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\begin{document}
\title{Research statement}
\author{Christian Schnell}
\maketitle
I am an algebraic geometer working on Hodge theory and its applications to the
study of the geometry and topology of complex algebraic varieties. My research over
the past few years has focused on degeneration problems---how the cohomology groups
of a family of nonsingular algebraic varieties behave as the varieties in question
degenerate to a singular one. I have applied this knowledge to the construction of
extension spaces that control the degenerations, and that can be used to define
numerical invariants. I am also very interested in derived categories and
equivalences between them, and in the interactions between Hodge theory and derived
categories that occur in several places in algebraic geometry.
The geometry of complex projective varieties is a rich and classical subject. On the one
hand, such a variety $X$ can be defined by homogeneous polynomial equations inside
complex projective space, and can be studied using methods from algebraic geometry.
On the other hand, when $X$ is nonsingular, it is also a compact complex manifold with a
natural K\"ahler metric, and can be studied using methods from differential geometry.
An example of this is the Hodge decomposition
\[
H^k(X, \CC) = \bigoplus_{p+q=k} H^{p,q}(X)
\]
of the cohomology groups of a complex projective manifold $X$; classes in the
subspace $H^{p,q}(X)$ are represented by closed differential forms of
type $(p,q)$, harmonic with respect to the K\"ahler metric. Such a
decomposition is called a Hodge structure of weight $k$, and \emph{Hodge theory} is
the study of Hodge structures and their interactions with the geometry of the
underlying varieties.
The most important open problem in Hodge theory is the Hodge conjecture \cite{Hodge},
which predicts the existence of algebraic subvarieties of $X$ corresponding to
certain cohomology classes called Hodge classes. At its heart is the question to what
extent the geometry of an algebraic variety is determined by its cohomology groups.
This question is of great importance to algebraic geometry: for instance, Grothendieck's
conjectural theory of motives \cite{Grothendieck}, which attempts to unify several
subfields, depends on a positive solution to the Hodge conjecture.
The conjecture remains unproved except for the case of divisors ($k=1$), and in a
small number of very special cases \cite{Lewis}. As I will explain below, many of the
problems that I have worked on are related, directly or indirectly, to
the Hodge conjecture.
\section*{Three of my results}
Before going into details, I would like to summarize three results that give a good
idea of the scope and nature of my work.
\subsection{Hyperplane sections and residues}
\label{subsec:residues}
The study of hyperplane sections of smooth projective varieties is important both for
an inductive approach to the Hodge conjecture, and as a testing ground for questions
about more general degenerations. By work of Griffiths \cite{RatInt} and Green
\cite{Green-Period}, the cohomology groups of nonsingular hyperplane sections can be
computed via residues of meromorphic forms (generalizing the classical theory of residues in
one complex variable); moreover, the Hodge filtration is essentially the filtration
by pole order. This result has found numerous applications, such as Green's proof of
the Torelli theorem for hyperplane sections \cite{Green-Period}. It is known that
this nice description breaks down when the hyperplane section has singularities \cite{DimcaSaito}.
In my dissertation and in \cite{Residues}, I used filtered $\Dmod$-modules to extend
the residue calculus to the family of all hyperplane sections. I showed that the
resulting filtered $\Dmod$-module on the dual projective space is a minimal extension
in the sense of Saito's theory of mixed Hodge modules \cite{Saito-MHM}, and this
remains one of the few examples where a concrete description of the minimal extension
is possible. As an application, I found a new proof for the results of Brosnan, Fang,
Nie, and Pearlstein \cite{BFNP} on the topology of the family of hyperplane sections.
The new method also lead to a big improvement in their basic vanishing theorem,
answering a question left open in their paper. (This improvement was independently
discovered by Beilinson.)
\subsection{N\'eron models and admissible normal functions}
\label{subsec:Neron}
The Hodge conjecture is closely connected to the theory of normal functions, which
are holomorphic sections of certain bundles of complex tori called intermediate
Jacobians. In fact, a theorem of Zucker \cite{Zucker-NF} shows that there is
a one-to-one correspondence between Hodge classes and normal functions for
the family of nonsingular hyperplane sections. A new approach to the Hodge
conjecture, through ``singularities'' of normal functions, has recently been
proposed by Green and Griffiths \cite{GG1,GG2}. The singularity of a normal function
is a locally defined cohomological invariant, and roughly speaking, the Hodge
conjecture becomes equivalent to proving that normal functions associated to
primitive Hodge classes have nontrivial singularities \cites{BFNP,dCM-singularities}.
To study the boundary behavior of normal functions, many people have worked on the
construction of N\'eron models, which here means spaces that extend the bundles of
intermediate Jacobians \cites{Zucker-NF,Clemens-Neron,Saito-ANF,GGK,BPS}. However,
those constructions only worked in special cases, and the resulting spaces were
poorly understood. In \cite{Neron}, I solved this problem by constructing
N\'eron models for arbitrary families of intermediate Jacobians; not only that, but
the new models are analytic spaces, instead of just topological spaces.
The construction uses methods from Saito's theory, and is based on a new and delicate
norm estimate for variations of Hodge structure of odd weight. In a subsequent joint
paper with Saito \cite{SS}, we described explicitly how the new model is related to
that of \cite{GGK} in the one-variable case.
My construction gives geometric meaning to the cohomological notion of singularities,
because an admissible normal function locally extends to a holomorphic section if and
only if its singularity is zero. More importantly, I showed that the graph of any admissible
normal function has an analytic closure in my model; one consequence is a
more conceptual proof for the conjecture of Green and Griffiths that the zero locus
of an admissible normal function on an algebraic variety is algebraic (proved shortly
before by Brosnan and Pearlstein \cite{BP-zero3}). This result provides the best
evidence so far for the existence of the conjectural Bloch-Beilinson filtration on
Chow groups, a big open problem in the study of algebraic cycles \cite{Jannsen}.
\subsection{Derived invariance of the first Betti number}
One can think of the derived category of coherent sheaves on a smooth projective
variety as a replacement, in algebraic geometry, for the ring of smooth functions on
a manifold. Just as kernel functions give rise to integral transforms, objects in the
derived category of a product $X \times Y$ give rise to so-called Fourier-Mukai
transforms between the derived categories of $X$ and $Y$. The most famous example is
the work of Mukai \cite{Mukai}, who showed that the derived category of an abelian
variety is equivalent to that of its dual. Derived categories also play an important
role in birational geometry \cite{Kawamata}, for instance in the work of Bridgeland
on threefold flops \cite{Bridgeland}, and in the mathematics used in string theory
(Kontsevich's homological mirror symmetry conjecture \cite{Kontsevich}).
Two varieties with isomorphic derived categories are said to be derived equivalent;
they share many invariants, such as dimension, Kodaira dimension, or minimality, and
it is expected \cite{BarannikovKontsevich} that they also have the same Hodge numbers
$h^{p,q} = \dim H^{p,q}(X)$. The latter is known for curves and surfaces through the
derived invariance of Hochschild cohomology.
In joint work with Popa \cite{PopaSchnell}, we proved that derived equivalent
varieties have isogenous Picard varieties and automorphism groups. This implies in
particular that the Hodge number $h^{1,0}$ is invariant under derived equivalences.
It follows that all Hodge numbers of derived equivalent threefolds are the same,
confirming the first nontrivial case of the general conjecture. We conjectured that
the Picard varieties should themselves be derived equivalent, and this question in
the surface case is currently the subject of a Ph.D thesis by Pham. Our method is
also being used in another Ph.D thesis by Lombardi to study the behavior of the
Albanese dimension under derived equivalences.
After our result, another natural question is whether being simply connected or not
is a derived invariant. Here the answer is no, even for Calabi-Yau threefolds.
I recently found an example of a simply connected Calabi-Yau threefold that is
derived-equivalent to one with fundamental group $(\ZZ/8 \ZZ)^{\oplus 2}$, using a
class of Calabi-Yau threefolds constructed by Gross and Popescu.
\section*{My other research work}
As I wrote in the introduction, my work has mostly focused on degeneration problems,
from the point of view of Hodge theory. Families of projective varieties, or
morphisms between projective varieties, are one important source of degeneration
problems. Given a morphism $f \colon X \to B$ between two smooth projective
varieties, all fibers $X_t = f^{-1}(t)$ of the same dimension form a family of
projective varieties. A general member $X_t$ is smooth, because $f$ is generically
submersive, but there are usually singular fibers, too. The cohomology groups $H^k(X_t,
\CC)$ of the smooth fibers, with their individual Hodge structures, are an example of
a geometric variation of Hodge structure. There is also a theory of abstract
variations of Hodge structure, and their degenerations have been studied intensively
\cites{CKS,Kashiwara-study}. A modern framework for Hodge theory in families, based
on perverse sheaves and $\Dmod$-modules, has been created by Saito
\cites{Saito-HM,Saito-MHM}.
\subsection{N\'eron models and the Green-Griffiths program}
As mentioned above, my main contribution to the program of Green and Griffiths
has been the construction of complex-analytic N\'eron models for
families of intermediate Jacobians, and the resulting geometric interpretation for
singularities of normal functions. To make further progress, it is necessary to
understand the local behavior of an admissible normal function near a singularity. In
particular, an interesting question is whether singularities are stable under
birational maps. For families of hypersurfaces, this is true at boundary points
corresponding to nodal hypersurfaces by \cite{BFNP}; on the other hand, Saito and
Fakhruddin have given an abstract example of a singularity that disappears after
blowing up the base manifold. I believe that classifying singularities by relating the structure of the
cohomological invariant to the geometry of the graph closure inside the N\'eron model
is a very promising direction.
In addition to the joint paper with Saito \cite{SS} where we explain how my N\'eron model is
related to that of \cite{GGK} in the one-dimensional case, I also clarified the
relationship with the topological N\'eron model introduced by Brosnan, Pearlstein,
and Saito \cite{BPS}. Another interesting question, which has only been answered in
dimension one, is how the complex-analytic construction is related to the
log-geometric N\'eron model of Kato, Nakayama, and Usui \cites{KNU,KNU-weak}.
\subsection{Zero loci of normal functions}
Green and Griffiths conjectured that the zero locus of an admissible normal function
on an algebraic variety should be algebraic. There are now two proofs of this
conjecture: one by Brosnan and Pearlstein \cite{BP-zero3}, as a culmination of a
series of papers \cites{BP-zero1,BP-zero2}; the other by myself, as one application
of my work in \cite{Neron}. In fact, I proved the following stronger statement:
the graph of any admissible normal function has an analytic closure inside my N\'eron
model. This follows from my norm estimate, with the help of a special case
of the mixed $\SL(2)$-orbit theorem \cite{KNU-SL}.
That result leads to a host of interesting questions about normal functions. How is
the geometry of the graph closure related to the cohomological invariant of a normal
function? For normal functions on algebraic varieties, is the graph closure again an
algebraic variety? (This question is related to the work of Sommese \cite{Sommese} on
images of period maps.) For normal functions coming from primitive Hodge classes, can
the closure be interpreted directly in terms of the Hodge class? Can one develop an
intersection theory that would assign an intersection number to two such graph
closures? Some parts of these questions could be good thesis projects for graduate
students.
The zero locus conjecture has the following implications for the Hodge conjecture. In
\cite{Observations}, I gave a positive answer to a question of Pearlstein: if the
normal function associated to a primitive Hodge class has a zero locus of positive
dimension, then it must have a singularity at one of the points whether the closure
of the zero locus meets the boundary. This means that one could, in theory, prove the
Hodge conjecture by showing that the zero locus has positive dimension (to be
technically accurate, provided the embedding of the variety into projective space is
of sufficiently high degree). However, I also gave examples to show that already in
the case of smooth projective surfaces (where the Hodge conjecture is known), there
are many primitive Hodge classes whose normal functions do not have any zeros.
\subsection{Poincar\'e bundles}
Topologically trivial line bundles on a complex torus $T$ are naturally parametrized
by the dual complex torus $\hat{T}$. Consequently, there is a universal line bundle
on $T \times \hat{T}$, the so-called Poincar\'e bundle; it plays an crucial role in
the study of abelian varieties \cites{Mukai,Beauville}. In the case of intermediate
Jacobians, the associated principal $\CC^{\ast}$-bundle has a Hodge-theoretic
interpretation in terms of biextensions of mixed Hodge structures \cite{Hain}. One
would like to extend the Poincar\'e bundle from a family of intermediate Jacobians to
its N\'eron model, in order to define numerical invariants of admissible normal
functions. In this direction, \cite{GG2} described a Poincar\'e bundle on the
one-parameter N\'eron model of \cite{GGK}.
In recent work, I constructed a $\CC^{\ast}$-bundle on the complex-analytic N\'eron
model of \subsecref{subsec:Neron}, given an admissible normal function for the dual
variation of Hodge structure. This solves ``half'' of the problem, since it produces
a line bundle on one of the factors; I am optimistic that it will lead to a
construction of the full Poincar\'e bundle. The degree of that bundle on the graph of
a second admissible normal function without singularities (or on the graph closure in
general) is then a numerical invariant associated to the pair.
For functions associated to primitive Hodge classes, I showed that one recovers the
intersection number of the Hodge classes (a similar calculation has been done by
Caibar and Clemens). Among other things, this gives another proof for my theorem that
normal functions with positive-dimensional zero locus necessarily have singularities.
I am currently working on constructing the full Poincar\'e bundle, and on
generalizing the correspondence with biextensions to this setting.
\subsection{The locus of Hodge classes}
A Hodge class on a smooth projective variety is a class in the $2k$-th integral
cohomology of the variety that can be represented by a differential form of type $(k,k)$.
The Hodge conjecture predicts that such classes come from algebraic cycles. Since
algebraic cycles can be parametrized by algebraic varieties, one consequence of the
conjecture is that in a family of varieties, the Hodge loci (the set of
points where the flat translate of a given Hodge class remains a Hodge class) should
be algebraic varieties. This is known to be true by the famous theorem of Cattani, Deligne, and
Kaplan \cite{CDK}. The Hodge conjecture also makes predictions about the field of
definition of Hodge loci, but not much is known about this part of the problem
\cite{Voisin-AH}.
In joint work with Brosnan and Pearlstein \cite{BPSchnell}, I extended the theorem of
Cattani, Deligne, and Kaplan to the locus of Hodge classes in any admissible
variation of mixed Hodge structure. The proof works by combining the existing result
in the pure case with our recent work on zero loci of admissible normal functions.
Our result has been used by Lewis in the study of arithmetically defined candidates
for the Bloch-Beilinson filtration on Chow groups.
A very interesting direction is to compactify the locus of Hodge classes, and to use the
compactification to get numerical invariants. My idea is to use similar methods as in
the construction of the N\'eron model, building analytic spaces from filtrations on
certain $\Dmod$-modules. I have succeeded in carrying out this program for variations
of Hodge structure of weight two; as I will explain in \subsecref{subsec:Maulik}
below, this has applications to the study of Noether-Lefschetz loci on Calabi-Yau
threefolds, and to verifying certain numerical predictions made by physics.
\subsection{Primitive cohomology}
The Lefschetz theorems show that there is a close relationship between the cohomology
groups of a smooth projective variety $X$ and those of a smooth hyperplane section
$D$. Except for the so-called primitive part, all the cohomology of $X$ comes from
the lower-dimensional variety $D$; when arguing by induction on the dimension, the
key point is therefore to understand the primitive part. As a consequence of his
connectedness theorem, Nori \cite{Nori} showed that it is isomorphic to the group
cohomology of the variable part of $H^{\dim D}(D, \QQ)$, viewed as a representation of the
monodromy group for the family of all smooth hyperplane sections.
In \cite{Tube}, I gave a new topological description of the primitive cohomology
through the so-called tube mapping. This mapping produces a primitive class from an
element of the monodromy group and a variable cohomology class invariant under that
element, and I showed that such classes generate the entire primitive cohomology. If
we consider the family of all smooth hyperplane sections, the variable cohomology
groups form a local system, i.e., a covering space of the parameter space, and my
theorem says that the primitive cohomology of $X$ is naturally embedded into the
first cohomology of this covering space. This means that its topology is fairly
complicated, which is what we want to happen because the same covering space also
contains the locus of Hodge classes.
When $X$ is a Calabi-Yau threefold, Clemens \cite{Clemens-CY} has defined local
potential functions on the local system whose gradients give equations for the locus
of Hodge classes. As an application of \cite{Tube}, I proved that there is no
globally defined potential function. In my opinion, this circle of ideas will be
useful for the generalized Hodge conjecture for Calabi-Yau threefolds, which again
relates the geometry of $X$ to sub-Hodge structures in $H^3(X, \QQ)$.
The conjecture is equivalent to proving that if there is a sub-Hodge
structure of the form $H = H^{2,1} \oplus H^{1,2}$, then the locus of Hodge classes
contains an abelian variety of dimension $\dim H^{2,1}$ with certain properties; on a
purely topological level, my result in \cite{Tube} provides evidence that this is the
case. The next question to be answered is how to recover the Hodge structure on the
primitive cohomology from the topology of the local system.
\subsection{Duality for filtered $\Dmod$-modules}
\label{subsec:duality}
Saito's theory of mixed Hodge modules provides a powerful and convenient framework for
Hodge theory. It associates to every complex algebraic variety an abelian category of
mixed Hodge modules, and morphisms of algebraic varieties give rise to pullback and
pushforward functors. A mixed Hodge module on a smooth variety consists roughly of a
perverse sheaf \cite{BBD} and a filtered $\Dmod$-module $(\Mmod, F)$ that correspond
to each other under the Riemann-Hilbert correspondence. Generically, meaning on a
dense open subset, a mixed Hodge module is just a variation of mixed Hodge structure.
The associated graded $\Gr_{\bullet}^F \! \Mmod$ for the filtration $F_{\bullet}
\Mmod$ on the $\Dmod$-module defines a coherent sheaf on the cotangent bundle of the
variety, which encodes in a rather subtle way how the variation of mixed Hodge
structure degenerates. Except for the pushforward by a proper morphism
\cite{Saito-HM}, it is a difficult problem to determine how the associated graded
behaves under the various operations on mixed Hodge modules.
In \cite{MHMduality}, I described the associated graded of the Verdier dual of a
mixed Hodge module, in terms of local cohomology (relative to the zero section of the
cotangent bundle). I had originally found a special case of the formula in my thesis,
after a lengthy computation using my work on residues (see
\subsecref{subsec:residues}). The resulting exact sequence reminded me of an exact
sequence that occurs in the theory of local cohomology, and this observation
eventually suggested the correct result for arbitrary mixed Hodge modules. As an
application, I obtained formulas for ext-sheaves of the graded pieces of the Hodge
filtration, which lead to necessary and sufficient conditions for the sheaves
$\Gr_k^F \! \Mmod$ to be locally free. In \cite{Residues}, I applied this theorem to
show that, in the important case of the family of hyperplane sections of a smooth
projective variety $X$, the sheaves $\Gr_k^F \! \Mmod$ in the interesting range
$-\dim X \leq k \leq 0$ are always reflexive. Among other things, this shows that the
analytic space underlying the N\'eron model (see \subsecref{subsec:Neron}) is only
mildly singular in that case.
\subsection{Surfaces with big anticanonical class}
One aim of the Minimal Model Program is the classification of algebraic varieties
based on the behavior of the canonical line bundle. At one end of the spectrum are
varieties of general type (whose canonical model has ample canonical bundle), at the
other end are so-called Fano varieties with ample anticanonical bundle. On Fano
varieties, and more generally on varieties with big and nef anticanonical bundle, the
operations of the Minimal Model Program can be performed for arbitrary divisor
classes; they are Mori dream spaces in the terminology of Hu and Keel \cite{KeelHu}.
On moduli spaces of stable maps, the anticanonical class is often big but not nef, and
this led Chen to ask whether varieties with big anticanonical class are still Mori
dream spaces. In a joint paper, we proved that this is true for rational surfaces,
and noted that it fails in higher dimensions. The same theorem, however, was shortly
afterwards proved by another group \cite{Testa}, and so we decided to leave our paper
unpublished.
\section*{Two ongoing projects}
To close this summary of my work, here are two of the projects that I am currently
working on.
\subsection{Noether-Lefschetz loci on Calabi-Yau threefolds}
\label{subsec:Maulik}
Calabi-Yau threefolds are three-dimensional projective manifolds with trivial
canonical bundle; by Yau's famous theorem \cite{Yau}, they admit canonical Ricci-flat
metrics, one in each K\"ahler class. They have been the subject of intense study in
the past 25 years, because of the role they play in string theory. Counting
the number of holomorphic curves with given invariants is an important problem both
in algebraic geometry and in physics, and physicists have found many amazing formulas
for generating functions of such numbers, starting with \cite{COGP}.
Noether-Lefschetz loci for families of surfaces in the threefold are an interesting
special case; they have expected dimension zero, and should therefore have a virtual
number of points attached to them. Physicists predict that a certain generating
function created from such numbers is a modular form, and a result for families of
K3-surfaces has been proved by Maulik and Pandharipande \cite{MP}. In joint work with
Maulik, we are studying this question for the family of hyperplane sections of a
Calabi-Yau threefold. Using the methods I developed for compactifying Hodge loci, we
are able to assign a virtual number of points to each Noether-Lefschetz locus that
takes into account positive dimensional components and points at infinity. Currently,
we are investigating the invariance of these numbers under deformations of the
Calabi-Yau threefold, and their relationship with moduli spaces of sheaves on the
threefold. The virtual numbers can also be computed from certain Hodge modules that
are supported on boundary strata (obtained by deformation to the normal cone), and we
are also working on the relationship between these Hodge modules and the original
variation of Hodge structure. Our eventual goal is to prove the modularity of the
resulting generating functions, for instance with the help of \cite{KudlaMillson}.
\subsection{Canonical transforms for mixed Hodge modules}
Together with Cautis, I am working on describing the effect of the pullback functor
on the associated graded of a mixed Hodge module (see \subsecref{subsec:duality}).
Our goal is to get a satisfactory formalism for canonical transforms on the derived
category that relates the transform of a complex of mixed Hodge modules to that of
the corresponding complex of coherent sheaves on the cotangent bundle. This is
motivated by work of Cautis, Kamnitzer, and Licata \cite{CKL} on derived equivalences
between the cotangent bundles of Grassmanians; the results about the family of
hyperplane sections described in \subsecref{subsec:residues} may also be seen as a special
case. There is already a theory of canonical transforms for filtered $\Dmod$-modules,
developed by Laumon \cite{Laumon}, but the resulting formalism is unfortunately not
compatible with the theory of mixed Hodge modules. It also does not give the correct
result for the pullback functor, except in the non-characteristic case.
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\end{bibsection}
\end{document}