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Hodge Theory via Deformations of Affine Cones and Fano Threefolds
Enrico Fatighenti
University of Warwick
Supervisor: Miles Reid
[email protected]
Abstract
Deformation Theory and Hodge Theory are known to be closely related.
Examples of this friendship can be found, for example, in the theory of the
Variation of Hodge Structures, or in the Griffiths’s Residue Theory. The latter identifies the Hodge Structures of a hypersurface with the Milnor Algebra of the hypersurface itself - an important deformation-theoretical invariant.
The aim of the present work is to show how to compute effectively (part of)
the Hodge Structure of a smooth projective variety X, under some mild conditions, by looking at some graded component of the infinitesimal deformation/obstruction module of the affine cone over X. As an application of this
result, we computed the whole Hodge Structure of Fano Threefolds.
Back in the day: the hypersurface case
One of the most important contact points between Hodge Theory and
Deformation Theory is the Griffiths’s Residue Theorem: if X is a
smooth projective hypersurface, it establishes an isomorphism, given
by a generalisation of the Poincar´e Residue map,
p−1,n−p
Hprim
(X) ∼
= Mf
where the subscript prim stands for the primitive cohomology, and
the object on the right is called the Milnor Algebra, an important
deformation-theoretical invariant for the hypersurface, defined as
Deformations of the affine cone vs Deformations of the projective variety
Hodge Theory kicks in: Primitive Cohomology of X inside TA1
Up to now we considered the deformations of A, so a pretty natural
question arises: what do they shares with the deformations of X?
Now, the TA1 module is naturally Z-graded, and Schlessinger first
showed that the degree 0 component of the TA1 represents the projective
deformations of X inside the fixed Pn (and similar for the TA2 ). This
is quite evident if one considers the case of a hypersurface: it is quite
easy to show that, in this case,
Hypotheses first: let X be (quasi)smooth in a (weighted) projective
space and assume
• ωX ∼
= OX (α), some α ∈ Z, some D ∼
= OX (1) ample;
TA1 := Mf (d)
where d is the degree of the defining polynomial. This simply means
that, in order to deform the projective hypersurface X = V (f ), we may
deform the affine cone simply adding a polynomial of the same degree.
Any other choice of degree will bring the affine cone to something that
is not a cone (think to the example xy + ε = 0 in C2.)
or, more conceptually, as
TA1 := Ext1A(Ω1A, OA)
In a similar fashion, we can define the T 2-module, that measures the
infinitesimal obstructions to extending deformations of A, as
TA2 := Ext2A(Ω1A, OA)
and same for TA0 , that measures the infinitesimal automorphisms of A.
• Quasi-linear sections of weighted Grassmannians.
If the hypotheses above are verified, we can identify
TA1 ∼
= H 1(U, ΘU )
where U is the complement of the vertex in the affine cone and Θ is the
holomorphic tangent sheaf. The key step is now to relate the data on
X to the one on U . This is done via the exact sequence
. . . → H 1(X, OX (α)) → (TA1 )α → H 1(X, ΘX (α)) → H 2(X, OX (α)) → . . .
and, thanks to our vanishing hypotheses, Serre duality and diagram
chasing, we show
1
In order to generalise the previous setting, we looked back to some
old papers by Schlessinger, Wahl et al. (see[2] and [3]), in which they
considered the deformations of an affine isolated singularity. In particular, if X is a smooth projective variety, we can consider the affine
cone A over X: under the identification D ∼
= OX (1), for some D ∈
Pic (X), we can think to A as Spec(R(X, D)), whereas X =Proj (A).
We can define the module TA1 (the first cotangent sheaf), the set of isomorphism classes of first order infinitesimal deformations of A, by the
sequence
0 → ΘA → ΘCn+1|A → NA → TA1 ,
• Smooth Fano varieties;
Passing in cohomology and using some standard argument it is possible to express everything in terms of the cohomology of X. Hence we
have a sequence of Z-graded modules and, restricting to degree α, we
get
with Jf = (f0, . . . , fn), that is the Jacobian ideal of f , generated by
its partial derivatives. As said, the Milnor Algebra Mf contains the
deformation data for X. The purpose of this work is to generalise this
setting to any codimension case.
Our Champion: The T
• hn−1,0(X) where dim X = n ≥ 2;
What type of varieties satisfies the previous hypotheses? We identify a
bunch of significative examples (altough there are many others)
• Complete intersections;
0 → ΘU |X → ΘU → π ∗(ΘX ) → 0.
Mf := C[x0, . . . , xn]/Jf
Theorem. Let X be smooth, projective satisfying the hypotheses
above. Then
n−1,1
(TA1 )α ∼
= Hprim (X)
and totally similar for the TA2 , with
n−1,2
(TA2 )α ∼
= Hprim (X).
Figure 2: Hodge Diamond of a Fano Threefold
In particular we looked at the Graded Ring Database (see [1]) by Gavin
Brown, Alexander Kasprzyk, Miles Reid et al., that contains several
thousands of examples of quasi-smooth Fano Threefolds of various
codimensions in weighted projective spaces: some months ago, using
some birational tricks (namely, the famous Unprojections) we were
able to compute the Euler characteristic of all the Fano in codimension
2 and most of the codimension 3. Computing the T 1 (this is enough,
being all Fano of index 1) with SINGULAR we are able to confirm
these results. In particular we are now able to effectively compute
the full Hodge Structure for more than one hundred example of Fano
Threefolds (and, at a theorical level, several thousands).
What happens next?
As we saw, we have been able to identify some part of the Hodge Theory of X by looking to some graded component of a deformation module. A pretty natural question is: where are the others?We are currently
thinking about this
V• question. One possibility could be to look at the
ΘX (α) instead of ’just’ the sheaf ΘX (α), but this
whole complex
is totally work in progress...
References
Figure 1: Deformations of affine cones into affine cones
Application to Fano Threefolds
Up to now everything is very classical, but now we add Hodge Theory to the recipe, and figure out something new. We said above that,
amongst the various deformation-components of the T 1-module of A,
the degree 0 gives information on X, and in particular takes care of its
own deformation. What we want to do now is identify another special
component in the TA1 , and find there (part of) the Hodge Theory of X.
since in their case the knowledge of H 2,1(X) and H 1,1(X), predicted
by the T 1 and the T 2 is enough to determine the whole of the Hodge
Structure.
Up to now, we described a theorical method to find (part of) the Hodge
structure of a smooth projective variety inside some (not too) weird
module. The question is: can we compute anything? The answer is
positive, thanks to Computer Algebra software such as SINGULAR
that are particularly good in computing the TA1 . As an immediate application, we decide to turn our attention to the Fano Threefold case,
[1] Gavin Brown and Alexander Kasprzyk. The graded ring database,
online, access via http://grdb.lboro.ac.uk/.
[2] Michael Schlessinger. On rigid singularities. The Rice University
Studies, 19(1):147–162, 1973.
[3] Jonathan Wahl. The jacobian algebra of a graded gorenstein singularity. Duke Mathematical Journal, 55(4):843–871, 1987.