X) as well, contradiction with P2(X) = 1.

• Use result of Ein-Lazarsfeld proved in a di↵erent context, saying that if 0 2V 0(!X) is isolated, then X is birational to an abelian variety.

E↵ective results on pluricanonical maps. By a result of Hacon-McKernan,Takayama, and Tsuji, for every dimension n there exists f(n) such that the linearsystem |mKX | gives birational map for m � f(n). However, for varieties ofmaximal Albanese dimension the situation is much better (for an analogy, thinkof the Lefschetz theorem on abelian varieties):

Theorem. (Chen-Hacon, ’01; Jiang-Lahoz-Tirabassi, ’11) Let X be of maximalAlbanese dimension. If X is of general type, then |3KX | is birational. If not,|4KX | gives the Iitaka fibration.

Barja-Lahoz-Naranjo-Pareschi, ’10: strong results towards the classification ofthose varieties of general type and maximal Albanese dimension for which |2KX |is not birational.




0 �! OA �! OA(⇥)⌦ Adj(⇥) �! f⇤!X �! 0,



• Apply the Dimension Theorem to !X (X is of maximal Albanese dimension).

• Conclude Adj(⇥) ' OA since its zero locus has to satisfy Z ⇢ Ta2A(⇥+a) = ;.

Characterization of abelian varieties. A conjecture of Kollar improving pre-vious numerical characterizations of abelian varieties:

Theorem. (Chen-Hacon, ’01) A smooth projective variety is birational to anabelian variety if and only if q(X) = dimX and P1(X) = P2(X) = 1.

Notation: q(X) = h0(X,⌦1X); Pm(X) = h0(X,!⌦m

X ).

Idea: • Show that 0 is an isolated point in V 0(!X): use GV methods to seethat otherwise there is a positive dimensional component 0 2 Z such that �Z ⇢V 0(!X) as well, contradiction with P2(X) = 1.

• Use result of Ein-Lazarsfeld proved in a di↵erent context, saying that if 0 2V 0(!X) is isolated, then X is birational to an abelian variety.

E↵ective results on pluricanonical maps. By a result of Hacon-McKernan,Takayama, and Tsuji, for every dimension n there exists f(n) such that the linearsystem |mKX | gives birational map for m � f(n). However, for varieties ofmaximal Albanese dimension the situation is much better (for an analogy, thinkof the Lefschetz theorem on abelian varieties):

Theorem. (Chen-Hacon, ’01; Jiang-Lahoz-Tirabassi, ’11) Let X be of maximalAlbanese dimension. If X is of general type, then |3KX | is birational. If not,|4KX | gives the Iitaka fibration.

Barja-Lahoz-Naranjo-Pareschi, ’10: strong results towards the classification ofthose varieties of general type and maximal Albanese dimension for which |2KX |is not birational.

E↵ective results on pluricanonical maps. By a result of Hacon-McKernan,Takayama, and Tsuji, for every dimension n there exists f(n) such that the linearsystem |mKX | gives birational map for m � f(n). However, for varieties ofmaximal Albanese dimension the situation is much better:

Theorem. (Chen-Hacon, ’01; ... ; Jiang-Lahoz-Tirabassi, ’11) Let X be of maxi-mal Albanese dimension. If X is of general type, then |3KX | is birational. If not,|4KX | gives the Iitaka fibration.

Barja-Lahoz-Naranjo-Pareschi, ’10: results towards classifying varieties of generaltype and maximal Albanese dimension for which |2KX | is not birational.

Idea: Use generic vanishing to deduce the continuous global generation of sheavesof the form

OX(mKX)⌦ I�X, k (m� 1)KX k �

and of related gadgets, where the ideal above is an asymptotic multiplier ideal.Then generically apply a global generation criterion for varieties that are finiteover abelian varieties coming from the theory of M -regularity (Pareschi – P.).

Numerical inequalities for irregular varieties. (1) Higher dimensional Castelnuovo-de Franchis theorem.

Theorem. (Pareschi – P. , 2009) If X is a smooth projective variety withoutirregular fibration, i.e. with no morphisms f : X ! Y with Y normal and ofmaximal Albanese dimension, 0 < dimY < dimX, then

�(!X) � q(X)� dimX.

This is an extension to arbitrary dimension of the classical Castelnuovo-de Fran-chis inequality pg � 2q � 3 (which can be rewritten as �(!X) � q(X) � 2) forsurfaces without irrational pencils.

Idea: Hypothesis implies that X is of maximal Albanese dimension, so we haveseen that in any case �(!X) � 0. Point of the Theorem is that we can be moreprecise. Define the Generic Vanishing index of X as

gv(X) := mini>0 {codim V i(!X)� i} � 0.

The Theorem we really prove is

• X of maximal Albanese dimension =) �(!X) � gv(X).

To show this, GV theory implies that the Fourier-Mukai transform �P (OX) isa gv(X)-syzygy sheaf around the origin, of rank �(!X); then apply the Evans-Gri�th Syzygy Theorem.

Recent developments:

• Generic Vanishing theory has been strengthened by homological and Hodge theoretic methods; new classes of applications.

• (1) Derived category approach (Hacon, 2003): allows for substantial extension of the theory. Hacon used the Fourier-Mukai transform, duality and commutative algebra to reduce the Dimension Theorem to Kawamata-Viehweg vanishing.

4

X ⇥ bAp1

xx

✏✏

p2

&&

X

a

✏✏

A⇥ bAp1

xx

p2

&&

bA

'✏✏

Alb(X) = A bA ' Pic0(X)

Consider P a Poincare line bundle on X ⇥ bA. Define integral functor

�P : D(X) ! D( bA), F 7! Rp2⇤(p⇤1F ⌦ P ).

Hacon showed using vanishing that

(1) �P (OX) 2 D[dim a(X),n]( bA).

In particular, if X is of maximal Albanese dimension, it is a sheaf in degree n, i.e.

Rip2⇤P = 0, for all i 6= n,

a conjecture of Green-Lazarsfeld, and this implies the Dimension Theorem.

Opened the door for putting Generic Vanishing in a much larger picture, formalizingit as a property satisfied by many other sheaves (GV sheaves). Also, relationship withother fields. E.g.:

• F satisfies GV () �P (F) is a perverse coherent sheaf + study of t-structures ( ,2010)

• dimension conditions on V i(F) () comm. algebra syzygy conditions on �P (R�F)(Pareschi – , 2008)

• GV really does complement Kodaira vanishing: for instance !X ⌦L with arbitrary Lnef still satisfies the Dimension Theorem (Pareschi – , 2008)

• Dimension Theorem is in fact equivalent to (1); in particular the Green-Lazarsfeldconjecture holds in the compact Kahler case as well (Pareschi – , 2009)

(2) Extension to Hodge modules. ( – Schnell, 2011). Generic Vanishing applies tofiltered D-modules arising from the Hodge theory of algebraic maps.

For instance: let a : X ! A = Alb(X) be the Albanese map of X. By the Decomposi-tion Theorem, one has

a⇤QX [n] 'M

i

Ei[�i],

Opened the door for putting Generic Vanishing in a larger picture, formalizing it asa property satisfied by many objects (GV-sheaves) with respect to integral functors(Pareschi – P., ’08-’10).

• relationship with perverse coherent sheaves + other moduli spaces...

• Dimension Theorem really does complement Kodaira vanishing: it is satisfiesd forinstance by ⇥X ⇤ L with arbitrary L nef.

• Local study of the GV condition; e.g. Green-Lazarsfeld conjecture in the Kahler case.


For instance: let a : X ⌃ A = Alb(X) be the Albanese map of X. By the Decomposi-tion Theorem, one has

a�QX [n] ⌥�

i

Ei[�i],

with Ei (topological) perverse sheaves. Using the Riemann-Hilbert correspondence andM. Saito’s theory, we have a correspondence

Ei ⇧⌃ (Mi, F ),

where (Mi, F ) is a filtered (regular, holonomic) DA-module. For such an (M, F ), theassociated graded object is

gr•FM =�

k

grkFM,

which can be seen as a graded module over Sym•TA or, forgetting the grading, a coherentsheaf on the cotangent bundle T �A ⌥ A⇥ V , where V = H0(X,�1

X).

Theorem. For each Hodge D-module (M, F ) and each k, the sheaf grkFM satisfies theGeneric Vanishing package.

Example: Ria�⇥X arise in this fashion, and this case recovers the previous GV-theorems of Green-Lazarsfeld and Hacon.

However, this can be used in new situations. For instance, it leads to a Nakano-typegeneric vanishing theorem (which can be shown to be optimal):

Theorem. One hascodim V q(�p

X) ⌅ |p+ q � n|� �(a),

where �(a) is the defect of semismallness of the Albanese map, i.e.

�(a) := maxl⇥N (2l � dimX + dimAl),

with Al the locus of points over which the fibers have dimension at least l.

Singularities of theta divisors. The initial result that really showed the type ofpossible birational geometry applications:

6and of related gadgets, where the ideal above is an asymptotic multiplier ideal. Thengenerically apply a global generation criterion for varieties that are finite over abelianvarieties coming from the theory of M -regularity (Pareschi – P.).

Numerical inequalities for irregular varieties. (1) Higher dimensional Castelnuovo-de Franchis theorem.

Theorem. (Pareschi – P. , ’09) If X is a smooth projective irregular variety withoutirregular fibrations, i.e. with no morphisms f : X ⇤ Y with Y normal and of maximalAlbanese dimension, 0 < dimY < dimX, then

�(⇥X) ⇥ q(X)� dimX.

This is an extension to arbitrary dimension of the classical Castelnuovo-de Franchisinequality pg ⇥ 2q � 3 (which can be rewritten as �(⇥X) ⇥ q(X) � 2) for surfaceswithout irrational pencils.

Idea: Hypothesis implies that X is of maximal Albanese dimension, so we have seenthat in any case �(⇥X) ⇥ 0. Point of the Theorem is that we can be more precise.Define the Generic Vanishing index of X as

gv(X) := mini>0 {codim V i(⇥X)� i} ⇥ 0.

The Theorem we really prove is

• X of maximal Albanese dimension =⌅ �(⇥X) ⇥ gv(X).

To show this, GV theory implies that the Fourier-Mukai transform �P (OX) is a gv(X)-syzygy sheaf around the origin, of rank �(⇥X); then apply the Evans-Gri⇥th SyzygyTheorem.

(2) Inequalities for Hodge numbers; regularity.

Theorem.(Lazarsfeld – P., ’10) Too long to state, but point is that if X has no irregularfibrations, then

hp,0(X) ⇥ f(q(X)).

(Generalization of various results of Castelnuovo-de Franchis, Catanese, Green-Lazarsfeld.Extended in some cases to arbitrary hp,q(X) by Lombardi.)

Same method gives: if k is the dimension of the general fiber of the Albanese map,then:

n�

i=0

H i(X,⇥X) is k�regular over E = ⌃•H1(X,OX).

Idea: Use intersection theory and BGG correspondence for the globalized derivativecomplex governing the deformation theory of H i(X,L) with L ⇧ Pic0(X).

6Numerical inequalities for irregular varieties. (1) Higher dimensional Castelnuovo-de Franchis theorem.

Theorem. (Pareschi – P. , ’09) If X is a smooth projective irregular variety withoutirregular fibrations, i.e. with no morphisms f : X ⇤ Y with Y normal and of maximalAlbanese dimension, 0 < dimY < dimX, then

⇥(⇤X) ⇥ q(X)� dimX.

This is an extension to arbitrary dimension of the classical Castelnuovo-de Franchisinequality pg ⇥ 2q � 3 (which can be rewritten as ⇥(⇤X) ⇥ q(X) � 2) for surfaceswithout irrational pencils.

Idea: Hypothesis implies that X is of maximal Albanese dimension, so we have seenthat in any case ⇥(⇤X) ⇥ 0. Point of the Theorem is that we can be more precise.Define the Generic Vanishing index of X as

gv(X) := mini>0 {codim V i(⇤X)� i} ⇥ 0.

The Theorem really is

• X of maximal Albanese dimension =⌅ ⇥(⇤X) ⇥ gv(X).

To show this, GV theory implies that the Fourier-Mukai transform �P (OX) is a gv(X)-syzygy sheaf, of rank ⇥(⇤X); then apply the Evans-Gri⇥th Syzygy Theorem.



hp,0(X) ⇥ f(q(X)).



n�

i=0

H i(X,⇤X) is k�regular over E = ⌃•H1(X,OX).

Idea: Use intersection theory and BGG correspondence for the globalized derivativecomplex governing the deformation theory of H i(X,L) with L ⇧ Pic0(X).

Ueno’s Conjecture. Recently Chen and Hacon have proved part of a fundamentalconjecture of Ueno regarding varieties with �(X) = 0.


Theorem.(Lazarsfeld – P., 2010) Too long to state, but point is that if X has noirregular fibrations, then

hp,0(X) � f(q(X)).

(Generalization of various results of Castelnuovo-de Franchis, Catanese, Green-Lazarsfeld. Extended in some cases to arbitrary hp,q(X) by Lombardi.)


nM

i=0

H i(X,!X) is k�regular over E = ^•H1(X,OX).

Idea: Use intersection theory and BGG correspondence for the globalized deriva-tive complex governing the deformation theory of H i(X,L) with L 2 Pic0(X).

Ueno’s Conjecture. Recently Chen and Hacon have proved part of a funda-mental conjecture of Ueno regarding varieties with (X) = 0.

Theorem. (Chen-Hacon, 2011) Let X be a smooth projective variety with(X) = 0, and a : X ! A its Albanese map. Denote by F the general fiberof a. Then:

(0) (Kawamata, ’80) a is surjective and has connected fibers.

(1) (F ) = 0.

(1’) More generally, the Cn,m conjecture holds over a base of maximal Albanesedimension, i.e.

(X) � (F ) + (Y )

if f : X ! Y is a fiber space with Y of maximal Albanese dimension.

Idea: Naively, one encodes Pm(F ) in the rank of a⇤(!⌦mX ). More sophisticated

approach is to look at

Vm := a⇤�!⌦mX ⌦ I(k (m� 1)KX + a⇤(✏H) k)�

where H is an ample line bundle on A. Roughly speaking, apply generic vanishingto Vm (much more complicated in reality...)

Holomorphic one-forms on varieties of general type. More recent devel-opments have led to a solution to the following conjecture of Hacon-Kovacs andLuo-Zhang, partially due to Carrell as well.

Theorem. (P. – Schnell, ’12) If X is of general type, then there are no nowherevanishing holomorphic one-forms on X.



hp,0(X) � f(q(X)).



nM

i=0

H i(X,!X) is k�regular over E = ^•H1(X,OX).

Idea: Use intersection theory and BGG correspondence for the globalized derivativecomplex governing the deformation theory of H i(X,L) with L 2 Pic0(X).

Ueno’s Conjecture. Recently Chen and Hacon have proved part of a fundamentalconjecture of Ueno regarding varieties with (X) = 0.

Theorem. (Chen-Hacon, ’11) Let X be a smooth projective variety with (X) = 0,and a : X ! A its Albanese map. Denote by F the general fiber of a. Then:


(1) (F ) = 0.

(1’) More generally, the Cn,m conjecture holds over a base of maximal Albanese dimen-sion, i.e.

(X) � (F ) + (Y )

if f : X ! Y is a fiber space with Y of maximal Albanese dimension.

Idea: Naively, one encodes Pm(F ) in the rank of a⇤(!⌦mX ). More sophisticated approach

is to look at

Vm := a⇤�!⌦mX ⌦ I(k (m� 1)KX + a⇤(✏H) k)�

where H is an ample line bundle on A. Roughly speaking, apply generic vanishing toVm (much more complicated in reality...)

Holomorphic one-forms on varieties of general type. More recent developmentshave led to a solution to the following conjecture of Hacon-Kovacs and Luo-Zhang,partially due to Carrell as well.


• (II) Extension to Hodge modules (P. - Schnell, 2011): the Generic Vanishing package applies to filtered D-modules arising from the Hodge theory of algebraic maps.

(2) Extension to Hodge modules. ( – Schnell, 2011). Generic Vanishing appliesto filtered D-modules arising from the Hodge theory of algebraic maps.

For instance: let a : X ! A = Alb(X) be the Albanese map of X. By theDecomposition Theorem, one has

a⇤QX [n] 'M

i

Ei[�i],

with Ei (topological) perverse sheaves. Using the Riemann-Hilbert correspon-dence and M. Saito’s theory, we have a correspondence

Ei ! (Mi, F ),

where (Mi, F ) is a filtered (regular, holonomic) DA-module. For such an (M, F ),the associated graded object is

gr•FM =M

k

grkFM,

which can be seen as a graded module over Sym•TA or, forgetting the grading, acoherent sheaf on the cotangent bundle T ⇤A ' A⇥ V , where V = H0(X,⌦1

X).

Theorem. For each Hodge D-module (M, F ) and each k, the sheaf grkFM satis-fies the Generic Vanishing package.

Example: Ria⇤!X arise in this fashion, and this case recovers the previous GV-theorems of Green-Lazarsfeld and Hacon.

However, this can be used in new situations. For instance, it leads to a Nakano-type generic vanishing theorem (which can be shown to be optimal):

Theorem. One has

codim V q(⌦pX) � |p+ q � n|� �(a),


�(a) := maxl2N (2l � dimX + dimAl),





0 �! OA �! OA(⇥)⌦ Adj(⇥) �! f⇤!X �! 0,



For instance: let a : X ! A = Alb(X) be the Albanese map of X. By the Decomposi-tion Theorem, one has

a⇤QX [n] 'M

i

Ei[�i],

with Ei (topological) perverse sheaves. Using the Riemann-Hilbert correspondence andM. Saito’s theory, we have a correspondence

Ei ! (Mi, F ),

where (Mi, F ) is a filtered (regular, holonomic) DA-module. For such an (M, F ), theassociated graded object is

gr•FM =M

k

grkFM,

which can be seen as a graded module over Sym•TA or, forgetting the grading, a coherentsheaf on the cotangent bundle T ⇤A ' A⇥ V , where V = H0(X,⌦1

X).

Theorem. For each Hodge D-module (M, F ) and each k, the sheaf grkFM satisfies theGeneric Vanishing package.

Example: Ria⇤!X arise in this fashion, and this case recovers the previous GV-theorems of Green-Lazarsfeld and Hacon.

However, this can be used in new situations. For instance, it leads to a Nakano-typegeneric vanishing theorem (which can be shown to be optimal):

Theorem. One hascodim V q(⌦p

X) � |p+ q � n|� �(a),


�(a) := maxl2N (2l � dimX + dimAl),


Singularities of theta divisors. The initial result that really showed the type ofpossible birational geometry applications:

Theorem.(Ein-Lazarsfeld, ’97) If (A,⇥) is an irreducible principally polarized abelianvariety, then ⇥ is normal and has rational singularities.

Idea: Use the adjoint ideal Adj(⇥), i.e. an ideal sheaf on A sitting in an exact sequence

0 �! OA �! OA(⇥)⌦ Adj(⇥) �! f⇤!X �! 0,

where f : X ! ⇥ is a resolution of singularities. It is a fact that ⇥ satisfies the requiredconclusion if and only if Adj(⇥) ' OA. Method:


7Theorem. (Chen-Hacon, ’11) Let X be a smooth projective variety with ⇥(X) = 0,and a : X ⌅ A its Albanese map. Denote by F the general fiber of a. Then:


(1) ⇥(F ) = 0.

(1’) More generally, the Cn,m conjecture holds over a base of maximal Albanese dimen-sion, i.e.

⇥(X) ⇤ ⇥(F ) + ⇥(Y )

if f : X ⌅ Y is a fiber space with Y of maximal Albanese dimension.

Idea: Naively, one encodes Pm(F ) in the rank of a�(⇤⇥mX ). More sophisticated approach

is to look atVm := a�

�⇤⇥mX ⇥ I(⇧ (m� 1)KX + a�(�H) ⇧)

⇥

where H is an ample line bundle on A. Roughly speaking, apply generic vanishing toVm (much more complicated in reality...)

Holomorphic one-forms on varieties of general type. More recent developmentshave led to a solution to the following conjecture of Hacon-Kovacs and Luo-Zhang,partially due to Carrell as well.


One interesting corollary of this, combined with Kollar’s C+n,m for morphisms with fibers

of general type, is the following (special cases previously due to Kovacs, Viehweg-Zuo):

Corollary. If f : X ⌅ A is a smooth family of varieties of general type over an abelianbase, then f is birationally isotrivial.

Idea: Use the Generic Vanishing package for Hodge D-modules. Main point is astronger statement about the total associated graded object gr•FM, or even betterabout the Laumon-Rothstein version of the Fourier-Mukai transform applied to OX asa D-module.

HAPPY BIRTHDAY ROB!

VANISHING THEOREMS FOR TOPOLOGICALLY TRIVIAL LINE …mpopa/talks/robfest.pdf · Univ. of Illinois at Chicago ... Happy birthday! Mention that I ﬁnd it a little bit embarassing to

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