# A Multi-Scale Deep Learning Framework for Projecting Weather Extremes

Antoine Blanchard
Verisk Analytics
Boston, MA 02115
<EMAIL_ADDRESS>
&Nishant Parashar
Verisk Analytics
Boston, MA 02115
<EMAIL_ADDRESS>
&Boyko Dodov
Verisk Analytics
Boston, MA 02115
<EMAIL_ADDRESS>
&Christian Lessig
Otto-von-Guericke Universität
Magdeburg, Germany
<EMAIL_ADDRESS>
&Themistoklis Sapsis
Massachusetts Institute of Technology
Cambridge, MA 02139
<EMAIL_ADDRESS>

###### Abstract

Weather extremes are a major societal and economic hazard, claiming thousands
of lives and causing billions of dollars in damage every year. Under climate
change, their impact and intensity are expected to worsen significantly.
Unfortunately, general circulation models (GCMs), which are currently the
primary tool for climate projections, cannot characterize weather extremes
accurately. To address this, we present a multi-resolution deep-learning
framework that, firstly, corrects a GCM’s biases by matching low-order and
tail statistics of its output with observations at coarse scales; and
secondly, increases the level of detail of the debiased GCM output by
reconstructing the finer scales as a function of the coarse scales. We use the
proposed framework to generate statistically realistic realizations of the
climate over Western Europe from a simple GCM corrected using observational
atmospheric reanalysis. We also discuss implications for probabilistic risk
assessment of natural disasters in a changing climate.

##  1 Introduction

The development of extreme weather under climate change is of growing concern
for many stakeholders across the globe, including civil society, governments,
and businesses [1, 2, 3]. The assessment of the risks associated with weather
extremes is thereby highly critical. Quite often, the most unlikely events in
the tail of a hazard’s probability distribution are also the most devastating.
Because of their unlikely nature, their statistics are difficult to predict,
even more so on a regional scale [4]. In addition, the effects of climate
change are expected to vary across weather perils, making exploration of
catastrophic scenarios even more challenging.

Extreme weather risk is often estimated by generating ensembles of possible
climate states using global circulation models (GCMs). Unfortunately, ensemble
size and resolution are insufficient to characterize tail events properly.
Even at high resolution, GCMs suffer from statistical biases [5] which can
lead to the incorrect characterization of extreme-weather inducing processes
(e.g., blocking events [6]). Moreover, high-resolution GCMs require vast
computing resources. Hence, to create large ensembles that provide a better
statistical sampling, simple GCMs and coarse resolutions have to be employed.
However, these simulations will be polluted with severe biases.

To remedy these issues, one strategy is to leverage the wealth of
observational data available about the Earth system and the power of machine
learning [7]. This can be pursued either in a purely data-based fashion [8],
or by augmenting the GCM dynamical equations with data-driven terms [9] or
post-processing the GCM output to make it consistent with observations [10,
11]. The main challenges are thereby the strongly nonlinear energy scattering
across scales as well the non-deterministic nature of the very fine scales
[12]. These two issues often compromise the stability of machine-learning
schemes [13, 14] and the physical consistency of the output [15, 16].

We introduce a multi-resolution deep learning framework for accelerating the
simulation of weather extremes. We combine a physics-based GCM run at coarse
resolution with machine-learning models trained on observational data to
reduce the biases and enhance the resolution of the GCM simulation. The key
ingredients are a) a compact, multi-scale representation of physical processes
on the sphere which efficiently captures energy transfers across scales; b)
novel statistical loss functions which place emphasis on extreme values and
space–time coherency; and c) a divide-and-conquer training strategy which
enables training of regional models at high spatial resolution. The full-scale
debiased simulation provides a complete view of risk at arbitrary resolution
which decision makers can use to investigate near-present scenarios and assess
their vulnerability to catastrophic weather events.

##  2 Methods

We start from coarse-resolution runs of a simple but fast GCM polluted by
strong biases. Based on these, we proceed in two steps; see Figure 1. First,
we correct the low-order and tail statistics of the GCM output at coarse
scales to match the statistics of observations. Second, we enhance the
resolution of the debiased GCM output by reconstructing the finer scales as a
function of the coarse ones. In both steps, the ground truth is reanalysis
data, a blend of observations and past forecasts which provides the most
realistic picture of past atmospheric states that is available.

Figure 1: Overview of framework. Reanalysis data is used as benchmark as it
provides the most comprehensive description of the observed climate.

#### Decomposition of spatial scales

The dynamics of the atmosphere are characterized by a wide range of scales,
from thousands of kilometers to a few meters. To delineate coarse- and fine-
scale phenomena, we use a discrete spherical wavelet frame [17] which provides
a mathematically robust framework for scale separation and allows us to
achieve a good balance between spatial and frequency localization. Each
physical variable is decomposed on a set of wavelet levels of increasing
granularity, with lower levels capturing large-scale spatial features (Figure
LABEL:fig:fig2a). At each level, the locations of the wavelet centers are
chosen as in [18] so as to guarantee the existence of a fast transform. This
is important for manipulating very large climate ensembles at high spatial
resolution. This also enables a highly efficient divide-and-conquer strategy
whereby a multitude of small, local neural networks (one for each wavelet
location) are trained concurrently, with communication between models being
through a custom loss function (described below). More details are available
in Section A.1.

(a)  (b)

Figure 2: (a) Wavelet decomposition for ERA5 reanalysis and (b) cone of
influence for model input.

#### Step 1: Bias-correction of coarse GCM output

The wavelet coefficients of the coarse, biased GCM output are mapped into a
different set of wavelet coefficients whose statistics match those of coarse-
scale observations; specifically, ERA5 reanalysis [19]. The bias-correction
step is formulated as a semi-supervised learning problem whereby reanalysis
data is used as target for all GCM ensemble members. For each wavelet center,
we employ an LSTM-based neural network whose input consists of time series for
that center and its nearest neighbors. To capture energy transfers across
scales—a landmark of turbulent flows—the input to the network also includes
wavelet centers that belong to coarser wavelet levels. This defines a cone of
influence for the center under consideration (Figure LABEL:fig:fig2b). From
this input, the model outputs a time series whose statistics are required to
match those of reanalysis data. Once the models have been trained, the
statistically-corrected time series are assembled and transformed back from
wavelet to physical space to produce bias-corrected realizations of coarse-
scale atmospheric dynamics.

#### Step 2: Reconstruction of small-scale features

The ensemble obtained from Step 1 has the same resolution as the GCM on which
it is based (typically, cells of size 250–500 km). To achieve a higher
resolution and be able to describe the local effects of extreme events, small-
scale structures must be added. This is done with a second machine-learning
step whose task is to reconstruct the fine scales (i.e., those below the GCM
resolution) as a function of the coarse scales (i.e., those resolved by the
GCM). The algorithm is trained on ERA5 reanalysis data in a supervised setting
and tested on the debiased output from Step 1. Each wavelet center is assigned
a TCN-based model that operates on a multi-scale cone of influence. Wavelet
levels can be trained in parallel, but this might lead to error accumulation
across levels at testing time. To combat this, we employ a sequential strategy
akin to imitation learning [20] whereby the input to each model is replaced
with model predictions from previous levels. This improves robustness of the
predictions, especially for the coarser levels.

#### Statistical loss functions

In Step 1, the models are trained using statistical loss functions; this is a
key innovation of our framework. We consider two classes of statistical
losses: quantile losses, which capture heavy tails and extreme values; and
spectral losses, which capture cross-correlations between wavelet centers. The
former guarantee that the frequency and amplitude of extreme weather events in
the output is consistent with reanalysis data. The latter guarantee space–time
coherency of the output with respect to wave propagation and transport
phenomena. These losses are computed over several decades of weather data to
ensure convergence of the statistics with respect to sample size (see Section
A.2). Crucially, enforcing statistical consistency with reanalysis data still
allows for variability of events in the output, especially the most extreme
ones. In Step 2, statistical losses are used as regularizers, with the MSE
loss (or a variant of it that emphasizes extreme values [21]) being the main
driver for the optimization.

##  3 Results

#### Experimental protocol

To demonstrate the potential of the framework, we use SPEEDY [22, 23] as our
simplified GCM. The GCM runs are at the T47 resolution (about 300 km), thereby
covering wavelet levels 1–5 in full. The target resolution for the downscaled
output is about 80 km, corresponding to wavelet level 8. Our models are
trained on vorticity and divergence close to the ocean surface ($\sigma$-level
0.95). For the debiasing step (Step 1), each model consists of a one-layer
LSTM cell with hidden size 60 followed by a linear layer and a residual block.
Step 1 is trained on 5 GCM runs, each spanning 10 years; validation and
testing are performed on two additional runs. For the downscaling step (Step
2), each model consists of a TCN block with 4 channels of size 64 and kernel
size 2. Step 2 is trained on 30 years of reanalysis data, and the remaining 10
years are used for validation and testing. Experiments are performed on Tesla
V100 GPUs hosted on AWS. Training time on a single GPU is about 24 hours for
Step 1 and about 60 hours for Step 2.

#### Validation

To assess performance, we transform the full-resolution wavelet time series
that are produced by our algorithm back to physical space and consider a
number of metrics defined at fixed locations across Western Europe. Since our
main goal is risk quantification, our success metrics are largely statistical
in nature. We also rely on visual inspection of movies by domain experts to
evaluate realism of the machine-learning output. Figure 3 shows that the
debiasing models are able to correct the statistics of the GCM output at
coarse scales. The fine-scale features added by the models from Step 2 bring
the statistics of the full-resolution output even closer to the ERA5 ground
truth. The machine-learning simulation looks remarkably realistic. Numerous
fronts, which are not present in the coarse-scale GCM simulation, are clearly
visible in the machine-learning output, emerging, propagating, and breaking as
in the reanalysis. Additional results are presented in Section A.3.

(a)

(b)

(c)

Figure 3: Comparison of full-scale debiased simulation produced by the
machine-learning algorithm to the SPEEDY GCM and reanalysis data. Top panel:
Temporal statistics (left: pdf; right: spectral density) of vorticity computed
over a one-year period for Paris and London. Bottom panel: End-to-end
illustration of the machine-learning pipeline tested on two snapshots of
vorticity from the coarse-scale GCM; the ERA5 reanalysis snapshots on the far
right are provided for qualitative comparison only.

##  4 Discussion

With the proposed framework, million-year simulations of weather extremes are
becoming within reach, enabling more accurate quantification of catastrophic
weather events. This will allow us to meet the growing demand from decision
makers for global simulations that capture all types of dependencies across
multiple regions and perils. These large ensembles can also be subsampled to
create event simulations conditioned on a particular climate-change scenario
(e.g., how would the combination of La Niña with a positive AMO index affect
hurricane activity?). This will considerably improve the estimation of the
tail risks associated with weather extremes as they are affected by climate
change. From a technical standpoint, several aspects warrant further
investigation. In particular, it would be desirable to incorporate more
physics into the machine-learning models, either in the form of soft
constraints (e.g., a shallow-water PDE loss) or directly in the network
architecture.

## Acknowledgments and Disclosure of Funding

Funding in direct support of this work was provided by Verisk Analytics. CL
acknowledges support from the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) – Project-ID 422037413 – TRR 287. The authors declare
that they have no known competing financial interests or personal
relationships that could have appeared to influence the submitted work.

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##  Appendix A Appendix

###  A.1 Training strategy

The spherical wavelet decomposition provides a multi-scale representation of
physical variables on the sphere. Beside scale separation, it is also used to
reduce the dimensionality of the input. At full ERA5 resolution, the spatial
field at each time step is represented on a 721-by-1440 grid. In wavelet
space, the same spatial field can be represented almost losslessly using 9
wavelet levels and fewer than 700,000 wavelet coefficients, a nearly 30%
reduction compared to the physical grid.

Despite this compression, the size of the datasets on which the machine-
learning algorithm is trained can be large due to the temporal resolution
(3-hourly) and extent (several years) of the time series for each wavelet
coefficient. This bottleneck is exacerbated by the fact that the GCM dataset
is an ensemble of realizations. This makes it impossible to fit the entire
dataset into the memory of a single GPU. Multi-GPU training would be one
solution but would also ultimately become impractical as more physical
variables are added to the training. Another alternative would be to split the
time series into small (e.g., monthly) chunks for mini-batching. However, this
is incompatible with the statistical loss functions used in Step 1 as the
latter require thousands of samples.

To circumvent the memory issue, we deploy a divide-and-conquer strategy
(Figure 4) whereby a multitude of small, local models (one for each wavelet
center) are trained concurrently, with communication between models being
through the loss function. This strategy is computationally more efficient and
less memory-intensive than one in which all wavelet centers are handled by one
large model. To minimize the memory footprint, each optimization step consists
of one gradient-free forward pass through all the models, combined with a
series of gradient-equipped forward passes executed in sequence for each
individual model (Figure 5). In this way, we never store more than one
computational graph in memory at any given time. Conceptually, this strategy
is similar to locally freezing certain components of one large model.

Figure 4: Overview of the divide-and-conquer strategy used for training. In
addition to alleviating the memory load, this approach allows the training of
regional neural network models at high spatial resolution. Figure 5: Sketch of
one optimization step with forward passes applied sequentially without and
with gradient computations.

###  A.2 Statistical loss functions

#### Computational considerations

We use statistical loss functions that ensure that the temporal statistics of
the machine learning output are consistent with the observation-based
reanalysis. Consistency is enforced in wavelet space, i.e., between the time
series of wavelet coefficients. The statistical losses operate on a single
wavelet center $y(t)$, where we have a quantile loss, as well on neighborhoods
of wavelet centers $\mathbf{y}(t)$, where we use a cross-spectrum loss. For a
given statistical property $S$, our loss functions $\mathcal{L}$ take the form

$\mathcal{L}(y_{\textrm{out}},y_{\textrm{obs}})=\textrm{MSE}(S(y_{\textrm{out}}),S(y_{\textrm{obs}})),$
(1)

with the subscripts “out” and “obs” denoting the machine-learning output and
the observed value from reanalysis, respectively. To capture the temporal
statistics accurately, the time series used in the loss function must contain
a sufficiently large number of samples. In Step 1, the times series used to
compute the losses extend over a 10-year interval, which is computationally
made possible by the divide-and-conquer strategy described above that
alleviates memory issues. In Step 2, the statistical losses are mainly used as
regularizers and shorter (1-year) time series can be used.

#### Quantile loss

The quantile loss matches the quantiles of an individual machine-learned time
series with its reanalysis counterpart. It is defined as

$\mathcal{L}(y_{\textrm{out}},y_{\textrm{obs}})=\textrm{MSE}(\mathbf{q}(y_{\textrm{out}}),\mathbf{q}(y_{\textrm{obs}})),$
(2)

where $\mathbf{q}$ is a vector containing the q-th quantiles of $y(t)$. If
$\mathbf{q}$ is specified as a vector of evenly spaced numbers between 0 and
1, then the quantile loss is equivalent to matching the marginal distributions
of the individual time series. In practice we have found that no more than a
dozen quantiles are required to properly capture the distributions.
Importantly, since we are interested in extreme events, we place special
emphasis on the distribution tails by using a non-uniform spacing of the
quantiles; for instance,

$\mathbf{q}=\setcounter{MaxMatrixCols}{11}\begin{bmatrix}0&0.001&0.01&0.1&0.3&0.5&0.7&0.9&0.99&0.999&1\end{bmatrix}.$
(3)

We also tried a version of (2) similar to the cross-entropy loss but the
results were inferior compared to (2).

#### Cross-spectrum loss

Temporal dependencies between two real time series $x(t)$ and $y(t)$ is
measured by the cross-correlation, defined as the temporal convolution between
$x$ and $y$:

$a_{x,y}(\tau)=\int_{-\infty}^{\infty}x(t)y(t+\tau)\,\mathrm{d}t.$ (4)

Since the Fourier transform diagonalizes the convolution operator, one can
equivalently compute the cross-spectrum $\Gamma_{x,y}$ between two signals $x$
and $y$:

$\Gamma_{x,y}(\omega)=\mathcal{F}[a_{x,y}](\omega),$ (5)

where $\mathcal{F}$ denotes the Fourier transform and $\omega$ a temporal
frequency. The cross-spectrum is a complex-valued quantity and therefore can
be written in polar form as

$\Gamma_{x,y}(\omega)=A_{x,y}(\omega)\exp(\mathrm{j}\Phi_{x,y}(\omega)),$ (6)

with $A$ denotes the real-valued cross-spectrum amplitude and $\Phi$ the
phase. The former quantifies how energy is distributed among frequencies and
the latter contains information about how these frequencies align with each
other in time. Hence, the cross-spectrum can be used to capture propagation of
wave and energy between neighboring wavelet centers.

We therefore define the cross-spectrum loss between a particular center and
its neighbors as

$\mathcal{L}(y_{\textrm{out}},y_{\textrm{obs}})=\textrm{MSE}(A_{y_{\textrm{out}},\mathbf{y}_{\textrm{out}}}(\omega),A_{y_{\textrm{obs}},\mathbf{y}_{\textrm{obs}}}(\omega))+\textrm{MSE}(\Phi_{y_{\textrm{out}},\mathbf{y}_{\textrm{out}}}(\omega),\Phi_{y_{\textrm{obs}},\mathbf{y}_{\textrm{obs}}}(\omega)).$
(7)

In practice, we found that penalizing real and imaginary parts to be more
robust during training, and we therefore prefer

$\displaystyle\mathcal{L}(y_{\textrm{out}},y_{\textrm{obs}})$
$\displaystyle=\textrm{MSE}(\mathrm{Re}[\Gamma_{y_{\textrm{out}},\mathbf{y}_{\textrm{out}}}(\omega)],\mathrm{Re}[\Gamma_{y_{\textrm{obs}},\mathbf{y}_{\textrm{obs}}}(\omega)])$
$\displaystyle+\textrm{MSE}(\mathrm{Im}[\Gamma_{y_{\textrm{out}},\mathbf{y}_{\textrm{out}}}(\omega)],\mathrm{Im}[\Gamma_{y_{\textrm{obs}},\mathbf{y}_{\textrm{obs}}}(\omega)]).$
(8)

Additional emphasis can be placed on the high frequencies by using a log-
transform of $\Gamma$ in the loss function. Note that if $y$ is appended to
the vector of neighbors $\mathbf{y}$, then the cross-spectrum loss also
captures the auto-correlation between the wavelet center $y$ and itself.

###  A.3 Additional results

In this appendix we present additional results that consider the two steps of
our algorithm separately.

#### Step 1: Debiasing the coarse scales

Figure 6 demonstrates how the debiasing algorithm is able to correct the
statistical biases of the SPEEDY model in terms of temporal probability
distribution function and spectrum. The agreement of the debiased output with
coarse-scale observations is very good, especially in the tails of the
distributions where extreme events reside. Figure 7 shows that the machine-
learning algorithm is able to correct the monthly distributions of vorticity
and match them with reanalysis data. The top and bottom monthly quantiles are
captured well, which is critical for extreme events. The monthly mean trends
of the debiased output are also consistent with observations. This is
confirmed in the seasonal averages shown in Figure 8. Finally, in Figure 9 we
compare two snapshots from the SPEEDY simulation with their debiased versions
obtained with the machine-learning algorithm. We clearly see that the
algorithm maintains the overall consistency of the field and “deforms” the
SPEEDY simulation as little as possible in order to correct the statistics.

(a)  (b)

(c)  (d)

Figure 6: Temporal statistics (left: probability density; right: spectrum
amplitude) of coarse-scale vorticity at select cities computed over a 10-year
period for SPEEDY GCM, machine-learning debiased simulation (ML), and
reanalysis data (ERA5).

(a)  (b)

(c)  (d)

Figure 7: Monthly distributions of coarse-scale vorticity at select cities
computed over a 10-year period for SPEEDY GCM, machine-learning debiased
simulation (ML), and reanalysis data (ERA5). The top, middle, and bottom
notches denote the 0.9, 0.5, and 0.1 quantiles, respectively. Figure 8:
Seasonal averages of coarse-scale vorticity computed over a 10-year period for
original SPEEDY simulation (top), machine-learning debiased simulation
(middle), and reanalysis data (bottom); DJF: December–February; MAM:
March–April; JJA: June–August; SON: September–November.

(a)

(b)

Figure 9: Two snapshots of coarse scale vorticity for SPEEDY and the
corresponding machine-learning debiased simulations (ML) resulting from Step 1
of our algorithm. The machine learning algorithm deforms the SPEEDY simulation
in order to match its statistics with observations at coarse scales but
retains the overall structure of the flow.

#### Step 2: Reconstructing the small scales

To test the performance of the resolution-enhancing second machine learning
step in isolation, we used reanalysis data represented on the same coarse
wavelet level that is used for Step 1. Visually, Figure 10 shows that the
algorithm is able to reconstruct the fine scales with high fidelity. The
temporal statistics of the fine scales are also captured very well; see Figure
11.

(a)

(b)

Figure 10: Two snapshots of fine scale vorticity reconstructed from the coarse
scales. The resolution-enhancing machine learning models use low resolution
reanalysis data as input to show the effect of the second step in isolation.

(a)

(b)

Figure 11: Temporal statistics of vorticity computed over a one year period
for select cities. The resolution-enhancing machine learning models use low
resolution reanalysis data as input to show the effect of the second step in
isolation.

#### Steps 1 and 2 combined: Generating a full-scale debiased simulation

We present additional results for the full scale simulation at seven other
cities besides Paris and London which are already shown in Figure 3. The
probability density functions and spectra in Figure 12 show good agreement
between the full resolution debiased machine learning output and observational
(reanalysis) data.

(a)

(b)

Figure 12: Temporal statistics of vorticity computed over a one year period
for select cities. he resolution-enhancing machine-learning models (“ML”) are
tested on the debiased output of Step 1 to produce a full scale debiased
simulation.


Holomorphic Lagrangian subvarieties in holomorphic symplectic manifolds with
Lagrangian fibrations and special Kähler geometry

Ljudmila Kamenova111Partially supported by a grant from the Simons
Foundation/SFARI (522730, LK), Misha Verbitsky222Partially supported by the
Russian Academic Excellence Project ’5-100’, FAPERJ E-26/202.912/2018 and CNPq
- Process 313608/2017-2.

Abstract. Let $M$ be a holomorphic symplectic Kähler manifold equipped with a
Lagrangian fibration $\pi$ with compact fibers. The base of this manifold is
equipped with a special Kähler structure, that is, a Kähler structure
$(I,g,\omega)$ and a symplectic flat connection $\nabla$ such that the metric
$g$ is locally the Hessian of a function. We prove that any Lagrangian
subvariety $Z\subset M$ which intersects smooth fibers of $\pi$ and smoothly
projects to $\pi(Z)$ is a toric fibration over its image $\pi(Z)$ in $B$, and
this image is also special Kähler. This answers a question of N. Hitchin
related to Kapustin-Witten BBB/BAA duality.

##  1 Introduction

The present paper is motivated by the observations made by N. Hitchin [H2] who
worked on the Kapustin-Witten version of the geometric Langlands
correspondence, interpreted as Montonen-Olive generalization of electric-
magnetic duality. This theory originates in 1977, when P. Goddard, J. Nuyts
and D. Olive discovered that magnetic sources in gauge theory with gauge group
$G$ are classified by irreducible representations of the Langlands dual group
${}^{L}G$ ([GNO]). Then C. Montonen and D. Olive conjectured that the Yang-
Mills theories with the gauge groups $G$ and ${}^{L}G$ are isomorphic on the
quantum level. The Montonen-Olive duality can be regarded as a quantum field
generalization of the usual electric-magnetic duality.

M. Atiyah had suggested that the Montonen-Olive conjecture ([MO]) might be
related to the Langlands duality, but it took many years until 2006, when A.
Kapustin and E. Witten explained this conjectural relation.

In their celebrated paper [KW], Kapustin and Witten produced a rich dictionary
of the correspondence between the geometric Langlands program and S-duality in
the four-dimensional N=4 gauge theory. This approach is based on the
comparison between two Hitchin systems (the spaces of Higgs bundles on a
curve) with values in Langlands dual groups. Both of these Hitchin systems are
equipped with a Lagrangian fibration. Reminiscent of the Strominger-Yau-Zaslow
interpretation of the Mirror Symmetry, the Langlands duality is interpreted as
a correspondence between certain categories on these two spaces, associated
with the duality of their fibers. For a less technical survey of the Kapustin-
Witten program, see [Ka].

The Kapustin-Witten interpretation of Montonen-Olive/geometric Langlands
duality can be understood as SYZ Mirror symmetry on the Hitchin space, but it
is firmly based on the hyperkähler geometry of the Hitchin space. In place of
the Fukaya category on the symplectic side of Mirror Symmetry, one has a
category associated with the holomorphic Lagrangian subvarieties (BAA, ABA and
AAB branes). In place of the derived category of coherent sheaves on the
complex side of Mirror Symmetry one has a category which has pairs
(trianalytic subvariety, hyperholomorphic bundle on it) as objects; these are
called BBB branes. Since the fiberwise duality should somehow exchange these
two categories, Hitchin argued, the fibers of the BAA brane under the Hitchin
fibration map should be tori, and its image should retain the special Kähler
structure which exists on the base of the Hitchin fibration. We define all
these notions and state this result rigorously in Section 2.

Hitchin stated his theorem in bigger generality than required by the Kapustin-
Witten theory: he expected it to be true for any hyperkähler manifold equipped
(such as the Hitchin system) with a ${\mathbb{C}}^{*}$-action rotating the
complex structures within the twistor family. We prove the same result without
a ${\mathbb{C}}^{*}$-action. Our main theorem is the following.

Theorem 1.1: (3.1) Let $(M,\Omega)$ be a holomorphic symplectic Kähler
manifold, and let $\pi:\;M{\>\longrightarrow\>}B$ be a proper Lagrangian
fibration. Consider an irreducible Lagrangian subvariety $Z\subset M$ such
that $\pi(Z)$ does not lie in the discriminant locus $D$ of $\pi$. Then for
any smooth point $x\in\pi(Z)\backslash D$ which is a regular value of
$\pi:\;Z{\>\longrightarrow\>}\pi(Z)$, the fiber $\pi^{-1}(x)\cap Z$ is a union
of translation equivalent subtori in the complex torus $\pi^{-1}(x)$, and the
regular part of $\pi(Z)$ is a special Kähler submanifold in $B\backslash D$.

##  2 Special Kähler manifolds

###  2.1 Special Kähler manifolds and Hessian manifolds

Special Kähler manifolds first appeared in physics, [dW-vP], [dWLP], as
allowed targets for the scalars of the vector multiplets of field theories
with $N=2$ supersymmetry on a 4-dimensional Minkowski space-time. Originally
they came in two flavours, the affine special Kähler manifolds associated with
rigid supersymmetry, and projective special Kähler manifolds associated with
the local supersymmetry. In the present paper we are interested only in the
affine version.

The first comprehensive mathematical exposition of this theory is due to Dan
Freed, [F]. After this geometric structure was presented to the general
mathematical readership, special Kähler manifolds became prominent in
differential geometry. In [BC], Baues and Cortés have shown that special
Kähler manifolds can be interpreted as “affine hyperspheres”. This classical
concept, going back to the work of Blaschke in affine geometry, is described
by solutions of real Monge-Ampère equation. This interpretation leads to a
classification of special Kähler manifolds. For more details on the
differential geometry of special Kähler manifolds, the reader is directed to
the survey [C].

Definition 2.1: A special complex manifold is a complex manifold $(M,I)$
equipped with a flat, torsion-free connection $\nabla$ such that the tensor
$\nabla(I)\in\Lambda^{1}(M)\otimes\Lambda^{1}(M)\otimes TM$ is symmetric in
the first two variables. A special complex manifold is special Kähler if it is
equipped with a Kähler form $\omega$ which satisfies $\nabla(\omega)=0$.

Let $(M,I,\nabla,g,\omega)$ be a special Kähler manifold. Since
$\nabla(\omega)=0$, and $\nabla(I)$ is symmetric in the first two variables,
the tensor

$\nabla(g)=\nabla(I\circ\omega)\in\Lambda^{1}(M)\otimes\Lambda^{1}(M)\otimes\Lambda^{1}(M)$
(2.1)

is symmetric in the first two variables. This tensor is symmetric in the last
two variables, because $g$ is symmetric. Therefore, $\nabla g$ is a symmetric
3-tensor.

Definition 2.2: Let $(M,\nabla)$ be a manifold equipped with a flat torsion-
free connection, and $g$ a Riemannian metric. It is called Hessian if
$\nabla(g)$ is symmetric in all 3 variables.

Remark 2.3: It is not hard to see that the Riemannian metric $g$ on
$(M,\nabla)$ is Hessian if and only if $g$ is locally the Hessian of a
function, which is called the potential of the Hessian metric. A priori the
potential exists only locally, but when $M$ is simply connected, it can be
defined globally on $M$.

This construction is due to N. Hitchin, [H1], who exhibited the special Kähler
structure on the moduli space of holomorphic Lagrangian subvarieties in a
hyperkähler manifold, and exhibited many interesting differential-geometric
properties of special Kähler manifolds.

Remark 2.4: Let $(M,I,\nabla,g,\omega)$ be a special Kähler manifold,
$\operatorname{Vol}M$ the Riemannian volume form, and $f$ the potential of its
Hessian metric. Since $\operatorname{Vol}M=\omega^{n}$, and
$\nabla(\omega)=0$, the function $f$ is a solution of the real Monge-Ampère
equation $\det\frac{d^{2}f}{dx_{i}dx_{j}}=\operatorname{\sf const}$. In the
paper [ChY], Cheng and Yau studied Hessian manifolds with a prescribed
Riemannian volume form, and proved an analogue of Calabi-Yau’s theorem for
such manifolds.

Claim 2.5: Let $(M,I,\nabla,g,\omega)$ be a Kähler manifold equipped with a
flat connection $\nabla$ which satisfies $\nabla(\omega)=0$. Then
$(M,I,\nabla,g,\omega)$ is special Kähler if and only if the metric $g$ is
Hessian.

Proof: Follows immediately from (2.1).

###  2.2 Special Kähler structure on the base of a complex Lagrangian
fibration

Special Kähler manifolds naturally occur in many situations associated with
the geometry of Calabi-Yau and hyperkähler subvarieties. For the present
paper, the following construction is most significant.

Definition 2.6: Let $(M,\Omega)$ be a holomorphic symplectic manifold. A
Lagrangian subvariety of $M$ is a subvariety such that its smooth part is a
Lagrangian submanifold in $M$. A (holomorphic) Lagrangian fibration on $M$ is
a proper holomorphic map $\pi:\;M{\>\longrightarrow\>}B$ with general fibers
being Lagrangian submanifolds in $(M,\Omega)$.

The following claim is well-known in classical mechanics.

Claim 2.7: A smooth fiber of a holomorphic Lagrangian fibration is always a
torus.

Proof: For any fibration $\pi:\;M{\>\longrightarrow\>}B$, any smooth fiber $F$
has trivial normal bundle $NF$. However, $NF$ is dual to the tangent bundle
$TF$ whenever $\pi$ is a Lagrangian fibration. Therefore, the bundle $TF$ is
also trivial. For any function on $B$, its Hamiltonian gives a section of
$TF$. Choose a collection of holomorphic functions such that their
Hamiltonians give a basis in $TF$. Since these Hamiltonians commute, the
corresponding vector fields in $TF$ also commute. This gives a locally free
action of an abelian Lie group on $F$, and therefore $F$ is a quotient of an
abelian group by a lattice.

Definition 2.8: Let $\pi:\;M{\>\longrightarrow\>}B$ be a proper fibration.
Consider the first derived direct image $R^{1}\pi_{*}({\mathbb{R}}_{M})$,
where ${\mathbb{R}}_{M}$ is the trivial sheaf on $M$. This is a constructible
sheaf; at any point $x\in B$, the fiber of $R^{1}\pi_{*}({\mathbb{R}}_{M})$ is
equal to the first cohomology of the fiber $\pi^{-1}(x)$. Outside of
singularities of $\pi$, this sheaf is locally constant. The flat connection on
the corresponding vector bunlde is called the Gauss-Manin connection. This
connection is defined in the complement to the set $\operatorname{Disc}(\pi)$
of all critical values of $\pi$; this set is called the discriminant locus of
$\pi$.

Definition 2.9:  Let $\pi:\;M{\>\longrightarrow\>}B$ be a Lagrangian
fibration, and let $F$ be the fiber over $x\in B$. Then $\pi^{*}TB=NF=T^{*}F$.
Identifying $H^{0}(NF)=T_{x}B$ with $H^{0}(T^{*}F)=H^{1}(F,{\mathbb{R}})$, we
obtain an identification of $TB$ and the bundle
$R^{1}\pi_{*}({\mathbb{R}}_{M})$ of the first cohomology constructed above.
Therefore, $TB$ is equipped with a natural flat connection, also called the
Gauss-Manin connection.

Remark 2.10:  Let $\pi:\;M{\>\longrightarrow\>}B$ be a holomorphic Lagrangian
fibration. A Kähler form $\omega$ on $M$ restricted to a smooth fiber $F$ of
$\pi$ defines a cohomology class $[\omega]\in H^{2}(F)$. Since $F$ is a torus,
we can consider $[\omega]$ as a 2-form on $R^{1}\pi_{*}({\mathbb{R}}_{M})=TB$.
This form is clearly parallel under the Gauss-Manin connection. Abusing the
notation, we denote this 2-form by the same letter $\omega$.

Theorem 2.11: ([F, Theorem 3.4], [H1, Theorem 3]) Let
$\pi:\;M{\>\longrightarrow\>}B$ be a holomorphic Lagrangian fibration on a
Kähler holomorphic symplectic manifold and let $B_{0}\subset B$ be the
complement to the discriminant locus of $B$. Consider the 2-form $\omega$ on
$B$ constructed in 2.2, and the Gauss-Manin connection $\nabla$ on $TB$
defined in 2.2. Then $(B,\nabla,\omega)$ is a special Kähler manifold.

##  3 Special Kähler geometry and holomorphic Lagrangian subvarieties

###  3.1 Holomorphic Lagrangian subvarieties: main theorem

In his talk [H2] at the SCGP in October 2018, Nigel Hitchin stated the
following theorem.

Theorem 3.1: (Hitchin, [H2]) Let $\pi\;:M{\>\longrightarrow\>}B$ be an
algebraically integrable system with a ${\mathbb{C}}^{*}$-action defining a
projective special Kähler structure. Then any ${\mathbb{C}}^{*}$-invariant
holomorphic Lagrangian submanifold has an open set with the structure of a
fibration over a projective special Kähler submanifold of $B$, and each fiber
is a disjoint union of translates of an abelian subvariety.

Hitchin asked whether there is an analogue of his result in the affine special
Käher setting. Here we prove it, and give examples of holomorphic Lagrangian
submanifolds projecting to special Kähler submanifolds.

Theorem 3.2:  Let $(M,\Omega)$ be a holomorphic symplectic Kähler manifold,
and let $\pi:\;M{\>\longrightarrow\>}B$ be a proper Lagrangian fibration.
Consider an irreducible Lagrangian subvariety $Z\subset M$ such that $\pi(Z)$
does not lie in the discriminant locus $D$ of $\pi$. Then for any smooth point
$x\in\pi(Z)\backslash D$ which is a regular value of
$\pi:\;Z{\>\longrightarrow\>}\pi(Z)$, the fiber $\pi^{-1}(x)\cap Z$ is a union
of translation equivalent subtori in the complex torus $\pi^{-1}(x)$, and the
smooth part of $\pi(Z)\backslash D$ is a special Kähler submanifold in
$B_{0}:=B\backslash D$.

Before we prove 3.1, we state the following elementary linear-algebraic lemma.

Lemma 3.3:  Suppose $V\subset W\oplus W^{*}$ is a Lagrangian vector subspace
in $W\oplus W^{*}$ with standard symplectic structure, and $\pi:\;W\oplus
W^{*}{\>\longrightarrow\>}W$ the projection. Then $\pi(V)^{\bot}=V\cap W^{*}$,
where $R^{\bot}\subset W^{*}$ denotes the annihilator of a subspace $R\subset
W$, and $W^{*}$ is considered as a subspace in $W\oplus W^{*}$.

Proof of 3.1. Step 1: Let $Z_{x}:=\pi^{-1}(x)\cap Z$, where
$x\in\pi(Z)\backslash D$ is a regular value of
$\pi:\;Z{\>\longrightarrow\>}\pi(Z)$. Denote by $T^{\pi}M$ the fiberwise
tangent bundle, and let $T^{\pi}_{z}M$ be its fiber over $z\in Z_{x}$. The
holomorphic symplectic form induces non-degenerate pairing between
$T^{\pi}_{z}M$ and $T_{x}B$. 3.1 applied to $V=TZ$ and $W\oplus W^{*}=T_{z}M$
implies $T_{z}Z_{x}=\pi(T_{z}Z)^{\bot}\cap T^{\pi}_{z}M$. Indeed, in this case
$T_{z}Z_{x}=V\cap W^{*}$ and $\pi(T_{z}Z)=\pi(V)$. However, $\dim\pi(Z)=\dim
Z-\dim Z_{x}=\dim\pi(T_{z}Z)$, hence in all points $x\in B\backslash D$ and
all $z\in\pi^{-1}(x)$, one has $T_{x}\pi(Z)=\pi(T_{z}Z)$. This gives

$T_{z}Z_{x}=T_{x}(\pi(Z))^{\bot}.$ (3.1)

Step 2: From (3.1), we obtain that $\pi(T_{z}Z)^{\bot}$ is constant: for
different $z,z^{\prime}\in Z_{x}$, the spaces $T_{z}Z_{x}$ and
$T_{z^{\prime}}Z_{x}$ are obtained by a translation within the torus
$\pi^{-1}(x)$. In other words, the space $T_{z}Z_{x}$ is constant in the
standard coordinates on the torus, and $Z_{x}\subset\pi^{-1}(x)$ is a union of
subtori which are translates of each other.

Step 3: Since $\pi(Z)\subset B$ is a complex subvariety, in order to prove
that it is special Kähler it suffices to show that it is totally geodesic
(that is, constant) with respect to the Gauss-Manin connection $\nabla$ on
$TB_{0}$. However, the connection $\nabla$ is identified with the Gauss-Manin
connection under the identification $TB_{0}=R^{1}\pi_{*}({\mathbb{R}}_{M})$,
and it preserves any sublattice in $T_{x}B_{0}=H^{1}(F,{\mathbb{Z}})$, where
$F=\pi^{-1}(x)$.

Since $Z_{x}\subset\pi^{-1}(x)$ is a subtorus, it corresponds to a sublattice
$H_{1}(Z_{x})\subset H_{1}(\pi^{-1}(x))$ in homology, and in a neighbourhood
$U\ni x$, all fibers $Z_{x^{\prime}}\subset\pi^{-1}(x^{\prime})$ correspond to
the same sublattice. Therefore, its orthogonal complement in
$R^{1}\pi_{*}({\mathbb{R}}_{M})$ is constant. However, by (3.1), this
orthogonal complement generates $T_{x}(\pi(Z))$. This implies that $TB_{0}$ is
constant with respect to the Gauss-Manin connection $\nabla$ on $B_{0}$.

##  4 Examples

Many (or most) holomorphic Lagrangian subvarieties of hyperkähler manifolds
occur as fibers of Lagrangian fibrations. Indeed, in [HwW] it was shown that
any Lagrangian subtorus in a hyperkähler manifold is a fiber of a holomorphic
Lagrangian fibration. However, for any two distinct Lagrangian fibrations over
a maximal holonomy hyperkähler manifold, the intersection index of their
fibers is positive ([KLV], second paragraph of the proof of Theorem 2.11).
Therefore, any fiber of the first fibration is projected to the base of the
second one surjectively and finitely in the general point. In this case, 3.1
is tautologically true, because the fibers of
$\pi{\left|{}_{{\phantom{|}\\!\\!}_{Z}}\right.}$ are 0-dimensional, and its
base coincides with $B$.

It is much harder to find examples where the special Kähler geometry of
$\pi(Z)$ is non-trivial. This is easy to explain. Indeed, $\pi(Z)$ gives a
flat submanifold in the special Kähler manifold $B_{0}=B\backslash D$.
Therefore, the tangent space $T_{x}\pi(Z)$ to any smooth point is fixed by the
monodromy of the Gauss-Manin connection on $B_{0}$. However, the monodromy
representation is quite often irreducible, or has very few subrepresentaions.

The Hitchin system (moduli of Higgs bundles over a curve) is equipped with a
Lagrangian fibration (“Hitchin fibration”), which has abelian varieties
(Jacobians of the “spectral curve”) as its fibers. For some examples of the
Hitchin system, the corresponding monodromy representation was computed in
[BS]. From this computations it follows that the monodromy representation is
reducible ([BS, Corollary 4.23]). This suggests that some interesting
Lagrangian subvarieties, not transversal to fibers of the Hitchin system,
might exist in this case. Two of the first papers to study the monodromy for
the Hitchin fibration are [Cop] and [Sch].

Holomorphic Lagrangian fibrations on a deformation of the second Hilbert
scheme of a K3 were studied by D. Markushevich and by L. Kamenova ([Ma], [K]).
They split into two distinct cases. In the first case, studied by D.
Markushevich, the Abelian fibers are Jacobians of smooth genus two curves and
the fibers of $\pi:\;M{\>\longrightarrow\>}B$ have no elliptic curve, because
these Jacobians are always simple ([M]). Therefore, all Lagrangian
subvarieties of $M$ either project to $B$ surjectively or lie in the fibers of
$\pi$. In the second case, studied by L. Kamenova, the fibers of $\pi$ are
products of two elliptic curves, i.e., the fibers are Jacobians of singular
genus two curves. This situation occurs, for example, when one takes the
punctual Hilbert scheme of $2$ points on an elliptic K3 surface
$S{\>\longrightarrow\>}{\mathbb{P}}^{1}$. Then the Hilbert scheme $S^{[2]}$ is
fibered over the base $({\mathbb{P}}^{1})^{[2]}={\mathbb{P}}^{2}$ with general
fibers that are products of the fibers of the ellitic fibration
$S{\>\longrightarrow\>}{\mathbb{P}}^{1}$. As shown in [K], under some
“genericity” hypotheses, all deformations of the second Hilbert scheme of a
K3, fibered with the fiber that is a product of two elliptic curves, are
obtained in this way.

The main (and, so far, the only) non-trivial example of the geometric
construction obtained in this paper is given by the Hilbert scheme of an
elliptic K3 surface as follows. Let
$\pi:\;M{\>\longrightarrow\>}S={\mathbb{C}}P^{1}$ be an elliptic fibration on
a K3 surface. Consider the corresponding fibration
$\pi^{[n]}:\;M^{[n]}{\>\longrightarrow\>}S^{[n]}={\mathbb{C}}P^{n}$ on its
Hilbert scheme. A multisection of $\pi$ is a curve which is transversal to the
fibers of $\pi$; a multisection exists if and only if $M$ is projective. Fix
points $s_{1},...,s_{k}\in S$, and let $C_{k+1},...,C_{n}\subset M$ be
multisections. Denote by
$\hat{L}_{k}(s_{1},...,s_{k},C_{k+1},...,C_{n})\subset\operatorname{Sym}^{n}M$
the set of $n$-tuples of points
$(e_{1},...,e_{k},c_{k+1},...,c_{n})\in\operatorname{Sym}^{n}M$, such that
$e_{i}\in\pi^{-1}(s_{i})$ and $c_{j}\in C_{j}$. Since the holomorphic
symplectic form on $\operatorname{Sym}^{n}M$ is locally a product of the
holomorphic symplectic form on $M$, and the curves $\pi^{-1}(s_{i})$ and
$C_{j}\subset M$ are Lagrangian, the subvariety

$\hat{L}_{k}(s_{1},...,s_{k},C_{k+1},...,C_{n})\subset\operatorname{Sym}^{n}M$

is Lagrangian. Then its proper preimage $L_{k}(s_{1},...,s_{k},C)\subset
M^{[n]}$ is also Lagrangian. Under the natural map
$\pi^{[n]}:\;M^{[n]}{\>\longrightarrow\>}S^{[n]}={\mathbb{C}}P^{n}$, this
subvariety is projected to a subset of
$\operatorname{Sym}^{n}S={\mathbb{C}}P^{n}$ consisting of all $n$-tuples which
contain $(s_{1},...,s_{k})$.

Another example is due to Richard Thomas (private communication). In the early
versions of this paper, we did not specify the behaviour of the restriction
$\pi{\left|{}_{{\phantom{|}\\!\\!}_{Z}}\right.}$ outside of its regular
values, and this example shows that it can be pretty wild.

Let $S$ be a compact complex torus, $\dim_{\mathbb{C}}S=n$, and $M=T^{*}S$ the
total space of its cotangent bundle. Since $T^{*}S$ admits a natural
trivialization, the manifold $M$ is equipped with a Lagrangian fibration
$M{\>\longrightarrow\>}{\mathbb{C}}^{n}$, with the fibers obtained as
translates of $S$.

Let $X\subset S$ be a complex submanifold, and $Z:=NS^{\bot}\subset M$ the
total space of its conormal bundle. It is always Lagrangian, and in many
situations the projection of $Z$ to the base ${\mathbb{C}}^{n}$ is
$n$-dimensional. To illustrate it, let us identify the base
$B={\mathbb{C}}^{n}$ of $\pi$ with $T^{*}_{s}S$, for some (or all) $s\in S$. A
vector $v\in B$ belongs to $\pi(Z)$ if and only if $v\in T_{x}X^{\bot}$ for
some $x\in X$, where $T_{x}X^{\bot}=\\{\zeta\in T^{*}_{x}S\ \ |\ \
\langle\zeta,T_{x}X\rangle=0\\}.$ Then $\pi(Z)$ is a union of hyperplanes
$T_{x}X^{\bot}$ parametrized by a 1-dimensional family of $x\in X$.

Unless the tangent space $T_{x}X$ stays constant as we vary $x\in X$, the
image $\pi(Z)$ is $n$-dimensional, and the corresponding fiber is
0-dimensional. However, the central fiber $\pi^{-1}(0)$ is $X$, not a torus
and of different dimensional from the general fiber. The proof of 3.1 fails
for the central fiber, because $\pi(Z)$ is not smooth in zero, and the
identification $T_{z}Z_{x}=\pi(T_{z}Z)^{\bot}\cap T^{\pi}_{z}M$ does not hold.

Acknowledgments. Great many thanks to Richard Thomas for his comments,
questions and examples. We are indebted to Nigel Hitchin for his interest and
inspiration, and to Laura Schaposnik for her comments and expertise. We
express our gratitude to Stony Brook University and to the SCGP, where this
paper was prepared, and for their hospitality.

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Ljudmila Kamenova
Department of Mathematics, 3-115
Stony Brook University
Stony Brook, NY 11794-3651, USA,
<EMAIL_ADDRESS>

Misha Verbitsky
Instituto Nacional de Matemática Pura e Aplicada (IMPA)
Estrada Dona Castorina, 110, Jardim Botânico, CEP 22460-320,
Rio de Janeiro, RJ - Brasil
also:
Laboratory of Algebraic Geometry, HSE University,
Department of Mathematics, 9 Usacheva str., Moscow, Russia,
<EMAIL_ADDRESS>