# HEP Software Foundation Community White Paper Working Group – Data and
Software Preservation to Enable Reuse

M. D. Hildreth b A. Boehnlein c K. Cranmer d S. Dallmeier-Tiessen e R. Gardner
f T. Hacker c L. Heinrich g I. Jimenez h M. Kane i D. S. Katz j T. Malik g C.
Maltzahn k M. Neubauer l S. Neubert m Jim Pivarski n E. Sexton-Kennedy d J.
Shiers b T. Simko o S. Smith p D. South q A. Verbytskyi r G. Watts e J.
Wozniak Working Group ConvenerUniversity of Notre Dame, Notre Dame, IN,
USAThomas Jefferson National Accelerator Facility, Newport News, VA, USANew
York University, New York, NY, USACERN, Geneva, SwizterlandUniversity of
Chicago, Chicago, IL, USAPurdue University, West Lafayette, IN, USAUniversity
of California Santa Cruz, Santa Cruz, CA, USASoundcloud, Berlin, GermanyNCSA,
Urbana-Champaign, IL, USADePaul University, Chigago, IL, USAUniversity of
Illinois Urbana-Champaign, Urbana-Champaign, IL, USAUniversity of Heidelberg,
Heidelberg, GermanyPrinceton University, Princeton NJ, USAFermi National
Accelerator Laboratory, Batavia, IL, USAMcMaster University, Hamilton,
Ontario, CanadaDESY, Hamburg, GermanyMax-Planck-Institut für Physik, Munich,
Germany University of Washington, Seattle, WA, USA

###### Abstract

In this chapter of the High Energy Physics Software Foundation Community
Whitepaper, we discuss the current state of infrastructure, best practices,
and ongoing developments in the area of data and software preservation in high
energy physics. A re-framing of the motivation for preservation to enable re-
use is presented. A series of research and development goals in software and
other cyberinfrastructure that will aid in the enabling of reuse of particle
physics analyses and production software are presented and discussed.

##  1 Overview

As part of the process to produce a Community White Paper (CWP)CWP  within the
efforts of the High Energy Physics Software Foundation (HSF)HSF , a working
group on Data and Software preservation was established to contribute to the
broader discussion of how efforts in this area tied in with the overall goals
of the HSF and the HEP community. This document is essentially the chapter
contributed to the CWP by this working group.

The Data and Software Preservation Working Group examined current preservation
practices and technologies in HEP and beyond with the goal of describing a set
of tools and procedures that will allow analysis preservation, discovery, and
reuse. In addition, the group examined the issues around preserving the
capabilities to read, simulate, and process the large experimental dataset
from the raw data through physics analysis level data. Clearly, the technical,
sociological, and financial issues surrounding this topic are complex. This
overview outlines the main points of the discussion, summarizes some
interesting new developments, and suggests several areas of research and
development that have the potential to advance the tools available for
preservation as well as providing more of an impetus to do so.

Several recent documents produced by current projects in this area are readily
available. Because of this, a larger “White Paper” on knowledge preservation
seems redundant. For more details, including use cases and a detailed
description of the efforts within and among the various HEP experiments, the
reader is urged to consult the 2015 DPHEP reportDPHEP16 , documentation
describing the CERN Open Data PortalCERNOpen  and the CERN Analysis
Preservation PortalCAP , the work of the DASPOS projectDASPOS , and a recent
report on public access to data in the Mathematical and Physical Sciences in
the United StatesMPSOpen , which provides many other references.

##  2 Introduction and Motivation

It is easy to state a simple motivation for the preservation of the data and
knowledge surrounding High Energy Physics (HEP) experiments. Given the large
monetary investments that have been and will be made in particle physics
experiments, it is incumbent upon physicists to preserve the data and the
knowledge that can lead to scientific results in a manner such that this
investment is not lost to future generations of analysts. The word“knowledge”
in this context includes processing and analysis software, documentation, and
other components necessary for reusing a given dataset. This includes
preserving processed data containing an appropriate level of detail to enable
new analysis (“Level 3” data as defined in the 2009 DPHEP reportDPHEP09  ).
Preservation of this type can enable, for example, new analyses on older data,
as well as a way to revisit the details of a result after publication. Among
other applications, the latter use can be especially important in resolving
conflicts between published results, applying new theoretical assumptions, or
evaluating different theoretical models. Preservation enabling reuse offers
specific benefits within a given experiment as well. The preservation of
software and workflows such that they can be shared enhances collaborative
work between analysts and analysis groups, provides a way of capturing the
knowledge behind a given analysis during the review process, enables easy
transfer of knowledge to new students or analysis teams, and could establish a
manner by which results can be generated automatically for submission to
central repositories such as HEPDataHEPDATA , etc. At a base level,
preservation within an experiment could provide ways of re-processing and re-
analyzing data that could have been collected more than a decade earlier. A
final series of motivations comes from the potential re-use by others outside
of the HEP experimental community. Significant outreach efforts bringing the
excitement of analysis and discovery to younger students has been enabled by
the preservation of experimental data and software in an accessible format.
Many examples also exist of phenomenology papers reinterpreting the results of
a particular analysis in a new context. This has been extended further with
published results based on the re-analysis of processed data by scientists
outside of the collaborationsMITQCD . Engagement of external communities, such
as machine learning specialists, can be enhanced by providing the capability
to process and understand low-level HEP data in portable and relatively
platform-independent packages. This allows external users direct access to the
same tools and data as the experimentalists working in the collaborations.
Connections with industrial partners, such as those fostered by CERN
OpenLabOpenlab  can be facilitated in a similar manner. Note that the focus of
the efforts described here is to both enable and facilitate knowledge
preservation so that becomes easy to do. Policy considerations on what
workflows, analyses, and other knowledge should be preserved will necessarily
be left to the scientists and the experiments to decide.

##  3 Overview of Working Group Considerations

In the process of evaluating what should be addressed within the scope of this
working group, a consensus emerged on a set of elements that should form the
basis of preservation systems that would enable HEP data to follow the FAIR
principles as defined by the Force11 working groupF11 , namely that they be
findable, accessible, interoperable, and reusable. Note that reusability
implies, to a large extent, the other attributes, which is why we propose the
goal of reuse. Here a “preservation system” is taken to mean a set of software
tools coupled with a searchable archive that enables reuse. To the extent that
they need to be developed, the elements that might comprise such a system are
reflected in the research road map discussed below. Throughout, the assumption
is that the bit-wise preservation of data and other knowledge products, while
neither a simple nor a zero-cost endeavor, is a solved problem in that items
stored on digital media will continue to be retrievable with an acceptably
small error rate long into the future. Clearly, a variety of preservation
systems will probably be appropriate given the different preservation needs
between large central experiment workflows and the work of an individual
analyst. The larger question is to what extent common low-level tools can be
provided that address similar needs across a wide scale of preservation
problems. The sections below contain some suggestions and examples of the
kinds of elements that may be useful.

###  3.1 Assessment Tools

When approaching the complex question of preservation, the question of “what
should be preserved to enable reuse?” must be answeredMPS2 . Of course, the
answer depends on the level of reuse that is desired and the potential
preservation framework in which the information will reside. For any set of
preservation systems, a set of guidelines should be developed that reflect the
base assumptions on which preservation choices are made, that outline the
features of each system, and that enable a user to choose a path toward
preservation that is appropriate for the use case. These guidelines or tools
incorporating them can be complemented by different components of the
preservation systems. One example could be a software tool executed at run
time that is able to automatically provide the user with information on which
external components the preserved elements depend. For example, to reuse a
piece of analysis software, a potential new user may need access to data
catalogues, conditions databases, software configurations, and maybe even
experiment-specific software build systems. A tool that is able to observe the
execution of the code before it is preserved could supply a list of these
external elements, potentially while packaging the code up for preservation. A
second example of a complementary tool is the design of elements that
determine capture fidelity directly into the preservation framework. These
elements can audit captured information, data, executables, etc., to ensure
that the stored elements are sufficient to satisfy the user’s desired level of
preservation.

###  3.2 Capture Tools

While a host of diverse tools exist for various aspects of digital
preservationWflwTools , very few provide a complete solution that enables
scientists to package up their data, workflows, codes, executables, and
documentation for preservation. And, none of these is suitable for the large-
scale computational workflows typical of HEP. The diversity of workflows,
analysis practices, scales of computation, and computing techniques makes this
a challenging area in which to suggest research. It is likely that different
tools may need to be developed to address the wide scale of preservation needs
and scales within HEP and beyond. It may be possible, however, to develop
scalable infrastructure building-blocks that can be used to construct these
tools. No matter what forms these tools take, it is desirable that the capture
of information, metadata, and the other necessary elements enabling
preservation be done automatically, with as little effort from the user as
possible.

The simplest need that should be served by these tools is the ability to run a
given executable,“wrap it up”, and then re-run it at any future time while
obtaining the same results. The manner in which this is done and the execution
environments could be quite varied. Techniques such as continuous integration
could be brought into analysis workflows, for example. Changes to code,
workflows, and data products would be tracked and tested automatically,
leaving a complete record of their evolution. This would naturally serve as a
”continuous preservation” environment, assuming there are sufficient personnel
to intervene and resolve any problems raised by the integration framework. At
the opposite end of the spectrum, yet still served by continuous integration,
is the infrastructure necessary for full-scale production of processed events.
Since reuse of data often implies a comparison of new theoretical models, the
infrastructure required to regenerate simulated events must also be preserved
in order to insure that these comparisons are possible. Currently, only
minimal central efforts are underway to preserve the capabilities for data
reprocessing or the full simulation chain in large HEP experiments. Looking 10
or even 5 years in the future, this may render the current data unusable. The
working group considered this a serious problem that should be addressed
before the expertise to do so becomes sparse.

While not directly related to capture tools, the availability and
sustainability of infrastructure like CVMFS, which provides so many services
to HEP experiments, is also a serious issue. Can we be sure that CVMFS will
provide the appropriate libraries needed to run older software versions
several years from now? Will CVMFS itself have a well-delineated future as
computing platforms become more heterogeneous? These are questions that should
be addressed.

Figure 1: A sample REANA workflow, constructed from elements specified in
Yadage, showing the different components and the order of operation.

###  3.3 Applications: Preservation-as-a-Service

In the space of knowledge preservation, several preservation systems have been
or are being built that provide infrastructure that users can employ in order
to share and reuse elements of workflows, data, etc. One of the first of these
systems in HEP is RECASTRECAST , which provides the capability to reinterpret
the constraints an analysis may place on new physics by comparing new models
against a what was found in a completed search analysis on experimental data.
The novelty here is that the RECAST infrastructure maintains the capability to
re-run the full simulation and analysis chain given the new physics model as
input. If run as a service for theorists, for example, the individual
components of the analysis workflow are not accessible. The re-analysis is
performed as a“black box” process, and the results are subsequently made
available. However, the technology for preserving, composing, and
orchestrating the processing of the different workflow steps within RECAST has
evolved to a more general system called REANA that is under construction
within/behind the CERN Analysis Preservation Portal (CAP)CAP . In this
framework, an analyst can preserve individual workflow steps in containers and
string them together with a series of scripts that enable quite complex
workflows. An example workflow represented by these scripts can be seen in
Figure 1. Within REANA, the preserved executables can be reinstantiated
according to the instructions in the workflow scripts by a job scheduler and
workflow system that can be attached to any generic computing resource that
supports container execution. Other means of instantiating executables, such
as the more general approach represented by the Umbrella projectUmbrella ,
have also been added to this infrastructure. A diagram of the REANA system is
shown in Figure 2. If appropriate, the executables representing individual
workflow steps can be shared, swapped out for different processes, and re-used
in different workflows. The infrastructure also provides a container registry
and will eventually have discovery tools that will enable users to find
executables that are suitable for their purposes. Finally, the infrastructure
under construction is built entirely from commodity elements, making it
possible to install at any site with appropriate computing capability.

Figure 2: The REANA framework being developed as part of the CERN Analysis
Preservation Portal.

A second type of preservation-as-a-service infrastructure is represented by
common analysis and workflow execution frameworks such as the ALICE“train
model”. In these projects, individual users can include their analysis
workflows in a central process that automatically runs their code as it makes
a pass through the experimental dataset. This requires more effort up front,
as the analysis codes need to be validated and placed in official repositories
before the analysis train is run. It does, however, provide preservation and
versioning of the analysis code and a means of automatic execution, something
that is unusual for analysis codes.

A third example of a preservation infrastructure is represented by the work
done at DESYDESY  in building a system that automatically performed periodic
validation of a given experimental code base that has been targeted for
preservation. As computing technology, software libraries, compilers, etc.,
are updated, standard validation scripts are run to assess if changes to the
computing environment have caused changes in the physics results or outputs,
or if work needs to be done to allow the code to even run after the underlying
changes. All major experiments now use such systems to track changes between
software releases during development. Almost none use these techniques to
monitor the viability of older code and software releases.

##  4 Current Policies of the LHC Experiments

Each of the LHC experiments has adopted a data access and/or data preservation
policy, all of which can be found here. All of the LHC experiments support
public access to some subset of the data in a highly-reduced data format for
the purposes of outreach and education. CMS has gone one step further,
releasing substantial datasets in an AOD format that can be used for new
analysis. The data release includes simulated data, virtual machines that can
instantiate the added analysis examples, and extensive
documentationCMSOpenData . ALICE has promised to release 10% of their
processed data after a five-year embargo and has released 2010 data at this
timeALICEOpenData . LHCb has promised to release 50% of the processed data and
associated software after five years contingent on having sufficient manpower
to produce this. So far, that has not been the case. ATLAS has chosen a
different direction for data release, as is evidenced by this excerpt from
their public access policyATLASOpenData :

> “Data associated with journal publications are also made available: tables
> and data from plots (e.g. cross section values, likelihood profiles,
> selection efficiencies, cross section limits, …) are stored in appropriate
> repositories such as HEPDATAHEPDATA . ATLAS also strives to make additional
> material related to the paper available that allows a reinterpretation of
> the data in the context of new theoretical models. For example, an extended
> encapsulation of the analysis is often provided for measurements in the
> framework of RIVETRivet . For searches information on signal acceptances is
> also made available to allow reinterpretation of these searches in the
> context of models developed by theorists after the publication. ATLAS is
> also exploring how to provide the capability for reinterpretation of
> searches in the future via a service such as RECAST. RECAST allows theorists
> to evaluate the sensitivity of a published analysis to a new model they have
> developed by submitting their model to ATLAS.”

None of the LHC experiments have made recent public statements addressing the
new capabilities of the CERN Analysis Portal and whether or not some use of it
will be required (or strongly encouraged). All of them support some mechanisms
for internal preservation of the knowledge surrounding a physics publication.

##  5 Key Challenges for Preservation and Reuse

Creating the infrastructure and tools that will allow HEP data to adhere to
the FAIR principles of being findable, accessible, interoperable, and,
reusable presents several challenges, which are outlined in the sections
below.

###  5.1 Interoperability

Almost all of the systems and techniques discussed in the previous section
rely on the capture of executables and their preservation for reuse. Today,
the optimal technology to facilitate this is the linux container. Even the
systems such as the analysis train model which start from preserved code are
generally compiled for a small number of hardware platforms. However, with the
proliferation of computing architectures and the potential customization
required for optimal performance, it may be quite difficult to guarantee that
any preserved executable can be run on an arbitrary compute node. The large
HEP experiments have already encountered many difficulties in this area when
trying to use large HPC facilities in an opportunistic fashion. As the
computing hardware becomes more heterogeneous, new solutions may need to be
found for sustainable computing.

An alternate approach would be to ensure that whatever tools are developed for
executable and workflow capture function on most platforms where computing is
done. This would guarantee that preservation could be done at the time of
execution. Of course, all such systems will face the challenges of tracking
the evolution of the underlying hardware once they function at one point in
time.

No matter what preservation tools are developed that might enable reuse of
software, analysis techniques, and data, if they are not conceived from the
beginning as an integral part of the standard frameworks, retrofitting will be
nearly impossible. Much of the information that is needed to enable reuse is
transient and available only at the time of processing. For example, an
analysis framework with embedded preservation infrastructure that enables the
automatic capture of appropriate details and metadata describing the
workflows, the data processed, and the output products would allow the user to
compile a record of that computation and preserve it for later reuse,
archiving, or sharing with others. Without automatic capture, assembling the
necessary information is painful and time-consuming, which dramatically raises
the effort for preservation.

###  5.2 Findability

For reuse within an experiment or reuse more broadly, the elements that have
been preserved need to be linked with some sort of metadata record in a
searchable database. While simple search-able parameters can be attached to
data and software objects, reuse places much more stringent requirements on
documentation and demands a sophisticated metadata description for data
objects, executables, workflow steps, and workflows themselves. Substantial
work has been done in this area in order to provide suitable metadata
descriptions of computation, workflows, etc.Ontology  which will be
incorporated into the CERN Analysis Preservation Portal, but the current
situation represents a work in progress rather than a fully-developed
solution. Further investment in metadata schema implementation, prototyping,
and testing will be required before a suitably-descriptive metadata vocabulary
can exist.

Once a metadata description that is suitably complex to enable re-use is in
place, a registry or database structure will also be required in order to both
store analysis elements and to enable users to search for and find them. While
the effort to create such an infrastructure is probably small compared to that
needed for the metadata description, it is still a necessary part of the
system enabling preservation and reuse.

###  5.3 Cost Justification

Clearly, adding the infrastructure necessary for preservation and reuse will
require a substantial investment in development and, to a lesser extent,
hardware for the implementation. As discussed above, the motivations for doing
so are strong. The premise of this document is that the motivations do indeed
justify the cost of development, especially since many of the basic elements
of the entire preservation and reuse ecosystem already exist or are currently
being envisioned, designed, and built. As the complexities of computational
workflows increase and become increasingly ubiquitous across science, there is
a exponentially growing need for the sorts of tools and infrastructure
described here.

There is, however, a difference between mandated re-use and reproducibility
and tools that enable these yet grow organically from the needs of the
analysts. If an infrastructure that enables preservation and reuse can be
embedded in the frameworks and analysis procedures in a way that makes the
analyst’s work more productive, then there is little cost to the analyst
beyond the adoption of the analysis framework. Increased productivity benefits
everyone, resulting in more rapid production of results. The barriers to
preservation are lowered by making it automatic, and automatically useful to
the individual physicist and the larger collaborations. Focusing on the
internal benefits of such systems may be the simplest means to accelerate
their acceptance.

###  5.4 Adoption Challenges and Incentive Structure

An additional challenge is the actual adoption of preservation tools and
techniques by the community of physicists. An advantage to embedding them in a
framework that is the implementation of a completely new analysis model is
that they are there from the start. As discussed elsewhere, many of the
standard techniques of software, such as continuous integration, could have a
profound impact on the way analysis is done as well as providing the means for
automatic preservation. Hence, efforts to design and implement new models of
data analysis offer a unique opportunity to realize the benefits of
preservation, sharing, and reuse as long as these elements are considered as
integral parts of the infrastructure.

There has been much discussion about what sorts of modifications might be
necessary to the scientific and academic incentive/reward structure in order
to create a “culture of preservation.” While such cultural and institutional
changes might drive the adoption of preservation techniques and make re-use of
scientific software and data widespread, a top-down approach is difficult to
implement and to ensure that it meets the needs of the community. An effort
originating in the community itself, however, and one that makes the community
more productive may well be the path of least resistance.

##  6 Connections with Other Working Groups

Aspects of preservation and reuse underlie many of the themes discussed across
a wide variety of working groups whose conclusions appear in the CWP report.
Some major overlapping themes are discussed here.

###  6.1 Data Analysis and Interpretation WG

One focus of the Data Analysis and Interpretation Working Group[cite report]
is designing an analysis framework that enables the analyst to remember and
reproduce a piece of an analysis or an entire workflow at some time removed
from its original creation. Here the emphasis is not so much on scientific
integrity but just on being able to pause and reliably restart an analysis
effort. Creating an infrastructure with this capability would incorporate
essentially all aspects of a larger preservation system including: provenance
tracking of results (which jobs ran when with which input and produced which
output?), a means of capturing an executable and its computation environment,
and some method of cataloguing the results. A system of this type may be
sufficiently modular to allow the arbitrary composability of different
analysis executables, as seen in the REANA framework. Clearly, the proposed
elements of this framework would be very much in alignment with the broad
outline of infrastructure discussed in this document. Many of the building
blocks that need to be developed can be both widely applicable and shared
among preservation systems.

###  6.2 Other Groups

Other elements of preservation and reuse infrastructure are clearly relevant
for other working groups within the CWP effort. For example, tools for
executable and environment packaging, workflow preservation and
reinstantiation have obvious overlaps with the interests of the Workflow and
Resource Management WG111For the detailed overview and statements of each of
the WGs, please see Reference CWP  and the individual WG chapters therein. and
the Facilities and Distributed Computing WG. To the extent that continuous
integration and other frameworks can contribute to software preservation,
there is a direct connection to the Software Development WG. Engagement with
the Event/Data Processing Frameworks WG will be required in order to address
how the production software for the experiments can be included in
preservation frameworks for long-term use. The Data Access, Organization, and
Management WG clearly overlaps with long-term preservation as well.

##  7 Meeting the Challenges: R&D Roadmap

While much of the infrastructure discussed in this document can be built today
from existing technologies, several topics for research and development
emerged from the discussions. These are outlined below, followed by a roadmap
for the next five years that will allow the exploration of many of these
issues. In proposing the R&D roadmap, we note that, for many of the issues
discussed, closer collaboration with industrial partners is absolutely
necessary. Not only have those in industry thought about many of these
problems, it is likely that some of them have already developed paths to
solutions. The rapid advancement and proliferation of various container
technologies is just one example of many. Whether these particular solutions
are appropriate for HEP and other scientific applications is an open question
and will need to be resolved on a case-by-case basis.

One issue that has broad implications for many of the projects in this report
is the projected longevity of executables stored in containers. This is, of
course, directly related to the rapid evolution of computing architectures. A
series of important questions in this domain are currently unanswered and
could form the basis of an important line of research. For example, how much
can one “future-proof” executables and software against hardware evolution? Is
there an invariant set of operating system “hooks” (similar to the POSIX
standardPOSIX ) that are necessary to guarantee that the executable within a
container will still run some years into the future no matter what the changes
in the hardware base? Will it be possible to“nest” the execution of containers
in VMs (in VMs) that preserve execution environments on new platforms? If so,
what are the compromises in performance?

A second, related issue is one of software design and implementation, and the
question of software sustainability in the modern era of heterogeneous
computing architectures. At what level is it possible to abstract the
functional needs of a piece of software from the highly-optimized
implementation that is required for computational speed and optimum memory use
on a given hardware platform? Is it possible (or desirable) to design low
level libraries that can offer these implementations in a modular fashion,
rather than at the level of single instructions? One possible solution would
be“toolkits” of numerical function libraries (e.g. a 21st century CERNLib)
that are optimized for a particular class of hardware and that could be called
from higher level code. Compilation systems could automatically recognize the
platform (or kernel version) and choose the appropriate optimizations.

As mentioned above, an area of investigation that links preservation and reuse
with questions raised by the Computing Models, Facilities, and Distributed
Computing WG is one of container orchestration. Clearly, it will be necessary
in the future to distribute and execute multiple different containers for
large multi-step HEP workflows. While many container orchestration tools have
been developed (e.g., Kubernetes, Helios), do they scale to the size of HEP
production needs, what new infrastructure is required in order to run multi-
step workflows within a typical HEP production system, and how can all of the
elements of the system be preserved and catalogued for re-use?

The evolution of analysis techniques to include“big data” applications such as
Apache Spark based on databases or dedicated big data ”appliances” raises a
host of new issues for preservation and reuse. Analysis done in this
environment follows a different paradigm from the skim/slim/ntuple processes
that has dominated HEP analysis for decades. However, there has been
significant development of HEP-specific analysis tools on these platformsDIANA
. In principle, analysis scripts or their sub-components can be reused, if
they are preserved in a discoverable manner along with the interfaces and
middleware that allow them to function. A separate question concerns the data
itself. In principle, one ingests or formats an entire dataset in order to
prepare it for analysis. Since there is no reduced data to associate with the
final results, a snapshot of the full dataset may be the most appropriate to
designate as the basis for the analysis. This becomes extremely unwieldy,
since either the entire (large) dataset in its ”ingested” state needs to be
kept, or the procedure by which it is ingested (and the hardware and database
configuration on which it was analyzed) needs to be reproducible. Clearly,
time-dependent datasets present additional challenges. The coupling of the
data and the underlying storage hardware in this paradigm and its implications
for preservation and reuse need to be better understood.

One concrete contribution that an R&D effort in this arena can make is the
development of tools that facilitate preservation and reuse. The high-level
goals addressed here require low-level tools that individuals can use to
easily capture their executable and a description of their computational
environment, for example, or their multi-step workflow. Currently, these
common tools do not exist in the HEP domain. In order to make progress in this
area, various use cases need to be developed so that tools can be tailored to
a particular task. An example set of these have already been enumerated in the
context of developing CAP/REANACAPExamp . Possible uses include those
mentioned in the introduction, such as having a complete set of code and
documentation for internal review of an analysis, evaluating new theoretical
models using the results of old analyses, preserving a benchmark analysis for
posterity or to pass to a new student, etc. Once several use cases are
established, the base or common elements that need to be captured and
described with metadata can be extracted and generic tools created. Ideally,
this would lead to metadata and provenance capture infrastructure in addition
to the preservation of the computations themselves. With diverse use cases,
tools that are applicable to more than one discipline can be developed. In the
case of HEP, the CAP/REANA infrastructure under construction at CERN can be
exploited as a test bed for different capture tools and discoverability,
making it the ideal development platform.

A potential second instance of development to fill a needs gap is one that
targets validation infrastructure. As mentioned above, despite the work done
at DESY directed at the continual evaluation of the viability of a preserved
code base, none of the current HEP experiments have implemented such a system.
This may be more a question of motivation rather than technical challenge. It
is an important issue, however, if a given experiment wishes to preserve the
capability to run the simulation software for an old detector configuration in
order to reinterpret an existing result or to re-analyze “old” data. An
initial investment in what would amount to a small extension of the standard
continuous integration and validation infrastructures that the experiments
already possess could potentially result in a dramatic reduction in the effort
required to resuscitate past software versions.

This leads to the larger question of the preservation of the production
workflows and executables of the HEP experiments. As mentioned above, none of
the LHC experiments have projects to preserve the ability to re-use old
software releases. This presents a major obstacle to the re-use of the data
for physics analysis, since, without further action, it will be impossible to
re-run the detector simulation to understand the response of the detector to
new physics signals. While the infrastructure to capture the executables and
the standard workflows exists, the lifetime of the candidate technologies is
unknown. That places this question in line with the rest of the R&D topics
presented in the above paragraphs. There is an added complexity, here, in the
external conditions databases and other external resources that are used to
provide constants to the simulation that must also be sustained or separately
preserved.

###  7.1 Research and Development Roadmap

Here, we present a progression of research topics that address the issues
discussed above. Note that many of them should be orchestrated in conjunction
with projects conducted by the R&D programs of other working groups, since the
questions addressed are common. The timescales indicated extend from the start
of the program.

Near term: (1 year)

  * •

Demonstrate a fully-functional version of the CERN Analysis Preservation
Portal (CAP), the container-based solution for workflow and executable
preservation. This includes the creation of a UI that allows a user to
reproduce a workflow with a single command, the implementation of a metadata
schema that allows the stored workflows and executables to be discovered with
a search engine, the implementation of this search engine, and the
provisioning of a scalable back-end system for computation. Demonstrate
composing new workflows from preserved executable steps.

  * •

Demonstrate the capability to provision and execute production workflows for
LHC experiments that are composed of multiple independent containers.
Investigate the modifications necessary to the underlying WLCG and workflow
management infrastructure in order to construct and deploy these workflows.
Begin work on large-scale executable/workflow preservation.

  * •

Begin the investigation of limitations on the longevity and ubiquity of
container-based solutions to executable preservation.

  * •

Collection of analysis use cases and which elements are necessary to preserve
to enable re-use. Begin the creation of a taxonomy of information that should
be captured during analysis steps. Determine if there are analysis workflows
that are so different that they represent examples that cannot be captured in
a format amenable to CAP.

  * •

Begin investigation of preservation issues in “big data” analysis
environments.

Medium Term: (3 years)

  * •

Develop prototype analysis ecosystem(s), including embedded elements for the
capture of preservation information and metadata, and tools for the archiving
of this information.

  * •

Evaluation of prototypes and suitability for re-use: determination of which
elements are missing in order to enable re-use, which elements are difficult
to capture. Needs assessment in order to determine future R&D necessary to
evolve prototypes.

  * •

Continue evaluation of the limits of container technologies in the
preservation arena.

  * •

Continue exploration of large-scale analytic workflows.

  * •

Develop prototype for the preservation and validation of large-scale
production executables/workflows.

Longer Term: (5 years)

  * •

Deploy analysis ecosystem that enables re-use for any analysis that can be
conducted in the ecosystem.

  * •

Deploy system for the preservation and validation of large-scale production
executables/workflows

##  8 Engaging Communities Beyond HEP

Clearly, the problems discussed in this document are common to many
applications in other areas of science; many industrial applications may also
present solutions or suggest ways forward to arrive at solutions. A particular
example of this is represented by the series of tools and procedures developed
by the DevOps community. The modification and adoption of some of these and
their application to physics analysis would naturally solve a series of
problems related to versioning and workflow preservation. Beyond this, many
examples also exist of, for example, workflow systems in other scientific
fields. Many may not scale to LHC-sized applications, but the system
architectures can be quite similar to those developed in HEP. Unfortunately,
few opportunities exist to discover and exploit commonalities in
conceptualization and implementation. This is one case where the strong HEP
commitment to these technologies could serve as a basis for reaching out to
other scientific disciplines to foster the creation of communities of
development centered around the advancement of this work.

## References

  * (1) A. A. Alves, Jr et al., “A Roadmap for HEP Software and Computing R&D for the 2020s,” arXiv:1712.06982 [physics.comp-ph].
  * (2) http://hepsoftwarefoundation.org/
  * (3) S. Amera, et al., “Status Report of the DPHEP Collaboration: A Global Effort for Sustainable Data Preservation in High Energy Physics,” arXiv:1512.02019v2.
  * (4) Accessible at http://opendata.cern.ch/about .
  * (5) Accessible at https://analysispreservation.cern.ch/  .
  * (6) Information available at daspos.crc.nd.edu .
  * (7) doi:10.7274/R0542KN7; information available at mpsopendata.crc.nd.edu .
  * (8) Z. Akopov, et al., “Status Report of the DPHEP Study Group: Towards a Global Effort for Sustainable Data Preservation in High Energy Physics”, https://arxiv.org/ftp/arxiv/papers/0912/0912.0255.pdf .
  * (9) http://hepdata.cedar.ac.uk/ .
  * (10) J. Thaler et al., 10.1103/PhysRevD.96.074003 .
  * (11) http://openlab.cern .
  * (12) M. D. Wilkinson, et al., “The FAIR Guiding Principles for scientific data management and stewardship,” Scientific Data 3, 160018 (2016). doi:10.1038/sdata.2016.18. Information available at https://www.force11.org/group/fairgroup/fairprinciples .
  * (13) For a lengthy discussion, see, for example, https://mpsopendata.crc.nd.edu/index.php/final-report.
  * (14) See, for example, the COPTR compilation at http://coptr.digipres.org/Main_Page or the POWRR Tool grid: http://digitalpowrr.niu.edu/tool-grid/ .
  * (15) Kyle Cranmer and Itay Yavin, “RECAST: Extending the Impact of Existing Analyses,” JHEP 1104:038, 2011; https://arxiv.org/abs/1010.2506 .
  * (16) http://reanahub.io/ .
  * (17) https://ccl.cse.nd.edu/software/umbrella/ .
  * (18) Dmitri Ozerov and David M South, 2014 J. Phys.: Conf. Ser. 513 042043\.
  * (19) http://opendata.cern.ch/research/CMS .
  * (20) http://opendata.cern.ch/education/ALICE .
  * (21) https://cds.cern.ch/record/2002139/files/ATL-CB-PUB-2015-001.pdf .
  * (22) http://rivet.hepforge.org/ .
  * (23) See, e.g., D. Carral et al., ”An Ontology Design Pattern for Particle Physics Analysis”, Proceedings of the 6th Workshop on Ontology and Semantic Web Patterns (WOP 2015) co-located with the 14th International Semantic Web Conference (ISWC 2015), Bethlehem, Pensylvania, USA, October 11, 2015.
  * (24) http://standards.ieee.org/develop/wg/POSIX.html .
  * (25) The DIANA-HEP project; http://histogrammar.org/ .
  * (26) http://doi.org/10.5281/zenodo.33693 .


the statement is already clear for $i=0$. Therefore, it suffices to show the
$\pi_{1}^{\textrm{{\'{e}}t}}(\mathcal{T}_{L},b_{\mathbb{C}})$-stablity of the
direct sum
$(\mathbb{V}^{0}_{B,b_{\mathbb{C}}}\oplus\mathbb{V}^{1}_{B,b_{\mathbb{C}}}){\otimes}\mathbb{Q}_{\ell}=\mathbb{V}^{f}_{B,b_{\mathbb{C}}}{\otimes}\mathbb{Q}_{\ell}\subseteq\mathrm{H}^{2}_{\textrm{{\'{e}}t}}(\mathcal{X}_{b_{\mathbb{C}}},\mathbb{Q}_{\ell})$.
Since $b_{L}$ determines a splitting
$\pi_{1}^{\textrm{{\'{e}}t}}(\mathcal{T}_{L},b_{\mathbb{C}})\,{\cong}\,\pi_{1}^{\textrm{{\'{e}}t}}(\mathcal{T}_{\overline{L}},b_{\mathbb{C}})\rtimes\mathrm{Gal}_{L}$
and
$\pi_{1}^{\textrm{{\'{e}}t}}(\mathcal{T}_{\overline{L}},b_{\mathbb{C}})=\pi_{1}^{\textrm{{\'{e}}t}}(\mathcal{T}_{\mathbb{C}},b_{\mathbb{C}})$
acts trivially on this direct sum, we reduce to showing that this direct sum
is $\mathrm{Gal}_{L}$-stable. This follows easily from the fact that
$\iota_{\mathbb{C},B}{\otimes}\mathbb{Q}_{\ell}$ is
$\mathrm{Gal}_{L}$-equivariant, as $\iota_{\mathbb{C}}$ is defined over $L$.

(b) Set
$\mathcal{H}_{\mathrm{dR},\mathbb{C}}:=\mathcal{H}_{\mathrm{dR}}|_{\mathcal{T}_{\mathbb{C}}}$
and
$\mathcal{H}_{\mathrm{dR},L}:=\mathcal{H}_{\mathrm{dR}}|_{\mathcal{T}_{L}}$.
Note that the horizontal subspace
$\mathrm{H}^{0}(\mathcal{H}_{\mathrm{dR},\mathbb{C}})^{\nabla=0}$ is invariant
under the $\mathrm{Aut}(\mathbb{C}/L)$-action on
$\mathrm{H}^{0}(\mathcal{H}_{\mathrm{dR},\mathbb{C}})$, so that
$\mathrm{H}^{0}(\mathcal{H}_{\mathrm{dR},L})^{\nabla=0}{\otimes}_{L}\mathbb{C}=\mathrm{H}^{0}(\mathcal{H}_{\mathrm{dR},\mathbb{C}})^{\nabla=0}$.
Consider now the de Rham realization of $\iota_{L}:=\iota{\otimes}L$:

$\iota_{L,\mathrm{dR}}:\mathrm{H}^{2}_{\mathrm{dR}}(\overline{\mathcal{X}}_{L}/L)\to\mathrm{H}^{2}_{\mathrm{dR}}(\mathcal{X}_{b_{L}}/L)$

This is a surjection onto the horizontal subspace
$\mathrm{H}^{0}(\mathcal{H}_{\mathrm{dR},L})^{\nabla=0}\subseteq\mathrm{H}^{2}_{\mathrm{dR}}(\mathcal{X}_{b_{L}}/L)$—one
can check this by tensoring with $\mathbb{C}$ and comparing with (10). Recall
that the theorem of the fixed part in particular says that there is a
canonical Hodge filtration on
$\mathrm{H}^{0}(\mathcal{H}_{\mathrm{dR},\mathbb{C}})^{\nabla=0}$. We observe
that this Hodge filtration descends (necessarily uniquely) to $L$, as we can
simply take the filtration on the image of $\iota_{L,\mathrm{dR}}$ inherited
from that on $\mathrm{H}^{2}_{\mathrm{dR}}(\overline{\mathcal{X}}_{L}/L)$.

Up to enlarging $N$, we may assume:

  * •

The direct summand
$\mathrm{NS}(\mathcal{X}_{\eta}){\otimes}_{\mathbb{Z}}\mathcal{O}_{\mathcal{T}_{L}}\subseteq\mathcal{H}_{\mathrm{dR},L}$
extends to a direct summand $\mathcal{H}^{0}_{\mathrm{dR}}$ of
$\mathcal{H}_{\mathrm{dR}}$ such that the natural map
$\mathrm{H}^{0}(\mathcal{H}^{0}_{\mathrm{dR}})^{\nabla=0}{\otimes}\mathcal{O}_{\mathcal{T}}\to\mathcal{H}^{0}_{\mathrm{dR}}$
is an isomorphism.

  * •

The direct summand
$\mathrm{H}^{0}(\mathcal{H}_{\mathrm{dR},L})^{\nabla=0}{\otimes}\mathcal{O}_{\mathcal{T}_{L}}\subseteq\mathcal{H}_{\mathrm{dR},L}$
extends to an orthogonal direct summand (as a filtered flat vector bundle)
$\mathcal{H}_{\mathrm{dR}}$, which we denote by
$\mathcal{H}^{f}_{\mathrm{dR}}$.

  * •

$\mathcal{H}^{0}_{\mathrm{dR}}$ is an orthogonal direct summand (as a filtered
flat vector bundle) of $\mathcal{H}^{f}_{\mathrm{dR}}$.

Finally, we may take $\mathcal{H}_{\mathrm{dR}}^{1}$ to be the orthogonal
complement of $\mathcal{H}^{0}_{\mathrm{dR}}$ in
$\mathcal{H}^{f}_{\mathrm{dR}}$.

(c) By the equivalence [NO85, Cor. 6.8], each piece
$\mathcal{H}^{i}_{\mathrm{dR}}|_{\hat{\mathcal{T}}_{\mathfrak{p}}}$, as a
vector bundle with an integrable quasi-nilpotent connection, gives rise to a
direct summand $\mathcal{H}^{i}_{\mathfrak{p},\mathrm{cris}}$. We need to
check that the decomposition
$\mathcal{H}_{\mathfrak{p},\mathrm{cris}}=\oplus_{i=0}^{2}\mathcal{H}^{i}_{\mathfrak{p},\mathrm{cris}}$
respects the Frobenius structure, i.e., the canonical Frobenius action
$F:F_{\mathcal{T}_{\mathfrak{p}}/\hat{\mathcal{O}}_{L,\mathfrak{p}}}^{*}\mathcal{H}_{\mathfrak{p},\mathrm{cris}}\to\mathcal{H}_{\mathfrak{p},\mathrm{cris}}$
sends each
$F_{\mathcal{T}_{\mathfrak{p}}/\hat{\mathcal{O}}_{L,\mathfrak{p}}}^{*}\mathcal{H}^{i}_{\mathfrak{p},\mathrm{cris}}$
into $\mathcal{H}^{i}_{\mathfrak{p},\mathrm{cris}}$. Since this statement can
be checked after inverting $p$, by [Ogu84, Thm 4.1] it suffices to show the
statement at one closed point on $\mathcal{T}_{\mathfrak{p}}$. Therefore, it
suffices to note that the Frobenius action on
$\mathrm{H}^{2}_{\mathrm{cris}}(\mathcal{X}_{b_{\mathfrak{p}}}/\hat{\mathcal{O}}_{L,\mathfrak{p}})=\mathcal{H}_{\mathfrak{p},\mathrm{cris}}|_{b_{\mathfrak{p}}}$
does send each
$\mathcal{H}^{i}_{\mathfrak{p},\mathrm{cris}}|_{b_{\mathfrak{p}}}$ into
$\mathcal{H}^{i}_{\mathfrak{p},\mathrm{cris}}|_{b_{\mathfrak{p}}}$. This is
clear for $i=0$; moreover, using the crystalline-de Rham comparison
isomorphisms one checks that
$(\mathcal{H}^{0}_{\mathfrak{p},\mathrm{cris}}\oplus\mathcal{H}^{1}_{\mathfrak{p},\mathrm{cris}})|_{b_{\mathfrak{p}}}\subseteq\mathrm{H}^{2}_{\mathrm{cris}}(\mathcal{X}_{b_{\mathfrak{p}}}/\hat{\mathcal{O}}_{L,\mathfrak{p}})$
is the image of
$\mathrm{H}^{2}_{\mathrm{cris}}(\overline{\mathcal{X}}_{\mathfrak{p}}/\hat{\mathcal{O}}_{L,\mathfrak{p}})\to\mathrm{H}^{2}_{\mathrm{cris}}(\mathcal{X}_{b_{\mathfrak{p}}}/\hat{\mathcal{O}}_{L,\mathfrak{p}})$
induced by $\iota{\otimes}k(\mathfrak{p})$, and hence is stable under the
Frobenius action.

Next, we check that
$(\mathcal{H}^{i}_{\mathfrak{p},\mathrm{cris}},\mathcal{H}^{i}_{\mathrm{dR}}|_{\hat{\mathcal{T}}_{\mathfrak{p}}})$
is a constant filtered F-crystal for $i=0,1$. This is clear for $i=0$. Since
$\mathcal{H}^{1}_{\mathfrak{p},\mathrm{cris}}$ is an F-crystal on
$\mathrm{Cris}(\mathcal{T}_{\mathfrak{p}}/\hat{\mathcal{O}}_{L,\mathfrak{p}})$,
its module of global sections
$\mathrm{H}^{0}(\mathcal{H}^{1}_{\mathrm{cris}})$ is an F-crystal on
$\mathrm{Cris}(k(\mathfrak{p})/\hat{\mathcal{O}}_{L,\mathfrak{p}})$, so that
there is a natural morphism

$\mathrm{H}^{0}((\mathcal{T}_{\mathfrak{p}}/\hat{\mathcal{O}}_{L,\mathfrak{p}})_{\mathrm{cris}},\mathcal{H}^{1}_{\mathfrak{p},\mathrm{cris}}){\otimes}\mathcal{O}_{\mathcal{T}_{\mathfrak{p}}/\hat{\mathcal{O}}_{L,\mathfrak{p}}}\to\mathcal{H}^{1}_{\mathfrak{p},\mathrm{cris}}$
(11)

of F-crystals, where
$\mathcal{O}_{\mathcal{T}_{\mathfrak{p}}/\hat{\mathcal{O}}_{L,\mathfrak{p}}}$
denotes the structure sheaf on
$\mathrm{Cris}(\mathcal{T}_{\mathfrak{p}}/\hat{\mathcal{O}}_{L,\mathfrak{p}})$.
Evaluating at the (pro)-object
$\hat{\mathcal{T}}_{\mathfrak{p}}\in\mathrm{Cris}(\mathcal{T}_{\mathfrak{p}}/\hat{\mathcal{O}}_{L,\mathfrak{p}})$,
we obtain

$\mathrm{H}^{0}(\mathcal{H}^{1}_{\mathrm{dR}})^{\nabla=0}{\otimes}\mathcal{O}_{\hat{\mathcal{T}}_{\mathfrak{p}}}\to\mathcal{H}^{1}_{\mathrm{dR}}|_{\hat{\mathcal{T}}_{\mathfrak{p}}}$
(12)

which is an isomorphism of flat vector bundles. Therefore, (11) is also an
isomorphism. Since (12) also preserves the filtrations, (11) and (12) together
give a trivialization of the filtered F-crystal when $i=1$.

(d) Assume first that $\mathscr{P}(\mathcal{T},s,\xi)$ dominates
$\mathcal{T}$. By (2.1.6), we can find a lifting
$(\widetilde{s},\widetilde{\xi})$ of $(s,\xi)$ over $\mathcal{O}_{K}$, where
$K$ is a finite extension of $W(k(s))[1/p]$. Choose an embedding
$K\hookrightarrow\mathbb{C}$. Then
$\mathscr{P}(\mathcal{T},\widetilde{s}_{\mathbb{C}},\widetilde{\xi}_{\mathbb{C}})$
dominates $\mathcal{T}_{\mathbb{C}}$. Using (2.1.10)(d), we infer that
$c_{1,\mathrm{dR}}(\widetilde{\xi}_{\mathbb{C}})$ extends to an element of
$\mathrm{H}^{0}(\mathbb{V}^{0})$. This implies that
$c_{1,\mathrm{dR}}(\widetilde{\xi})\in\mathcal{H}^{0}_{\mathrm{dR},\widetilde{s}}\subseteq\mathcal{H}_{\mathrm{dR},\widetilde{s}}$,
and hence
$c_{1,\mathrm{cris}}(\xi)\in\mathcal{H}^{0}_{\mathfrak{p},\mathrm{cris}}|_{s}$.
Conversely, suppose that
$c_{1,\mathrm{cris}}(\xi)\in\mathcal{H}^{0}_{\mathfrak{p},\mathrm{cris}}|_{s}$.
To show that $\mathscr{P}(\mathcal{T},s,\xi)$ dominates $\mathcal{T}$ it
suffices to show that $\xi$ deforms to the formal completion of
$\mathcal{T}{\otimes}W(k(s))$ at $s$. This follows from [Ogu79, Prop. 1.12]
and the fact that $\mathcal{H}^{0}_{\mathfrak{p},\mathrm{cris}}$ is a trivial
F-crystal and
$\mathcal{H}^{0}_{\mathrm{dR}}=\mathrm{Fil}^{1}\mathcal{H}^{0}_{\mathrm{dR}}$.

The argument for the statement about $\mathcal{T}_{\mathfrak{p}}$ is entirely
similar. Assume now that $\mathscr{P}(\mathcal{T}_{\mathfrak{p}},s,\xi)$
dominates $\mathcal{T}_{\mathfrak{p}}$. Then using (2.1.10)(d) again we infer
that $c_{1,\mathrm{cris}}(\xi)$ extends as a global section of the isocrystal
$\mathcal{H}_{\mathfrak{p},\mathrm{cris}}[1/p]$. Therefore, by [Mor19, Thm
1.4], for some $m\in\mathbb{N}$, $\xi^{p^{m}}$ extends to a line bundle on
$\mathcal{X}_{\mathfrak{p}}$ and hence on
$\overline{\mathcal{X}}_{\mathfrak{p}}$. This implies that
$c_{1,\mathrm{cris}}(\xi)$ up to a multiple lies in the image of
$\mathrm{H}^{2}_{\mathrm{cris}}(\overline{\mathcal{X}}_{\mathfrak{p}}{\otimes}k(s)/W(k(s)))\to\mathrm{H}^{2}_{\mathrm{cris}}(\mathcal{X}_{s}/W(k(s)))$.
One easily checks that this image is precisely
$(\mathcal{H}^{0}_{\mathfrak{p},\mathrm{cris}}\oplus\mathcal{H}^{1}_{\mathfrak{p},\mathrm{cris}})|_{s}$,
which is a direct summand. Therefore, $c_{1,\mathrm{cris}}(\xi)$ must lie in
$(\mathcal{H}^{0}_{\mathfrak{p},\mathrm{cris}}\oplus\mathcal{H}^{1}_{\mathfrak{p},\mathrm{cris}})|_{s}$
without passing to a multiple. Conversely, assume that
$c_{1,\mathrm{cris}}(\xi)\in(\mathcal{H}^{0}_{\mathfrak{p},\mathrm{cris}}\oplus\mathcal{H}^{1}_{\mathfrak{p},\mathrm{cris}})|_{s}$.
Then by [Ogu79, Cor. 1.13] and the fact that
$\mathcal{H}^{0}_{\mathfrak{p},\mathrm{cris}}\oplus\mathcal{H}^{1}_{\mathfrak{p},\mathrm{cris}}$
is a constant filtered F-crystal, $\xi$ deforms to the formal completion of
$\mathcal{T}_{\mathfrak{p}}{\otimes}k(s)$ at $s$. Hence
$\mathscr{P}(\mathcal{T}_{\mathfrak{p}},s,\xi)$ dominates
$\mathcal{T}_{\mathfrak{p}}$. ∎

### 5.2 Non-liftable Line Bundles

In this section, we study the $h^{0,2}=2$ case in detail and prove (1.1.4).
Again, apply our set-up and notations in §4.1 to the case when
$R=\mathcal{O}_{L}[1/N]$ for some number field $L\subseteq\mathbb{C}$ and
$N\in\mathbb{N}$. Assume in addition that $h^{0,2}=2$ for the fibers of the
family and for every (not just a general) geometric point $s\to\mathcal{S}$,
the Kodaira-Spencer map

$T_{s}\mathcal{S}_{k(s)}\to\mathrm{Hom}(\mathrm{H}^{1}(\Omega_{\mathcal{X}_{s}}),\mathrm{H}^{2}(\mathcal{O}_{\mathcal{X}_{s}}))$

has rank at least $h^{0,2}(\mathcal{X}_{s})=2$. Let $\mathcal{E}$ be the
proper subscheme of $\mathcal{S}$ obtained by applying (4.3.2). Let
$\mathcal{E}_{i}$’s be the irreducible components of $\mathcal{E}$.

######  Remark (5.2.1).

Our assumption on the Kodaira-Spencer map means that $\mathcal{E}=E$ in
(4.3.2) and (4.2.3) up to increasing $N$. In particular, we have

$\dim\mathcal{E}_{i,\mathbb{C}}\geq
d=\dim_{R}\mathcal{S}-(h^{0,2}-1)=\dim_{R}\mathcal{S}-1$

by (4.2.3)(ii) for all $i$. But
$\dim\mathcal{E}_{i,\mathbb{C}}<\dim_{R}\mathcal{S}$, so all the components
$\mathcal{E}_{i,\mathbb{C}}$ have codimension $1$ in
$\mathcal{S}_{\mathbb{C}}$. Note also that the restriction of $\mathbb{V}$ to
(the smooth locus of) $\mathcal{E}_{i,\mathbb{C}}$ is non-isotrivial.

#### (5.2.2)

Consider in addition the following conditions.

  1. (I)

Each $\mathcal{E}_{i}$ has geometrically connected fibers over
$\mathcal{O}_{L}[1/N]$.

  2. (II)

There exists a scheme $\widetilde{\mathcal{E}}_{i}$ smooth over
$\mathcal{O}_{L}[1/N]$ with geometrically connected fibers such that
$\widetilde{\mathcal{E}}_{i}$ admits a surjective and generically étale
morphism to $\mathcal{E}_{i}$.

  3. (III)

When the base $\mathcal{T}$ considered in (5.1.1) is taken to be
$\widetilde{\mathcal{E}}_{i}$, the restricted family
$(\mathcal{X}/\mathcal{S})|_{\widetilde{\mathcal{E}}_{i}}$ satisfies the
assumptions in (5.1.1) and the conclusion of (5.1.2) holds.

We claim that the above conditions can always be satisfied up to replacing $L$
by a finite extension in $\mathbb{C}$ and increasing $N$. First, up to
extending $L$, we may assume that each $\mathcal{E}_{i,L}$ is geometrically
connected, so that (I) holds up to increasing $N$ by [Sta22, 055H]. Next, to
construct $\widetilde{\mathcal{E}}_{i}$ which satisfies (II) and (III), we
first apply Hironaka’s resolution of singularities to find a smooth
$\mathcal{E}^{\prime}_{i,L}$ admiting a proper birational map to
$\mathcal{E}_{i,L}$, then we find a connected étale cover
$\widetilde{\mathcal{E}}_{i,\mathbb{C}}$ of
$\mathcal{E}^{\prime}_{i,\mathbb{C}}:=\mathcal{E}^{\prime}_{i,L}{\otimes}\mathbb{C}$
so that (5.1.1)(iii) is satisfied, i.e.,
$\mathrm{Mon}(\mathbb{V}_{B}|_{\widetilde{\mathcal{E}}_{i,\mathbb{C}}},s)$ is
connected for some (and hence every)
$s\in\widetilde{\mathcal{E}}_{i,\mathbb{C}}$. Then we apply the argument of
(4.4.1): Up to replacing $L$ by a finite extension in $\mathbb{C}$,
$\widetilde{\mathcal{E}}_{i,\mathbb{C}}$ is defined over $L$, i.e.,
$\widetilde{\mathcal{E}}_{i,\mathbb{C}}=\widetilde{\mathcal{E}}_{i,L}{\otimes}\mathbb{C}$
for some étale cover $\widetilde{\mathcal{E}}_{i,L}$ of $\mathcal{E}_{i,L}$;
moreover, we may assume that $\widetilde{\mathcal{E}}_{i,L}(L)\neq\emptyset$.
Up to replacing $\widetilde{\mathcal{E}}_{i,L}$ (and hence also
$\widetilde{\mathcal{E}}_{i,\mathbb{C}}$) by a connected étale cover,
(5.1.1)(iv) is satisfied, i.e.,
$\mathrm{Mon}(\mathcal{H}_{\ell}|_{\widetilde{\mathcal{E}}_{i,L}},s)$ is
connected for some (and hence every) geometric point $s$ on
$\widetilde{\mathcal{E}}_{i,L}$ for every prime $\ell$. Up to increasing $N$
and applying [Sta22, 055H] again,
$\widetilde{\mathcal{E}}_{i,L}\to\mathcal{E}_{i,L}$ spreads out to a morphism
$\widetilde{\mathcal{E}}_{i}\to\widetilde{\mathcal{E}}$ which satisfies (II)
above. Finally, the rest of assumptions in (5.1.1) when $\mathcal{T}$ there is
taken to be $\widetilde{\mathcal{E}}_{i}$ can easily be satisfied up to
increasing $N$ again, so that (III) is met.

######  Proposition (5.2.3).

Assume that the conditions in (5.2.2) are satisfied. Let
$\mathfrak{p}\subseteq\mathcal{O}_{L}[1/N]$ be a finite prime. We denote by
$\mathcal{H}_{\mathfrak{p},\mathrm{cris}}|_{\widetilde{\mathcal{E}}_{i}}=\oplus_{j=0}^{2}\mathcal{H}^{j}_{i,\mathfrak{p},\mathrm{cris}}$
the decomposition provided by (5.1.2)(c) when $\mathcal{T}$ is taken to be
$\widetilde{\mathcal{E}}_{i}$. Then following are equivalent for a LB-pair
$(s,\xi)$ on $\mathcal{X}/\mathcal{S}$ when $s$ lies above
$\mathcal{S}_{\mathfrak{p}}$.

  1. (i)

There exists an $\mathcal{E}_{i}$ and a lift $u\to\widetilde{\mathcal{E}}_{i}$
lifting $s$ such that, identifying $\mathcal{X}_{s}$ with $\mathcal{X}_{u}$,
$c_{1,\mathrm{cris}}(\xi)\in\mathrm{H}^{2}_{\mathrm{cris}}(\mathcal{X}_{u}/W(k(s)))$
lies in
$(\mathcal{H}^{0}_{i,\mathfrak{p},\mathrm{cris}}\oplus\mathcal{H}^{1}_{i,\mathfrak{p},\mathrm{cris}})|_{u}$
but not in $\mathcal{H}^{0}_{i,\mathfrak{p},\mathrm{cris}}|_{u}$.

  2. (ii)

The pair $(s,\xi)$ is not strongly liftable in the sense of (2.1.3). In
particular, $\mathscr{P}(\mathcal{S},s,\xi)$ is not flat.

###### Proof.

Set $k:=k(s)$, $p:=\mathrm{char\,}k$, and $W:=W(k)$.

(i) $\Rightarrow$ (ii): Let $\eta_{i,\mathfrak{p}}$ be the generic point of
$\mathcal{E}_{i,\mathfrak{p}}$. We first claim that the image of
$\mathscr{P}(\mathcal{S},s,\xi)$ on $\mathcal{S}$ contains
$\eta_{i,\mathfrak{p}}$. Let $\Delta$ be the formal completion of
$\widetilde{\mathcal{E}}_{i,\mathfrak{p}}{\otimes}_{k(\mathfrak{p})}k$ at $u$.
Then $\Delta=\mathrm{Spf}(\mathcal{O}_{\Delta})$ for a formal power series
ring $\mathcal{O}_{\Delta}$ over $k$. As
$(\mathcal{H}^{0}_{i,\mathfrak{p},\mathrm{cris}}\oplus\mathcal{H}^{1}_{i,\mathfrak{p},\mathrm{cris}})$
is a constant filtered F-crystal over
$\widetilde{\mathcal{E}}_{i,\mathfrak{p}}$ by (5.1.2)(c), [Ogu79, Cor. 1.13]
implies that $\xi$ extends to $\Delta$. Therefore, we can find an LB-pair
$(t,\zeta)$ such that as an alg-geometric point on
$\mathscr{P}_{\mathcal{X}/\mathcal{S}}$, $(t,\zeta)$ specializes to $(s,\xi)$,
and $t\to\mathcal{S}$ factors through $\eta_{i,\mathfrak{p}}$.

Next, we argue that $(t,\zeta)^{\flat}\in\mathscr{P}(\mathcal{S},s,\xi)$ does
not lie in the Zariski closure of any characteristic $0$ point of
$\mathscr{P}(\mathcal{S},s,\xi)$. Suppose on the contrary that there exists an
LB-pair $(\widetilde{t},\widetilde{\zeta})$ in characteristic $0$ which
specializes to $(t,\zeta)$. Since $t^{\flat}\in\mathcal{S}$ is just
$\eta_{i,\mathfrak{p}}$, it is easy to see that the Zariski closure of
$\widetilde{t}^{\flat}$ in $\mathcal{S}$ is of codimension $1$ by [Sta22,
0B2J]. Note that $\widetilde{t}^{\flat}\in\mathcal{S}$ cannot be the generic
point of $\mathcal{E}_{i}$. Indeed, otherwise
$\mathscr{P}(\widetilde{\mathcal{E}}_{i},u,\xi)$ dominates
$\widetilde{\mathcal{E}}_{i}$, which by (5.1.2)(d) implies that
$\xi\in\mathcal{H}^{0}_{i,\mathfrak{p},\mathrm{cris}}|_{u}$ and hence
contradicts (i). By considering dimensions of Zariski closures, it is clear
that $\widetilde{t}^{\flat}\in\mathcal{S}$ cannot be the generic point of any
$\mathcal{E}_{j}$ for $j\neq i$, so we must have that
$\widetilde{t}^{\flat}\not\in\mathcal{E}$. However, this implies that the
deformation space $\mathrm{Def}(\zeta)$ of $\zeta$ in the formal completion of
$\mathcal{S}_{k(\widetilde{t})}$ at $t$ has codimension $1$ and
$\widetilde{t}\not\in\mathcal{E}_{k(\widetilde{t})}$. This contradicts
(4.3.2).

(ii) $\Rightarrow$ (i): Consider the deformation space $\mathrm{Def}(\xi)_{k}$
(resp. $\mathrm{Def}(\xi)$) in the formal completion $\hat{\mathcal{S}}_{s,k}$
(resp. $\hat{\mathcal{S}}_{s}$) of $\mathcal{S}{\otimes}k$ (resp.
$\mathcal{S}{\otimes}W$) at $s$. Recall that if
$\mathrm{Def}(\xi)_{k}\subseteq\hat{\mathcal{S}}_{s,k}$ has codimension
exactly $2$, then $\mathrm{Def}(\xi)$ is flat over $W$ by miracle flatness
[Sta22, 00R4], which will imply that $(s,\xi)$ is strongly liftable.
Therefore, we must have that $\mathrm{Def}(\xi)_{k}$ has codimension $1$ in
$\hat{\mathcal{S}}_{s,k}$. In turn, this implies that there exists an LB-pair
$(t,\zeta)$ which specializes to $(s,\xi)$ such that the Zariski closure $Z$
of $t^{\flat}$ in $\mathcal{S}$ lies in $\mathcal{S}_{\mathfrak{p}}$ and has
codimension $1$ in $\mathcal{S}_{\mathfrak{p}}$. If we form
$\mathrm{Def}(\zeta)_{k(t)}$ inside $\hat{\mathcal{S}}_{t,k(t)}$ analogously,
then $\mathrm{Def}(\zeta)_{k(t)}$ must have codimension $1$. By (4.3.2) (see
also (4.3.4)) we must have $t\in\mathcal{E}_{k(t)}$. Therefore, the Zariski
closure of $t^{\flat}$ in $\mathcal{S}$ must be $\mathcal{E}_{i,\mathfrak{p}}$
for some $\mathcal{E}_{i}$. Let
$u\in\widetilde{\mathcal{E}}_{i,\mathfrak{p}}(k)$ be any lift of $s$. Then
$\mathscr{P}(\widetilde{\mathcal{E}}_{i,\mathfrak{p}},u,\xi)$ dominates
$\widetilde{\mathcal{E}}_{i,\mathfrak{p}}$, but
$\mathscr{P}(\widetilde{\mathcal{E}}_{i},u,\xi)$ does not dominate
$\widetilde{\mathcal{E}}_{i}$. Hence the conclusion follows from (5.1.2)(d). ∎

Finally, we are ready to prove (1.1.4).

######  Theorem (5.2.4).

Consider the setting in (1.1.4). Namely, let $L\subseteq\mathbb{C}$ be a
number field, $N\in\mathbb{N}_{>1}$, $\mathcal{S}$ be a smooth quasi-
projective $\mathcal{O}_{L}[1/N]$-scheme with $\mathcal{S}_{\mathbb{C}}$
connected, and let $f:\mathcal{X}\to\mathcal{S}$ be a smooth projective family
with geometrically connected fibers. Assume the following conditions.

  * •

$f_{\mathbb{C}}$ has large monodromy.

  * •

For every $s\in\mathcal{S}(\mathbb{C})$, $h^{0,2}(\mathcal{X}_{s})=2$, and the
Kodaira-Spencer map
$T_{s}\mathcal{S}_{\mathbb{C}}\to\mathrm{Hom}(\mathrm{H}^{1}(\Omega_{\mathcal{X}_{s}}),\mathrm{H}^{2}(\mathcal{O}_{\mathcal{X}_{s}}))$
has rank $\geq 2$.

Then up to increasing $N$, the following holds: given any LB-pair $(s,\xi)$ on
$\mathcal{X}/\mathcal{S}$ with $k:=k(s)$, there exists another LB-pair
$(s^{\prime},\xi^{\prime})$ over $k$ with the following properties.

  1. (i)

There exists a smooth connected $k$-curve $C$ equipped with a morphism
$C\to\mathcal{S}$, a relative line bundle
$\boldsymbol{\xi}\in\mathscr{P}_{\mathcal{X}_{C}/C}(C)$, and geometric points
$c,c^{\prime}$ on $C$ lifting $s,s^{\prime}$, such that identifying
$\mathcal{X}_{c}$ with $\mathcal{X}_{s}$ and $\mathcal{X}_{c^{\prime}}$ with
$\mathcal{X}_{s^{\prime}}$ we have $\boldsymbol{\xi}_{c}\sim_{\mathbb{Q}}\xi$
and $\boldsymbol{\xi}_{c^{\prime}}\sim_{\mathbb{Q}}\xi^{\prime}$;

  2. (ii)

There exist $\xi^{\prime}_{m}\in\mathrm{Pic}(\mathcal{X}_{s^{\prime}})$ for
$m=1,2$ such that $\xi^{\prime}=\xi^{\prime}_{1}+\xi^{\prime}_{2}$ and each
$(s^{\prime},\xi_{m}^{\prime})$ is liftable.

###### Proof.

To prove the theorem, we are allowed to make replacement as in (4.4.1) so that
the family $f:\mathcal{X}\to\mathcal{S}$ satisfies the assumptions in §4.1.
Moreover, up to replacing $L$ by a further finite extension in $\mathbb{C}$
and increasing $N$, the family satisfies the assumptions in (5.2.2), so that
we are in the setting of (5.2.3). Let
$\mathbb{V}|_{\widetilde{\mathcal{E}}_{i,\mathbb{C}}}=\oplus_{j=0}^{2}\mathbb{V}^{j}_{i}$
be the decomposition obtained by applying the set-up in (5.1.1) when the base
$\mathcal{T}$ therein is taken to be $\widetilde{\mathcal{E}}_{i}$. We make
the following claim.

Claim: Up to increasing $N$, the following holds: For every $i$ such that
$\mathbb{V}^{1}_{i}\neq 0$ and for every finite prime
$\mathfrak{p}\subseteq\mathcal{O}_{L}[1/N]$, there exists a point
$s^{\prime}_{0}\in\mathcal{E}_{i,\mathfrak{p}}(\overline{k(\mathfrak{p})})$
such that there exists a lift
$u^{\prime}_{0}\in\widetilde{\mathcal{E}}_{i,\mathfrak{p}}(\overline{k(\mathfrak{p})})$
such that, identifying $\mathcal{X}_{s^{\prime}_{0}}$ with
$\mathcal{X}_{u^{\prime}_{0}}$, there exists a line bundle
$\zeta\in\mathrm{Pic}(\mathcal{X}_{s^{\prime}_{0}})$ for which
$c_{1,\mathrm{cris}}(\zeta)\in\mathcal{H}^{2}_{i,\mathfrak{p},\mathrm{cris}}|_{u^{\prime}_{0}}$.

Proof of claim. Note first that by (5.2.1)
$\mathbb{V}|_{\widetilde{\mathcal{E}}_{i,\mathbb{C}}}$, or equivalently,
$\mathbb{V}^{2}_{i}$ is a non-isotrivial $\mathbb{Q}$-VHS. Moreover, we argue
that
$h^{2,0}(\mathbb{V}^{2}_{i})=\mathrm{rank\,}\mathrm{Fil}^{2}\mathbb{V}^{2}_{i,\mathrm{dR}}=1$,
or equivalently, $h^{2,0}(\mathbb{V}^{2}_{i})=1$. Indeed, any global section
of $\mathbb{V}_{B}|_{\widetilde{\mathcal{E}}_{i,\mathbb{C}}}$ which is
everywhere of type $(1,1)$ must lie in $\mathbb{V}^{0}_{i}$. Since
$\mathbb{V}^{1}_{i}$ is assumed non-zero, $h^{2,0}(\mathbb{V}^{1}_{i})$ must
be nonzero, but since $\mathbb{V}^{2}_{2}$ is not isotrivial,
$h^{2,0}(\mathbb{V}^{1}_{i})$ cannot be $2$. Hence by [Moo17, Prop. 6.4] there
exists some $\mathbb{C}$-point $u^{\prime}_{\mathbb{C}}$ on
$\widetilde{\mathcal{E}}_{i}(\mathbb{C})$ such that
$\mathbb{V}^{2}_{i,u^{\prime}_{\mathbb{C}}}$ has a nonzero $(1,1)$-class,
which corresponds to a line bundle
$\zeta_{\mathbb{C}}\in\mathrm{Pic}(\mathcal{X}_{u^{\prime}_{\mathbb{C}}})$. Up
to increasing $N$, for any $\mathfrak{p}$ we may find some
$u^{\prime}_{0}\in\widetilde{\mathcal{E}}_{i,\mathfrak{p}}(\overline{k(\mathfrak{p})})$
lying in the Zariski closure of $(u_{\mathbb{C}}^{\prime})^{\flat}$ in
$\widetilde{\mathcal{E}}_{i}$ and take $s^{\prime}_{0}$ to be the image of
$u^{\prime}_{0}$. The desired line bundle $\zeta$ is constructed by
specializing $\zeta_{\mathbb{C}}$. ∎

Now suppose that we have already increased $N$ so that the claim holds and
$(s,\xi)$ is a LB-pair as in the statement of the theorem. If suffices to
treat the case when it is not strongly liftable. Let $\mathfrak{p}$ be the
image of $s$ on $\mathrm{Spec}(\mathcal{O}_{L}[1/N])$. By (5.2.3) and its
proof, there exists an $\mathcal{E}_{i}$ and a lift
$u\in\widetilde{\mathcal{E}}_{i,\mathfrak{p}}(k)$ such that
$c_{1,\mathrm{cris}}(\xi)$ lies in
$(\mathcal{H}^{0}_{i,\mathfrak{p},\mathrm{cris}}\oplus\mathcal{H}^{1}_{i,\mathfrak{p},\mathrm{cris}})|_{u}$
but not in $\mathcal{H}^{0}_{i,\mathfrak{p},\mathrm{cris}}|_{u}$. Note that
this in particular implies that $\mathbb{V}^{1}_{i}\neq 0$. Moreover,
$\mathscr{P}(\widetilde{\mathcal{E}}_{i,\mathfrak{p}},u,\xi)$ dominates
$\mathcal{E}_{i,\mathfrak{p}}$. Recall that in the construction of
$\widetilde{\mathcal{E}}_{i,\mathfrak{p}}$, we assumed that
$\mathrm{Mon}(\mathcal{H}_{i,\ell},u)$ is connected (this is subsumed into
(5.2.2)(III)). By (2.1.10)(d), there exists a relative line bundle
$\widetilde{\boldsymbol{\xi}}$ on
$\mathcal{X}|_{\widetilde{\mathcal{E}}_{i,\mathfrak{p}}}$ such that
$\widetilde{\boldsymbol{\xi}}_{u}\sim_{\mathbb{Q}}\xi$. Fix this
$\mathcal{E}_{i}$ and take $(s^{\prime}_{0},u^{\prime}_{0},\zeta)$ as above.
Let $s^{\prime}$ and $u^{\prime}$ be the base change of $s^{\prime}_{0}$ and
$u^{\prime}_{0}$ along some embedding
$\overline{k(\mathfrak{p})}\hookrightarrow k$ determined by $s$. Then
$\widetilde{\boldsymbol{\xi}}_{u^{\prime}}$ is a line bundle on
$\mathcal{X}_{u^{\prime}}=\mathcal{X}_{s^{\prime}}$. As
$c_{1,\mathrm{cris}}(\boldsymbol{\xi}_{u})\in\mathcal{H}^{2}_{i,\mathfrak{p},\mathrm{cris}}|_{u}$,
by [Ogu84, Thm 4.1] the relative crystalline Chern class of $\boldsymbol{\xi}$
must be a global section of the direct summand
$\mathcal{H}^{2}_{i,\mathfrak{p},\mathrm{cris}}$ in
$\mathcal{H}_{i,\mathfrak{p},\mathrm{cris}}$, and hence
$c_{1,\mathrm{cris}}(\boldsymbol{\xi}_{u^{\prime}})\in\mathcal{H}^{2}_{i,\mathfrak{p},\mathrm{cris}}|_{u^{\prime}}$
as well. By (5.2.3), $\zeta$ and
$\zeta-\widetilde{\boldsymbol{\xi}}_{u^{\prime}}$ are both line bundles on
$\mathcal{X}_{s^{\prime}}$ which are strongly liftable because their
$c_{1,\mathrm{cris}}$ in
$\mathcal{H}_{i,\mathfrak{p},\mathrm{cris}}|_{u^{\prime}}$ has a non-zero
component in $\mathcal{H}^{2}_{i,\mathfrak{p},\mathrm{cris}}|_{u^{\prime}}$.

To conclude the proof, it suffices to note that by [CP16, Cor. 1.9], we can
always find an irreducible $k$-curve
$C^{\prime}\subseteq\widetilde{\mathcal{E}}_{i,\mathfrak{p}}{\otimes}k$ which
contains both $u$ and $u^{\prime}$. Then we may simply take $C$ to be the
normalization of $C^{\prime}$, $\boldsymbol{\xi}$ to be the restriction of
$\widetilde{\boldsymbol{\xi}}$ to $C$, and $c,c^{\prime}$ to be any lifts of
$u,u^{\prime}$ respectively. ∎

##  6 Applications and Examples

We introduce the following notations: Let $B$ be a base scheme and
$\mathcal{V}$ be a vector bundle of rank $r$ over $B$. We denote by
$\mathbb{A}(\mathcal{V})$ the associated $\mathbb{A}^{r}$-bundle over $B$,
$\mathbb{A}(\mathcal{V})^{*}$ the open part $\mathbb{A}(\mathcal{V})$ minus
the zero section, and $\mathbb{P}(\mathcal{V})$ the projectivization of
$\mathcal{V}$.

### 6.1 Elliptic Surfaces

Let $\pi:X\to\mathbb{P}^{1}_{k}$ be an elliptic surface (with zero section)
over a field $k$. Define $L:=(R^{1}\pi_{*}\mathcal{O}_{X})^{\vee}$ to be the
fundamental line bundle and set $h:=\mathrm{deg}(L)$ to be the height of $L$.
Note that if $h=1$, then $X$ is rational, and if $h=2$, then $X$ is an
elliptic K3 surface. In general, we have that the inregularity $q=0$ and
$h=\mathcal{X}(\mathcal{O}_{X})=p_{g}-1$, where $p_{g}=h^{2,0}(X)$ is the
geometric genus. We say that $X$ is admissible if all fibers of $\pi$ are
irreducible.

In order to parametrize all $X$ with height $h$, we consider $\mathbb{P}^{1}$
over $\mathbb{Z}[1/6]$ and let $\varpi$ be the structural morphism
$\mathbb{P}^{1}\to\mathrm{Spec}(\mathbb{Z}[1/6])$. Set $\mathcal{V}_{i}$ to be
the vector bundle over $\mathbb{Z}[1/6]$ given by
$\varpi_{*}\mathcal{O}_{\mathbb{P}^{1}}(ih)$. Let the coordinates of
$\mathcal{P}:=\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})^{*}\times_{\mathbb{Z}[1/6]}\mathbb{P}(\mathcal{O}_{\mathbb{P}^{1}}(2h)\oplus\mathcal{O}_{\mathbb{P}^{1}}(3h)\oplus\mathcal{O}_{\mathbb{P}^{1}})$
be denoted by $((a_{4},a_{6}),[x:y:z])$ and set $\overline{\mathcal{X}}$ to be
the subscheme defined by (cf. [Kas77, Thm 1])

$y^{2}z=x^{3}-a_{4}xz^{2}-a_{6}z^{3}.$

Note that the morphism
$\mathcal{P}\to\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})^{*}$ carries a
natural action by $\mathrm{Aut}(\mathbb{P}^{1})$ and $\mathbb{G}_{m}$, the
latter is given by

$\lambda\cdot((a_{4},a_{6}),[x:y:z])\mapsto((\lambda^{4}a_{4},\lambda^{6}a_{6}),[\lambda^{2}x:\lambda^{3}y:z]).$

These actions commute and both stabilize $\mathcal{P}$. Let
$\mathcal{S}\subseteq\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})^{*}$ be
the open subscheme over which $\overline{\mathcal{X}}$ is smooth and
$\mathcal{X}$ be the restriction of $\overline{\mathcal{X}}$ to $\mathcal{S}$.
It is well known that a geometric point
$s\to\mathbb{A}(\mathcal{V}_{4}\oplus\mathcal{V}_{6})^{*}$ factors through
$\mathcal{S}$ if and only if the fiber $\mathcal{X}_{s}$ is an admissible
elliptic surface.

The moduli stack $\mathsf{M}$ over $\mathbb{Z}[1/6]$ of admissible elliptic
surfaces over $\mathbb{P}^{1}$ of height $h$ is given by the quotient of
$\mathcal{S}$ by the action of
$\mathbb{G}_{m}\times\mathrm{Aut}(\mathbb{P}^{1})$. One easily computes that
$\mathsf{M}$ has relative dimension $10h-2$ over $\mathbb{Z}[1/6]$. To avoid
working with stacks, we consider $\mathcal{S}$.

######  Lemma (6.1.1).

Assume that $h\geq 3$. Then for every $s\in\mathcal{S}(\mathbb{C})$, the
Kodaira-Spencer map
$T_{s}\mathcal{S}_{\mathbb{C}}\to\mathrm{Hom}(\mathrm{H}^{1}(\Omega_{\mathcal{X}_{s}}),\mathrm{H}^{2}(\mathcal{O}_{\mathcal{X}_{s}}))$
has rank $10h-2$.

###### Proof.

This is a consequence of [She21, Thm 2.10]. We just need to explain why the
hypothesis there is satisfied: Note that the Euler characteristic of a nodal
curve (resp. cuspidal) of $1$ (resp. $2$), and that of $\mathcal{X}_{s}$ is
$12\chi(\mathcal{O}_{X})=12h$. Therefore, the topological Hurwitz formula
tells us that there are at least $6h$ singular fibers. The line bundle $L$ in
loc. cit. is the same as our $L$, i.e., the fundamental line bundle. ∎

######  Theorem (6.1.2).

There exists an open dense subscheme $U\subseteq\mathcal{S}$ such that every
LB-pair $(s,\xi)$ with $s$ lying on $U$ is liftable.

###### Proof.

This follows from (1.1.1), (6.1.1) and the monodromy calculation [JF11, Thm
4.10]. ∎

######  Theorem (6.1.3).

For $p\gg 0$, the following is true: Let $k$ be an algebraically closed field
of characteristic $p$, $X$ be an admissible elliptic surface over
$\mathbb{P}^{1}$ of height $3$, and $\xi$ be any line bundle on $X$. Then
$\xi$ deforms to a sum of two liftable ones.

More precisely, there exists a smooth family $\mathcal{X}\to
C\times\mathbb{P}^{1}\to C$ whose geometric fibers over $C$ are height $3$
elliptic surfaces over $\mathbb{P}^{1}$ over a smooth connected $k$-curve $C$
such that:

  1. (i)

there exist $c\in C(k)$ and $\boldsymbol{\xi}\in\mathrm{Pic}(\mathcal{X})$,
and an isomorphism $\mathcal{X}_{c}\cong X$ through which
$\boldsymbol{\xi}_{c}\sim_{\mathbb{Q}}\xi$;

  2. (ii)

for some $c^{\prime}\in C(k)$,
$\boldsymbol{\xi}_{c^{\prime}}\sim_{\mathbb{Q}}\zeta_{1}+\zeta_{2}$ such that
$(\mathcal{X}_{c^{\prime}},\zeta_{i})$ for both $i=1,2$ are liftable.

###### Proof.

Apply (5.2.4) to the family $\mathcal{X}\to\mathcal{S}$ constructed above. ∎

### 6.2 Complete Intersection Surfaces

Fix some $r\in\mathbb{N}_{\geq 1}$ and a tuple
$(d_{1},\cdots,d_{r})\in\mathbb{N}_{\geq 1}^{\oplus r}$. Consider
$\mathbb{P}^{r+2}$ over $\mathbb{Z}$ and let
$\varpi:\mathbb{P}^{r+2}\to\mathrm{Spec}(\mathbb{Z})$ denote the structural
morphism. Let

$\overline{\mathcal{X}}\subseteq\mathbb{P}^{r+2}\times_{\mathbb{Z}}\prod_{i=1}^{r}\mathbb{P}(\varpi_{*}\mathcal{O}_{\mathbb{P}^{r+2}}(d_{i})),\hskip
20.00003pt\left(\textrm{resp.
}\mathcal{S}\subseteq\prod_{i=1}^{r}\mathbb{P}(\varpi_{*}\mathcal{O}_{\mathbb{P}^{r+2}}(d_{i}))\right)$

be the universal complete intersection of $r$ hypersurfaces of degree
$d_{1},\cdots,d_{r}$ (resp. the open subscheme over which
$\overline{\mathcal{X}}$ is smooth). Denote the restriction of
$\overline{\mathcal{X}}$ to $\mathcal{S}$ by $\mathcal{X}$.

######  Theorem (6.2.1).

Assume the following two numerical conditions:

  1. (i)

$\sum d_{i}>r+3$

  2. (ii)

for some (and hence every) $s\in\mathcal{S}(\mathbb{C})$,
$\dim\mathrm{H}^{1}(T_{\mathcal{X}_{s}})\geq h^{2,0}(\mathcal{X}_{s})$.

Then there exists an open dense $U\subseteq\mathcal{S}$ such that every LB-
pair $(s,\xi)$ with $s$ lying on $U$ is liftable.

###### Proof.

For every $s\in\mathcal{S}$, the canonical divisor $K_{\mathcal{X}_{s}}$ of
$\mathcal{X}_{s}$ is the restriction of the $(-r-3+\sum d_{i})$-multiple of
the hyperplane section on $\mathbb{P}^{r+2}_{k(s)}$, so condition $(i)$
ensures that $K_{\mathcal{X}_{s}}$ is ample. By [Pet76, Cor. 1.9, Thm 3.5] and
their proofs, one infers that the natural map
$T_{s}\mathcal{S}_{\mathbb{C}}\to\mathrm{H}^{1}(T_{\mathcal{X}_{s}})$ is
surjective for every $s\in\mathcal{S}(\mathbb{C})$. Then Thm 5.4 ibid further
tells us that the Kodaira-Spencer map
$\mathrm{H}^{1}(T_{\mathcal{X}_{s}})\to\mathrm{Hom}(\mathrm{H}^{1}(\Omega_{\mathcal{X}_{s}}),\mathrm{H}^{2}(\mathcal{O}_{\mathcal{X}_{s}}))$
is injective. Finally, in order to apply (1.1.1) to $\mathcal{S}$, it suffices
to check the large monodromy condinition on $\mathcal{S}_{\mathbb{C}}$. This
follows from [Bea86, Thm 5]. ∎

######  Remark (6.2.2).

The formulas for the numbers $h^{p,q}(\mathcal{X}_{s})$ and
$\dim\mathrm{H}^{1}(T_{\mathcal{X}_{s}})$ for some (and hence every)
$s\in\mathcal{S}(\mathbb{C})$, in terms of $d_{1},\cdots,d_{r}$ are completely
known [SGA7, éxposé XI, théorème 2.3], so (ii) is indeed a purely numerical
condition. In particular, when $r=1$, any $d_{1}\geq 5$ will satisfy the
hypothesis of (6.2.1). This gives (1.2.1).

### 6.3 Non-examples

We remark on the assumptions (a) $f_{\mathbb{C}}$ has large monodromy and (b)
$f_{\mathbb{C}}$ generically has sufficiently big period image in (1.1.1) and
(1.1.3). The upshot is that (a) should be relaxable whereas (b) is more rigid.

To begin, there is a quick counterexample to the conclusion when neither (a)
nor (b) holds: Suppose that $\mathcal{S}=\mathrm{Spec}(\mathcal{O}_{L}[1/N])$
and $\mathcal{X}$ is a K3 surface over $\mathcal{S}$. Then for infinitely many
primes $\mathfrak{p}$, the reduction $\mathcal{X}_{k(\mathfrak{p})}$ has
higher geometric Picard number than $\mathcal{X}_{L}$ ([Sha+22, Thm
1.1],[Tay22, Thm 1.1]). At such $\mathfrak{p}$, we can find an irreducible
component of $\mathscr{P}_{\mathcal{X}/S}$ which admits a
$\overline{k(\mathfrak{p})}$-point but not an $\overline{L}$-point, and hence
is not flat over $S$. Therefore, both (1.1.1) and (1.1.3) are false in this
case. On the other hand, for families with $h^{0,2}=1$, we can relax (a) to
“$\mathrm{Mon}(R^{2}f_{\mathbb{C}}\mathbb{Q},s)$ is positive dimensional for
some (and hence every) $s\in\mathcal{S}(\mathbb{C})$”, which is in fact
equivalent to (b). This is a key observation we used in [HYZ23] and interested
reader can take a look at the introduction.

The example below illustrates that even if (a) is satisfied, (b) cannot be
dropped:

######  Proposition (6.3.1).

Let $\mathcal{S}$ be as in (6.2.1) with $r=1$ and $d_{1}=d$ being an even
number $\geq 6$. Let $\mathcal{C}\subseteq\mathcal{S}$ be a closed subscheme
which is smooth and of relative dimension $1$ over an open part of
$\mathbb{Z}$, and such that $\mathcal{C}_{\mathbb{C}}$ contains a Hodge-
generic point for the VHS given by $\mathrm{H}^{2}$ of
$\mathcal{X}_{\mathbb{C}}$ over $\mathcal{S}_{\mathbb{C}}$.

Assume the Tate conjecture for hypersurfaces in $\mathbb{P}^{3}$ over finite
fields and the Zilber-Pink conjecture for families of hypersurfaces in
$\mathbb{P}^{3}$. Let $\mathcal{C}_{\mathrm{bad}}$ be the set of closed points
$s\in\mathcal{S}$ such that there exist some alg-geometric point $\bar{s}$
over $s$ and $\xi\in\mathrm{Pic}(\mathcal{X}_{\bar{s}})$ for which the pair
$(\bar{s},\xi)$ is non-liftable on $\mathcal{C}$. Then
$\mathcal{C}_{\mathrm{bad}}$ is Zariski dense on $\mathcal{C}$.

Note that if the Zariski closure of $\mathcal{C}_{\mathbb{C}}$ in
$\mathbb{P}(\varpi_{*}\mathcal{O}_{\mathbb{P}^{3}_{\mathbb{C}}}(d))$ is a
Lefschetz pencil, then condition (a) is satisfied.

###### Proof.

Let $s\in\mathcal{S}$ be any closed point and $\ell$ be a prime
$\neq\mathrm{char\,}k(s)$. The Weil conjectures imply that the Frobenius
eigenvalues on
$\mathrm{H}^{2}_{\textrm{{\'{e}}t}}(\mathcal{X}_{\bar{s}},\mathbb{Q}_{\ell}(1))$
which are not roots of unity come in pairs. As
$\mathrm{H}^{2}_{\textrm{{\'{e}}t}}(\mathcal{X}_{\bar{s}},\mathbb{Q}_{\ell}(1))$
is even dimensional by the assumption that $d$ is even, the Tate conjecture
implies that $\mathrm{rank\,}\mathrm{NS}(\mathcal{X}_{\bar{s}})$ is even.
Therefore, there exists a line bundle
$\xi\in\mathrm{Pic}(\mathcal{X}_{\bar{s}})$ which does not have the same class
as $\mathcal{O}_{\mathcal{X}_{\bar{s}}}(1)$.

Suppose now that the LB-pair $(\bar{s},\xi)$ lifts to characteristic $0$. Then
by choosing an isomorphism between $\mathbb{C}$ and the algebraic closure of
$W(\bar{s})[1/p]$, we obtain a $\mathbb{C}$-point on the NL locus. Because
$\mathcal{C}$ is Hodge-generic, the expected codimension of a NL locus in
$\mathcal{C}_{\mathbb{C}}$ is $h^{0,2}>1$. By the Zilber-Pink conjecture
formulated in [BKU22, Conj. 2.5] this implies the NL locus of $\mathcal{C}$ is
a finite set of points, which implies that $s$ lies in the Zariski closure of
finitely many $\mathbb{C}$-points on $\mathcal{C}_{\mathbb{C}}$. ∎

##  Appendix A A Lemma on Flat Frames

######  Lemma (A.0.1).

Let $k$ be a field and fix some $r\in\mathbb{N}$. Let
$A_{\infty}=k[\\![x_{1},\cdots,x_{n}]\\!]$ for some $n$. Set
$\mathbb{D}:=\mathrm{Spf}(A_{\infty})$, $\mathfrak{m}:=(x_{1},\cdots,x_{n})$,
$I:=(x_{1}^{r+1},\cdots,x_{i}^{r+1})$ and $A:=A_{\infty}/I$. Let
$(\hat{M},\hat{\nabla})$ be any flat vector bundle over $\mathbb{D}$ and
$(M,\nabla)$ be its restriction to $\mathrm{Spec}(A)$. Suppose that $r!$ is
invertible in $k$.

Then for every $m_{0}\in M/\mathfrak{m}M$, there exists a unique $m\in
M^{\nabla=0}$ such that $m_{0}=m\mod\mathfrak{m}M$. In particular,
$(M,\nabla)$ is trivial as a flat vector bundle.

###### Proof.

First, we note that $\hat{\nabla}_{\partial_{i}}:\hat{M}\to\hat{M}$ has a
standard meaning, where $\\{\partial_{1},\cdots,\partial_{n}\\}$ are dual to
the generators $dx_{1},\cdots,dx_{n}$ for $\Omega_{\mathbb{D}/k}$. Let us
simply write $\hat{\nabla}_{i}$ for $\hat{\nabla}_{\partial_{i}}$. The
integrability of $\hat{\nabla}$ implies that the operators
$\hat{\nabla}_{i}$’s commute. Now we make sense of $\nabla_{i}$. For each $i$,
set $I^{i}:=(x_{1}^{r+1},\cdots,x_{i}^{r},\cdots,x_{n}^{r+1})$ and
$A^{i}:=A/(x_{i}^{r})=A_{\infty}/I^{i}$. By applying the conormal sequence to
the surjection $k[x_{1},\cdots,x_{n}]\to A$, one checks that

$\Omega_{A/k}=\bigoplus_{i=1}^{n}A^{i}dx_{i}\simeq\bigoplus_{i=1}^{n}A^{i}$

Then we define $\nabla_{i}$ to be the composition
$M\stackrel{{\scriptstyle\nabla}}{{\to}}M{\otimes}_{A}\Omega_{A/k}\to
M{\otimes}_{A}A^{i}$, where the second map is the projection to the $i$th
factor. As the restriction of $\hat{\nabla}$, $\nabla$ is characterized by the
property that for any $\hat{m}\in\hat{M}$ with image $m\in M$, $\nabla(m)$ is
the image of $\hat{\nabla}(\hat{m})$ under the composition
$\hat{M}{\otimes}_{A_{\infty}}\Omega_{\mathbb{D}/k}\to
M{\otimes}_{A}(\Omega_{\mathbb{D}/k}{\otimes}_{A_{\infty}}A)\to
M{\otimes}_{A}\Omega_{A/k}$, so that the following diagram commutes:

${{\hat{M}}}$${{\hat{M}}}$${M}$${M{\otimes}_{A}A^{i}}$$\scriptstyle{\hat{\nabla}_{i}}$$\scriptstyle{\nabla_{i}}$
(13)

Define an additive map $\hat{P}:\hat{M}\to\hat{M}$ by the formula (cf. [Kat70,
(5.1.2)]):

$\hat{P}(\hat{m})=\sum_{\alpha}\frac{(-1)^{|\alpha|}}{\alpha!}x^{\alpha}\hat{\nabla}^{\alpha}\text{
for all }\hat{m}\in\hat{M}$ (14)

where $\alpha$ runs through tuples
$(\alpha_{1},\cdots,\alpha_{n})\in(\mathbb{N}_{\leq r})^{n}$,
$|\alpha|=\sum_{i}\alpha_{i}$, $\alpha!=\alpha_{1}!\cdots\alpha_{n}!$,
$x^{\alpha}=x_{1}^{\alpha_{1}}\cdots x_{n}^{\alpha_{n}}$ and
$\hat{\nabla}^{\alpha}=\hat{\nabla}^{\alpha_{1}}_{1}\cdots\hat{\nabla}^{\alpha_{n}}_{n}$.
We define an additive map $P:M\to M$ by the setting for each $m\in M$ and
$\hat{m}\in\hat{M}$ lifting $m$, $P(m)=\hat{P}(\hat{m})\mod IM$. This is well
defined because for each $i$, $\hat{P}(x_{i}^{r+1}\hat{m})$ is divisible by
$x_{i}^{r+1}\in I$. We claim that $P$ is a projection onto $M^{\nabla=0}$. By
the diagram (13), we reduce to showing the following two statements for
$\hat{m}\in\hat{M}$: (a) For every $i$, $\hat{\nabla}_{i}P(\hat{m})\in
I^{i}\hat{M}$. (b) If for every $i$, $\hat{\nabla}_{i}(\hat{m})\in
I^{i}\hat{M}$, then $\hat{P}(\hat{m})=\hat{m}\mod IM$. Both are readily
checked by a direct computation.

Finally, for any $m_{0}\in M/\mathfrak{m}M$, to construct $m$ as desired we
may start with any $m\in M$ with mod $\mathfrak{m}M$ class $m_{0}$ and replace
$m$ by $P(m)$—it is clear by (14) this does not change mod $\mathfrak{m}$
class. Now we already know that $(M,\nabla)$ is isomorphic to
$\mathrm{rank\,}M$ copies of $(A,d)$. To verify the uniqueness of $m$ if
suffices to note that if $df=0$ and $f(0)=0$ for $f\in A$, then $f=0$. This is
clear when $\mathrm{char\,}k=0$, and one uses the assumption $r<p$ when
$\mathrm{char\,}k=p>0$. ∎

######  Remark (A.0.2).

When one relates the above to (3.2.1), note that
$(x_{1}^{r+1},\cdots,x_{i}^{r+1})\subseteq(x_{1},\cdots,x_{i})^{r+1}$. Also,
recall that Cohen structure theorem [Sta22, 0C0S] tells us that Noetherian
complete local $k$-algebra with residue field $k$ of dimension $n$ is
isomorphic to $k[\\![x_{1},\cdots,x_{n}]\\!]$.

When $\mathrm{char\,}{k}=p>0$, the $k$-algebra $A$ as above is a PD-thickening
of $\mathrm{Spec}(k)$, so it is also possible to prove the lemma directly by
crystalline formalism (e.g., some appropriate variant of [Sta22, 07JH]).

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