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Hey ya'll. Its time for the next chapter. Its kind of short, but I am putting up chapters as much as I can.
"So have you got the list all ready?" Caroline nodded. She was a little disappointed at how long the list was. Liane had invited 14 girls from their grade at their school as well as 2 from the upper class. Her mother had also forced her to invite some family friends' daughters, 3 of them. "Are you sure we can do a party with 19 other girls?" Liane frowned at her. "My party is going to be huge. You of all people should understand the need. Besides its not my fault my mom made me invite 3 girls just because we know their families pretty well." Caroline stared at her knees. Liane began to talk again. "Anyways, there's no need for you to worry. I have everything ready. You will still get to put your sleeping bag next to mine. Don't ask who the replacement will be if you don't show up. How can my best friend miss one of my most important parties ever?" Caroline felt her stomach doing flip flops. How could she tell her?


Lapel was able to capture the Terre Haute North Invitational in dramatic fashion, pulling out a winner take all match at 1 doubles against Terre Haute North. The 1 doubles team of Zach Krebs and Reid Ratzlaff was down 3-6,2-5 and was on the brink of defeat when they turned things around and eventually pulled out the win in a third set tiebreaker! Joseph Conrad and Jesse McCurdy lead the Bulldogs on the day, both going 3-0.
The Bulldogs are now 11-1 on the year and are back in action tonight as they host the Muncie Central Bearcats. This is the first of five straight home matches. Come out watch these boys play!


text:
Maryvale to face Roland Park in Soccer Championship
Following a commanding win in the quarterfinals, Maryvale squared off against No. 3 Severn in the semifinals on Thursday, Nov. 1 at home. The game was tied at 1-1 for the majority of the contest, but, with around 10 minutes left in regulation, Severn scored to take the lead 2-1. Then, the Lions answered back with around three minutes left to tie the game again at 2-2 and send it into overtime. After playing for only a few minutes in overtime, Maryvale's Audrey Allan '22 launched a free kick over the keeper's head into the back of the net, securing the Lion's spot in the IAAM B Conference Championship Game. The No. 2 Maryvale Lions will take on the No. 1 Roland Park Reds on Saturday, Nov. 3 at 5 p.m. at Anne Arundel Community College (101 College Pkwy, Arnold, MD 21012). Admission is $8 per person. Join us this weekend as we help cheer on our Lions to victory!
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redpajama_set_name: RedPajamaCommonCrawl


Contact this reporter at mat.honan@buzzfeed.com.
Google is holding its annual developer's conference in Mountain View, California on Wednesday. The company is expected to talk about its new Android O mobile operating system, its efforts in artificial intelligence, a new version of its Assistant for Apple's iOS, and more.
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BuzzFeed News sat down for a wide-ranging interview with the CEO of Instagram for its sixth anniversary and looked back on the slew of changes the company has made to its product in recent months.
BuzzFeed News talked to Google's vice president of virtual reality, Clay Bavor, to find out all about the company's new Daydream virtual reality platform and the Daydream View headset.
Today, BlackBerry announced that it will no longer make hardware. Here's the definitive history of the once-dominant smartphone's downfall.
"The Republican Party, and Donald Trump in particular, is running on a zero-sum vision."
The new feature will let people post pictures and videos that disappear after 24 hours, and never appear in the main Instagram feed.
The Facebook-owned company and Brazil's judiciary have been locked in an epic struggle for months now.
BuzzFeed built a news-gathering bot for the convention. If you're in Cleveland — or watching at home — try it out! Here's how to get it. Bleep, bloop.
The company hints at plans for improved photo storage, organization, and sharing.
The tech billionaire says he only recently learned of Thiel's role in financing a series of lawsuits against Gawker Media.
It's more like a public Slack room.
Facebook threw the company behind a massive push to add a slew of new features to its Live video as it takes on Snapchat, Periscope, and traditional television itself. Get ready to broadcast.


Apache Powder Co.The Apache Powder Co. Superfund site is located in Cochise County, Arizona. It covers about 9 square miles, including 1,200 acres of land owned by Apache Nitrogen Products (ANPI), formerly the Apache Powder Company. ANPI began making industrial chemicals and explosives on site in 1922. ANPI now makes chemical products used by agricultural and mining customers. ANPI previously disposed of solid and liquid wastes on its property. In the late 1980s, metals and explosive wastes were detected in some on-site soils and unlined evaporation ponds. Nitrate and perchlorate were also detected in groundwater on and off site. EPA placed the site on the National Priorities List (NPL) in 1990. In 1997, ANPI constructed a wetlands treatment system to biodegrade nitrate on the northern part of the site. EPA's cleanup plan also included off-site disposal and on-site capping of contaminated soils. The constructed wetlands along the San Pedro River serve as habitat for area wildlife. Following a renewable energy feasibility analysis by EPA in 2008, ANPI installed a 41-kilowatt solar canopy on site in 2010. The canopy offsets about 45 percent of ANPI's annual office energy use, providing $6,000 in cost savings each year. ANPI also uses solar energy to power ongoing groundwater cleanup and monitoring efforts.
As of December 2018, EPA had data on 3 on-site businesses. These businesses employed 77 people and generated an estimated $96,407,000 in annual sales revenue. For additional information click here.
The Indian Bend Wash Area Superfund site is located in Scottsdale and Tempe, Arizona. EPA placed the site on the National Priorities List (NPL) in December 1982, after discovery of contaminants in groundwater at several public water supply wells. In 1987, EPA divided the site into two areas. The first area – North Indian Bend Wash (NIBW) – is a 10-mile long groundwater plume north of the Salt River in Scottsdale and the South Indian Bend Wash (SIBW) an approximately 3-square-mile groundwater plume south of the Salt River in Tempe. Land use in both the NIBW and SIBW area is a mix of residential, industrial, commercial and recreational uses and waterways. Cleanup in the NIBW area addressed soil contamination with soil vapor extraction. Five groundwater pump-and-treat systems continue to operate as part of ongoing cleanup efforts, which are estimated to take 70 years to meet cleanup goals. At the SIBW area, cleanup addressed soil contamination with soil vapor extraction. EPA approved monitoring of the natural breakdown of groundwater contamination at the three distinct groundwater plumes that constitute the site. A pilot test of in situ chemical oxidation has assisted in attaining remedial goals for monitored natural attenuation at SIBW. EPA is reviewing contaminant source areas for potential vapor intrusion risk. Vapor intrusion exposure was not a known concern when the remedies for NIBW and SIBW were selected in the early 1990s. Continued use at the site includes residential, commercial and industrial uses.
As of December 2018, EPA had data on 84 on-site businesses. These businesses employed 1,698 people and generated an estimated $329,003,950 in annual sales revenue. For additional information click here.
Motorola, Inc. (52nd Street Plant)The Motorola, Inc. (52nd Street Plant) Superfund site is located in Phoenix, Arizona. EPA defines the site generally by the extent of impacted groundwater that underlies a 7-mile stretch spanning from downtown Phoenix to just east of Sky Harbor Airport. Land uses in the area include a mix of residential, commercial and industrial uses. The former Motorola 52nd Street Plant, a 90-acre former semiconductor manufacturing plant, is part of the site. The Honeywell International facility, a 118-acre aerospace manufacturing plant, is also part of the site. In 1982, Motorola discovered an underground storage tank leaking at its plant. Further studies identified soil and groundwater contaminated with several solvents used by Motorola when it made semiconductors. EPA added the site to the National Priorities List (NPL) in 1989. Motorola (now Freescale Semiconductor) began operating one groundwater treatment system in 1992. Motorola and Honeywell International began operating another groundwater treatment system in 2001. Motorola also treated contaminated soil in 1996 and 1997. Since 2011, EPA and Freescale have studied potential movement of groundwater vapors into homes, acted to prevent vapors from entering 19 residences, and completed protective action at a commercial building in 2017. EPA, the State and the site's potentially responsible parties are also evaluating other potential sources of contamination. All land uses in the area will be able to remain in continued use during cleanup.
As of December 2018, EPA had data on 17 on-site businesses. These businesses employed 1,134 people and generated an estimated $267,892,599 in annual sales revenue. For additional information click here.
Phoenix-Goodyear Airport AreaThe Phoenix-Goodyear Airport (PGA) Area Superfund site is located in Maricopa County, Arizona about 17 miles west of downtown Phoenix. The site includes two areas – PGA-North and PGA-South. PGA-North is the location of the former Unidynamics facility. This part of the site includes three unused parcels owned by the Crane Company. The City of Goodyear zoned the parcels for commercial and industrial use. PGA-South includes the Phoenix-Goodyear Airport and adjacent commercial and industrial properties, which are owned by JRC Goodyear. PGA-South is the site of the former Litchfield Naval Air Facility. In 1981, the state identified contaminated groundwater and soil near the airport. Maintaining and operating military aircraft at the former Naval Air Facility and researching and manufacturing military detonators and explosives at the Unidynamics facility led to contamination. In 1983, EPA placed the site on the National Priorities List (NPL). Cleanup includes treating groundwater, extracting soil vapor, and removing and capping contaminated soil. Cleanup is ongoing. A manufacturer of prefabricated homes is located next to the airport. More recent airport tenants include two airline flight schools and an aircraft maintenance company. The Phoenix-Goodyear Airport contributes an estimated $200 million a year to the local economy.
As of December 2018, EPA had data on 34 on-site businesses. These businesses employed 949 people and generated an estimated $112,404,117 in annual sales revenue. For additional information click here.
Tucson International Airport AreaThe 10-square-mile Tucson International Airport Area Superfund site is located in Tucson, Arizona. The site includes Tucson International Airport, portions of the Tohono O'odham Indian Reservation, residential areas in Tucson and South Tucson, and the Air Force Plant #44 Raytheon Missile Systems Company (AFP44). At least 20 facilities have operated on site since 1942, including aircraft and electronics facilities, a fire-drill training facility, and unlined landfills. Industrial use and disposal of wastes began in the 1940s. Because of widespread groundwater and soil contamination, EPA placed the site on the Superfund program's National Priorities List (NPL) in 1983. Cleanup includes ongoing groundwater treatment at AFP44 and the airport. Soil treatment is also ongoing. Treated water is returned to the municipal water supply for use as drinking water. Tucson International Airport remains operational. Today, ongoing industrial activities at the site include machining, surface preparation, surface coating, metal plating and missile parts assembly. At least 124 businesses operate on site, generating over $458 million in annual sales. The businesses employ nearly 4,200 people. The Federal Aviation Administration awarded the Tucson Airport Authority a $5.7 million grant for a solar panel installation covering the entire parking area. The solar panels generate about two million kilowatt-hours of energy per year. The design also includes "green walls" vegetated with plants that provide cool climate conditions under the solar panel installation.
As of December 2018, EPA had data on 92 on-site businesses. These businesses employed 4,148 people and generated an estimated $998,939,104 in annual sales revenue. For additional information click here.


Spring is in the Air ...Well, maybe if you don't live in the Midwest ~ it's still FREEZING here! Anyhoo, check out these AWESOME porch ideas, you might decide you need to try out something new on our own porch!
In the South, deep porches with lofty ceilings provide cooling shade to the home. With their screened enclosures, they also serve as outdoor rooms — like this one in Charleston, S.C., fitted with cushioned rockers and even a coffee table with a potted orchid. The ceiling is painted in traditional Haint Blue, a shade thought to repel wasps (or, for the superstitious, "haints"—ghosts or lost spirits).
Looking for days when we had slower lifestyles? Check out this porch that looks like it could harken back to the days before cell phones and internet! It has all the right pieces ~ an Original Charleston Bed Swing lined with comfy pillows, retro 1950s metal patio chairs, a chevron-patterned indoor-outdoor rug and lime-green garden seats. Take me back!!!
In this cottage, the old-style shutters can be opened and closed for privacy and shelter from the weather. Imagine settling down on the cushy porch swing for an afternoon nap with the ceiling fan to keep you cool. Heaven!
This roomy front porch in Atlanta features a rustic cedar ceiling enhanced with a semitransparent Cabot stain, and painted brick arches that match the creamy exterior of the house. Softly lit by wall sconces and table lamps, it looks to be the ideal spot for after-dinner conversation.
Designer Eddie Rider used clay pots and red geraniums on this porch to contrast with the sage-green wood siding. In case anyone gets too comfortable in the wicker rockers, a big wall clock keeps track of the time. The flooring is ipê wood, which turns a soft weathered gray as it ages.
This beautiful porch in Charleston was constructed from the same bricks that are on the walls of the home (Boral Shadow Stone with Savannah ivory mortar). The traditional beadboard ceiling is covered in Duron Paint's Piazza Blue.
With a lakeside setting like this one in Upstate New York, it's not surprising that a knockout front porch was part of the plan for this new home. It has all the essential elements: boxed columns, a bluestone floor, a wood ceiling and wicker seating with lots of plush cushions.


text:
\section{Introduction}

A classical question in complex algebraic geometry asks which smooth manifolds carry a complex projective or a K\"ahler structure.
Once we know that at least one such structure exists, the next natural question is to ask how many there are.
In this paper we address both questions in the case where the underlying smooth manifold is a spin $6$-manifold.
The condition for a $6$-manifold $M$ to be spin depends only on its homotopy type; if $M$ carries a complex structure $X$, then $M$ is spin if and only if the mod $2$ reduction of $c_1(X)$ vanishes.

The spin condition, though of a purely topological nature, turns out to put strong restrictions on the bimeromorphic geometry of K\"ahler manifolds.
This makes that class of manifolds particularly accessible to the methods of the minimal model program, established for K\"ahler threefolds only very recently by H\"oring and Peternell  \cite{HP15,HP16}.
We will see that this leads to surprisingly strong topological constraints on this class of K\"ahler manifolds; our main results are Theorems \ref{thm:b_3=0}, \ref{thm:deformations} and \ref{thm:w2=0} below.

\subsection{Many spin $6$-manifolds are non-K\"ahler but have K\"ahler homotopy type}
Any closed spin $6$-manifold carries an almost complex structure, see Section \ref{subsubsec:almostcxstr}.
Moreover, a conjecture of Yau predicts that actually any closed spin $6$-manifold admits a complex structure, see \cite[p.\ 6]{kotschick-survey} and \cite[Problem 52]{yau-problem}.
On the other hand, our first main result produces many examples that do not carry any K\"ahler structure.
The interesting feature of our result is that it holds for a large class of prescribed homotopy types.

\begin{theorem} \label{thm:b_3=0}
Let $X$ be a simply connected K\"ahler threefold with spin structure and $H^3(X,\mathbb Z)=0$.
Then there are infinitely many pairwise non-homeomorphic closed spin $6$-manifolds $M_i$ with the same oriented homotopy type as $X$, but which
are not homeomorphic to a K\"ahler manifold.
\end{theorem}

To the best knowledge of the authors, the above theorem produces the first examples of manifolds with $b_2>1$ that are homotopy equivalent but not homeomorphic to K\"ahler manifolds.
This is interesting because most known topological constraints implied by the K\"ahler condition (e.g.\ formality \cite{deligne-etal}, restrictions on the fundamental group \cite{amoros-etal}, or constraints on the cohomology algebra that come from Hodge theory)
are only sensitive to the homotopy type, but not to the homeomorphism type.

Simple examples to which Theorem \ref{thm:b_3=0} applies are the blow-up of $\mathbb P^3$ in finitely many points, the product of a K3 surface with $\mathbb P^1$,
or, more generally, the $\mathbb P^1$-bundle $\mathbb P(K_S\oplus \mathcal O_S)$ over any simply connected K\"ahler surface $S$;
the surface $S$ may very well be non-spin.

Previously, only rather specific examples of non-K\"ahler manifolds with K\"ahler homotopy type had been found.
For example, Libgober and Wood \cite{LW90} proved that the homotopy type of $\mathbb P^n$ with $n\leq 6$ determines uniquely the K\"ahler structure.
This proves Theorem \ref{thm:b_3=0} in the special case $X=\mathbb P^3$, because it is well-known that there are infinitely many pairwise non-homeomorphic $6$-manifolds that are homotopy equivalent to $\mathbb P^3$.
A related result of Hirzebruch--Kodaira \cite{HK57} and Yau \cite{yau} asserts that for any $n$, the homeomorphism type of $\mathbb P^n$ determines the K\"ahler structure uniquely.

\subsection{Almost no spin $6$-manifold is a complex projective variety of general type} \label{subsec:bounded}
In analogy with the classification of curves or surfaces, one expects that most K\"ahler structures are of general type.
Any K\"ahler structure of general type is projective and it is natural to ask which smooth manifolds carry such a structure.

Our next result is the following strong finiteness statement.

\begin{theorem}\label{thm:deformations}
There are only finitely many deformation types of smooth complex projective spin threefolds of general type and with bounded Betti numbers.
\end{theorem}

The above theorem shows that the deformation type and hence the topological type of a smooth complex projective spin threefold $X$ of general type is determined up to finite ambiguity by its Betti numbers.
In particular, once we fix the Betti numbers, only finitely many fundamental groups $\pi_1(X)$, and only finitely many ring structures and torsion subgroups of $H^{\ast}(X,\mathbb Z)$ occur.
In fact, Theorem \ref{thm:deformations} implies the following.

\begin{corollary} \label{cor:deformations}
Only finitely many closed spin $6$-manifolds with bounded Betti numbers carry
a K\"ahler structure of general type.
\end{corollary}

Corollary \ref{cor:deformations} has the surprising consequence that almost no closed spin $6$-manifold is a complex projective variety of general type.
To see this, we start with the case of simply connected spin $6$-manifolds $M$, which are completely classified by the work of Wall, see Section \ref{subsubsec:classification}.
It turns out that for given Betti numbers with $b_2>0$, the ring structure on $H^*(M,\mathbb Z)$ may vary among infinitely many different isomorphism types and almost none of these manifolds can be a variety of general type.
In fact, even the corresponding homotopy types cannot be realized by a variety of general type; non-simply connected examples can be produced similarly by taking connected sums with simply connected ones.
Going further, it is also possible to produce many new examples of manifolds that are homotopy equivalent but not homeomorphic to a variety of general type, see Corollary \ref{cor:6mfd-not-gentyp} below.

Although it is well--known that Theorem \ref{thm:deformations} holds in dimension two, this cannot be exploited in a similar way.
The problem being that the oriented homeomorphism type of a simply connected smooth $4$-manifold $M$ is determined by the cup product pairing on $H^2(M,\mathbb Z)$ by Freedman's work, and only very few (in particular finitely many) isomorphism types occur by Donaldson's theorem.

Our arguments are new and do not seem to apply to other natural classes of K\"ahler manifolds.
Indeed, it is easy to see that Theorem \ref{thm:deformations} and Corollary \ref{cor:deformations} fail in higher dimensions.
Moreover, both statements are sharp in the sense that they fail if one drops either the spin assumption, or the general type assumption, see Example \ref{ex:deformations:gentype} and Proposition \ref{prop:dolgachev} below.
Nonetheless, Theorem \ref{thm:b_3=0} and \ref{thm:deformations} lead to the expectation that in some sense, most spin $6$-manifolds should not carry any K\"ahler structure at all.

Finiteness results like Theorem \ref{thm:deformations} seem to be quite rare; in fact, apart from the aforementioned case of surfaces, we are aware of only one more result in this direction.
That result is due to Koll\'ar, who showed \cite[Theorem 4.2.3]{kollar} that all K\"ahler manifolds with $b_2=1$, given cohomology ring and Pontryagin classes form a bounded family.
Minimal ruled surfaces show that this statement fails already for $b_2=2$.
Moreover, for applications like Theorem \ref{thm:b_3=0} or Corollary \ref{cor:deformations}, it is crucial not to fix the Pontryagin classes.

\subsection{K\"ahler structures on spin $6$-manifolds have bounded Chern numbers}
Let us now assume that a given spin $6$-manifold $M$ carries a K\"ahler structure and we ask how many there are.
More precisely, it is natural to ask which (homotopy classes of) almost complex structures can be realized by a K\"ahler structure.
Since $M$ is spin, the first Chern class induces a bijection between the homotopy classes of almost complex structures on $M$ and the subgroup $H^2(M,2\mathbb Z)$ of classes in $H^2(M,\mathbb Z)$ whose mod $2$ reduction vanishes, see Proposition \ref{prop:almostcx}.
It follows from Theorem \ref{thm:deformations} that only finitely many such classes come from K\"ahler structures of general type.
This fails without the general type assumption: there are spin $6$-manifolds that admit infinitely many K\"ahler structures with unbounded first Chern class, see Proposition \ref{prop:dolgachev}.
Nonetheless, our next result shows that the Chern numbers are always bounded.

\begin{theorem} \label{thm:w2=0}
Let $X$ be a smooth compact K\"ahler threefold with spin structure.
Then the Chern numbers of $X$ are determined up to finite ambiguity by the isomorphism type of the cohomology ring $H^{\ast}(X,\mathbb Z)$ and the first Pontryagin class $p_1(X)$.
\end{theorem}

The following corollary of the above theorem solves a problem of Kotschick in the case of spin structures \cite[Problem 1]{kotschick-JTOP}.

\begin{corollary} \label{cor:Spin}
The Chern numbers take on only finitely many values on the K\"ahler structures with the same underlying closed spin $6$-manifold.
\end{corollary}

While the boundedness of $c_3$ and $c_1c_2$ on K\"ahler structures with the same underlying smooth $6$-manifold is well-known, the significance of the above results concerns the boundedness of $c_1^3$.
For non-uniruled K\"ahler threefolds, our bounds for $c_1^3$ are effective and depend only on the Betti numbers, see Corollary \ref{cor:ineq}.

The original motivation for Kotschick's question goes back to a problem of Hirzebruch, asking which linear combinations of Chern and Hodge numbers of a smooth complex projective variety are determined by the underlying smooth manifold \cite{kotschick-JTOP,kotschick-PNAS,kotschick-advances,kotschick-schreieder}.

Interestingly, Theorem \ref{thm:w2=0} and Corollary \ref{cor:Spin} fail if one drops the K\"ahler assumption  \cite{lebrun}. %
Moreover, none of the statements generalizes to higher dimensions.
Indeed, for any $n\geq 4$ there are smooth $2n$-manifolds with infinitely many K\"ahler structures whose Chern numbers are unbounded, see \cite{schreieder-tasin}.
The method of that paper can be adapted to produce similar examples with spin structures; more precisely, \cite[Theorem 1]{schreieder-tasin} remains true for spin manifolds.

\begin{remark} \label{rem:diffeotypes}
Any topological $6$-manifold admits at most one equivalence class of smooth structures; the existence of such a structure is detected by the Kirby--Siebenmann invariant.
In particular, Corollary \ref{cor:Spin} remains true if one replaces smooth spin $6$-manifolds by topological ones.
\end{remark}

\subsection{Conventions}
All manifolds are smooth, closed and connected.
A K\"ahler manifold is a complex manifold which admits a K\"ahler metric.
All schemes are separated; a variety is an integral scheme of finite type over $\mathbb C$.
A family of varieties is a proper flat morphism $\pi:\mathcal X\longrightarrow B$ between finite type schemes over $\mathbb C$ whose fibres over closed points are varieties; in this paper, the base $B$ will always be assumed to be reduced.
For any (K\"ahler) manifold $X$, we denote by $H^\ast_{tf}(X,\mathbb Z)$ the quotient $H^\ast(X,\mathbb Z)/H^\ast(X,\mathbb Z)_{tors}$, where $ H^\ast(X,\mathbb Z)_{tors}$ is the torsion subgroup of $H^\ast(X,\mathbb Z)$.

\section{Preliminaries} \label{sec:preliminaries}

\subsection{Spin $6$-manifolds}
A smooth orientable $6$-manifold $M$ is spin if and only if the second Stiefel--Whitney class $w_2(M)\in H^2(M,\mathbb Z\slash 2\mathbb Z)$ vanishes.
It follows from the Wu formula that this condition depends only on the homotopy type of $M$.
Moreover, if $X$ is a complex structure on $M$, then the mod $2$ reduction of $c_1(X)$ coincides with $w_2(M)$ and so it vanishes if and only if $M$ is spin.

For example, by the adjunction formula, a smooth divisor $X$ on a compact complex fourfold $Y$ is spin if it is linearly equivalent to $K_Y+2L$ for some $L\in \operatorname{Pic}(Y)$.
In particular, any smooth complex projective fourfold contains (many) smooth hypersurfaces $X$ with spin structure.
The smooth $6$-manifold which underlies $X$, or the blow-up of $X$ in a finite number of points, is spin and so the results of this paper apply.

The following definition generalises the notion of spin threefold; note however that this generalization is not a topological notion anymore.

\begin{definition}\label{def:divisible}
Let $X$ be a smooth complex projective threefold.
We say that $c_1(X)$ is \emph{numerically divisible by $m \in \mathbb N$} if its class in $H^2_{tf}(X,\mathbb Z)$ is divisible by $m$.
Moreover, $c_1(X)$ is \emph{numerically divisible} if it is divisible by some natural number $m\geq 2$.
\end{definition}

\subsubsection{Almost complex structures.} \label{subsubsec:almostcxstr}
The obstruction for a closed oriented $6$-manifold $M$ to carry an almost complex structure is given by the image of the second Stiefel--Whitney class via the Bockstein homomorphism $H^2(M,\mathbb Z\slash 2\mathbb Z)\longrightarrow H^3(M,\mathbb Z)$ and hence it vanishes if $M$ is spin.
In fact, we have the following,
see proof of \cite[Proposition 8]{OV95}.

\begin{proposition} \label{prop:almostcx}
Let $M$ be a closed spin $6$-manifold.
Then the first Chern class induces a bijection between the homotopy classes of almost complex structures on $M$ and the subgroup $H^2(M,2\mathbb Z)\subset H^2(M,\mathbb Z)$ of classes whose mod $2$ reduction vanishes.
\end{proposition}

\subsubsection{Classification of simply connected spin $6$-manifolds} \label{subsubsec:classification}
Wall \cite{wall} showed that simply connected closed spin $6$-manifolds $M$ with torsion free cohomology are classified by the following tuple  of algebraic invariants
$$
(b_3(M),H^2(M,\mathbb Z),F_M,p_1(M)) ,
$$
where $F_M$ denotes the cubic form on $H^2(M,\mathbb Z)$, given by cup product, and $p_1(M)$ is the linear form on $H^2(M,\mathbb Z)$, given by cup product with the first Pontryagin class, see also \cite[Section 1]{OV95}.
Two such manifolds $M$ and $M'$ are orientation-preservingly homeomorphic (or diffeomorphic, see Remark \ref{rem:diffeotypes} above) if and only if there is an isomorphism between $H^2(M,\mathbb Z)$ and $H^2(M',\mathbb Z)$ which respects the above tuple of algebraic invariants.
A given tuple
$
(b_3,H,F,p_1)
$
can be realized by a simply-connected smooth spin $6$-manifold with torsion-free cohomology if and only if $b_3$ is even and
\begin{align} \label{eq:admissible}
4W^3\equiv p_1(M)W \mod 24
\end{align}
for all $W\in H$; such tuples are called admissible.

\subsection{Chern numbers of smooth threefolds}
Let $X$ be a smooth compact complex threefold.
Since $c_3(X)=\sum_i(-1)^ib_i(X)$ coincides with the topological Euler number, it is a topological invariant of $X$.
Moreover, by Riemann--Roch, $c_1c_2(X)=24\chi(X,\mathcal O_X)$ is bounded by the Hodge numbers, hence by the Betti numbers if $X$ is K\"ahler.
In particular, in order to prove Theorem \ref{thm:w2=0}, we only need to prove the boundedness of $c_1^3$.
This is known if $K_X$ is nef.

\begin{proposition} \label{prop:gentype}
Let $X$ be a smooth K\"ahler threefold such that $K_X$ is nef.
Then $c_1^3(X)$ is bounded by the Betti numbers of $X$.
\end{proposition}

\begin{proof}
If $X$ is of general type, then it is projective because it is Moishezon and K\"ahler.
Following an observation of Kotschick \cite{kotschick-JTOP}, the boundedness of $c_1^3(X)$ follows then from the Miyaoka--Yau inequality
\begin{align} \label{eq:Yauineq}
0>c_1^3(X)\geq \frac{8}{3}c_1c_2(X),
\end{align}
see \cite{tsuji,zhang}.
If $X$ has Kodaira dimension $0,1$ or $2$, then $c_1^3(X)=0$, cf.\ \cite{CHP16}.
\end{proof}

\section{The bimeromorphic geometry of K\"ahler threefolds with spin structure} \label{sec:MMP}

One of the key ideas of this paper is the observation that for a K\"ahler threefold, the purely topological condition of being spin puts strong restrictions on its bimeromorphic geometry.
This is a consequence of the minimal model program for K\"ahler threefolds \cite{HP15,HP16}, together with some classical results of Mori \cite{mori-annals}.

Before we state the result, let us recall that a K\"ahler manifold $Y$ is a Mori fibre space, if there is a proper morphism $f:Y\longrightarrow B$ with positive dimensional connected fibres onto a normal $\mathbb Q$-factorial K\"ahler space $B$ with at most klt singularities, such that $-K_Y$ is $f$-ample and the relative Picard number is one, see \cite{HP15}.
In this paper, we will only need to deal with the special case where $B$ is smooth.

\begin{theorem}[\cite{HP15,HP16,mori-annals}]  \label{thm:MMP}
Let $X$ be a smooth K\"ahler threefold such that $c_1(X)$ is numerically divisible by $2$ (e.g.\ a spin threefold).
Then there is a finite sequence of blow-downs to smooth points
$$
X=Y_r\longrightarrow Y_{r-1}\longrightarrow \dots \longrightarrow Y_1\longrightarrow Y_0=Y,
$$
such that $Y$ is a smooth K\"ahler threefold with $c_1(Y)$ numerically divisible by $2$, which is either a minimal model (i.e.\ $K_Y$ is nef) or a Mori fibre space.
Moreover, if $Y$ is a Mori fibre space, then it is one of the following:
\begin{enumerate}
\item an unramified conic bundle over a smooth K\"ahler surface;
\item a quadric bundle over a smooth curve;
\item a smooth Fano threefold.
\end{enumerate}
\end{theorem}

\begin{proof}
By \cite{HP15,HP16}, we can run a minimal model program on $X$.
If $X$ is not a minimal model, nor a Mori fibre space, then, since there are no flips in the smooth category \cite{mori-annals, mori-flips}, there is a divisorial contraction $f:X\longrightarrow Y$ with exceptional divisor $E$.
Since $c_1(X)$ is numerically divisible by 2, we get that $K_X.C\leq -2$ for every contracted curve $C\subset E$.
It is known to experts, that the theorem follows now from \cite{mori-annals} and \cite{mori-prokhorov}; we sketch the proof for convenience of the reader.

Since $-K_X$ is $f$-ample, Mori's classification of all possible exceptional divisors applies \cite[Theorem 3.3]{mori-annals}.
In particular, $f$ contracts $E$ either to a smooth curve, or to a point.
In the former case, $f$ is the blow-up of a smooth curve of $Y$ and so the general fibre of $E\longrightarrow f(E)$ is a rational curve $C\subset E$ with $K_X.C=-1$, which contradicts our assumptions.
In the latter case, $E$ together with its normal bundle in $X$ is isomorphic to $(\mathbb P^2,\mathcal O(1))$, $(\mathbb P^2,\mathcal O(2))$ or $(Q,\mathcal O(1))$, where $Q\subset \mathbb P^3$ is an integral quadric.
If $(E,N_{E\slash X})=(Q,\mathcal O(1))$, then $K_X=f^{\ast}K_Y+E$ which implies $K_X.\ell=-1$ for a line $\ell\subset Q$.
If $(E,N_{E\slash X})=(\mathbb P^2,\mathcal O(2))$, then $K_X=f^{\ast}K_Y+\frac{1}{2}E$ which implies again $K_X.\ell=-1$ for a line $\ell\subset \mathbb P^2$.
Since $K_X$ is numerically divisible by 2, none of the two possibilities occur, which shows $(E,N_{E\slash X})=(\mathbb P^2,\mathcal O(1))$, and so $f$ is the blow-down to a smooth point of $Y$.
In particular, $K_X=f^{\ast}K_Y+2E$ and so $Y$ is a smooth K\"ahler threefold  such that $c_1(Y)$ is numerically divisible by 2 and $b_2(Y)=b_2(X)-1$.
After a finite number of such blow-downs, we may assume that $Y$ is either a minimal model, or a Mori fibre space.

It remains to deal with the case where $f:Y\longrightarrow B$ is a Mori fibre space.
If $\dim(B)\leq 1$, then $h^{2,0}(Y)=0$ and so $Y$ is projective by Kodaira's criterion.
It thus follows from \cite[Theorem 3.5]{mori-annals} that $B$ is smooth.
Moreover, if $\dim(B)=1$, then the general fibre of $f$ is a smooth del Pezzo surface with spin structure, hence a smooth quadric in $\mathbb P^3$.
This concludes the case $\dim(B)\leq 1$.

If $\dim(B)=2$, then, by \cite[Corollary 2.4.2]{mori-prokhorov}, $B$ is smooth and there is a rank $3$ vector bundle $\mathcal E$ on $B$ together with an inclusion $Y\subset \mathbb P(\mathcal E)$ which realises each fiber $f^{-1}(b)$ as a conic in $\mathbb P(\mathcal E_b)$.
If $f^{-1}(b)$ is singular, then it contains a line $\ell$ with $K_{Y}.\ell=-1$, which contradicts the assumption that $c_1(Y)$ is numerically divisible.
This proves that $f$ is an unramified conic bundle over a smooth K\"ahler surface, which finishes the proof.
\end{proof}

The same argument as above, together with the classification of smooth Fano threefolds \cite[\S 12]{iskovski}, shows the following.

\begin{theorem}[\cite{HP15,HP16,mori-annals}] \label{thm:p>2
Let $X$ be a smooth K\"ahler threefold such that $c_1(X)$ is numerically divisible by some integer $m \ge 3$.
Then one of the following
holds:
\begin{enumerate}
\item $X$ is a minimal model, i.e.\ $K_X$ is nef;
\item $m=3$ and $X$ is a $\mathbb P^2$-bundle over a smooth curve;
\item $m=3$ and $X$ is isomorphic to a smooth quadric in $\mathbb P^4$;
\item $m=4$ and $X$ is isomorphic to $\mathbb P^3$.
\end{enumerate}
\end{theorem}

\section{Non-K\"ahler manifolds with K\"ahler homotopy types} \label{sec:b_3=0}

In this section we produce many non-K\"ahler manifolds with the homotopy type of K\"ahler manifolds.
The main results are Theorem \ref{thm:b_3=0:2}
and Proposition \ref{prop:b_2=1} below.
We start with the following lemma.

\begin{lemma} \label{lem:p1(Mfs)}
Let $X$ be a smooth K\"ahler threefold admitting the structure of an unramified conic bundle $f:X\longrightarrow S$ over a smooth K\"ahler surface $S$.
Then, $p_1(X)\in f^{\ast}H^4(S,\mathbb Z)$.
\end{lemma}

\begin{proof}
If $X$ is projective, then the first statement follows from the observation that for any smooth curve $C\subset S$, the preimage $R:=f^{-1}(C)$ is a ruled surface with normal bundle $f^{\ast}\mathcal O_S(C)|_R$ and so $p_1(X).R=p_1(R)+p_1(f^{\ast}\mathcal O_S(C)|_{R})=0$.
The second statement can be proven similarly.

In general, the lemma follows from the following topological argument, suggested to us by M.\ Land respectively.
By \cite{smale}, any smooth $S^2$-bundle $f:M\longrightarrow S$ can be realized as the sphere bundle of an oriented real rank three vector bundle $E$ on $S$.
Since the tangent bundle of a sphere is stably trivial, $TM$ is stably isomorphic to $f^{\ast}(TS\oplus E)$ and so $p_1(M)=f^*p_1(TS\oplus E)$.
\end{proof}

The next result implies Theorem \ref{thm:b_3=0} stated in the introduction.

\begin{theorem} \label{thm:b_3=0:2}
Let $X$ be a simply connected spin 3-fold with $b_2(X)\geq 1$ and $H^3(X,\mathbb Z)=0$.
Then there are infinitely many pairwise non-homeomorphic spin $6$-manifolds $M_i$ with the oriented homotopy type of $X$, and which are not homeomorphic to a K\"ahler manifold.
\end{theorem}

\begin{proof}
Since $X$ is simply connected with $H^3(X,\mathbb Z)=0$, $X$ has torsion free cohomology, and so the classification of Wall applies, see Section \ref{subsubsec:classification}.
Let
$$
(0,H^2(X,\mathbb Z),F_X,p_1(X))
$$
be the admissible tuple which corresponds to $X$.

In order to construct examples that are homotopy equivalent to $X$, we fix a general element
$$
\omega\in H^2(X,\mathbb Z)^{\vee} \ \ \text{with}\ \ \omega \equiv 0 \mod 48 .
$$
General means here that its image in rational cohomology lies outside a finite number of proper subvarieties that we will encounter in the process of the proof; it is important to note that these subvarieties are going to depend only on the ring structure of $H^*(X,\mathbb Q)$.

For any integer $r\equiv 1 \mod 48$, the tuple
$$
(0,H^2(X,\mathbb Z),F_X,r(p_1(X)+\omega))
$$
is admissible, see (\ref{eq:admissible}).
Let $M_{r}$ be the corresponding spin $6$-manifold.
By work of Zhubr (see \cite{zhubr} or \cite[Theorem 2]{OV95}), the homotopy type of $M_{r}$ depends only on the reduction modulo $48$ of $p_1(M_r)$.
Our choices ensure therefore that $M_{r}$ is homotopy equivalent to $X$.
Since the image of the first Pontryagin class in $H^4_{t.f.}(X,\mathbb Z)$ ($=H^4(X,\mathbb Z)$) is a homeomorphism invariant by Novikov's theorem \cite{novikov}, $M_{r}$ and $M_{r'}$ are not homeomorphic if $r\neq r'$, see also Remark \ref{rem:diffeotypes}.
In order to conclude Theorem \ref{thm:b_3=0:2}, it thus suffices to show that for infinitely many values of $r$, $M_{r}$ is not homeomorphic to a K\"ahler manifold.

For a contradiction, we assume from now on that $r>>0$ is sufficiently large and that there is a K\"ahler manifold $X_{r}$ which is homeomorphic to $M_{r}$.
By construction, there is a natural isomorphism
\begin{align}\label{H*X}
H^*(X_{r},\mathbb Z)\cong H^*(X,\mathbb Z).
\end{align}
By Novikov's theorem, or Remark \ref{rem:diffeotypes}, the above isomorphism identifies $p_1(X_r)$ with $p_1(M_r)=r(p_1(X)+\omega)$.
Since the first Pontryagin class of a blow-up of a complex threefold in a point is not divisible by any integer greater than $4$, see \cite[Proposition 13]{OV95}, it follows from Theorem \ref{thm:MMP} that $X_{r}$ is minimal or a Mori fibre space.

Since $X_{r}$ is K\"ahler with $b_1(X_{r})=b_3(X_{r})=0$, Riemann--Roch says
\begin{align}\label{eq:RR}
c_1c_2(X_{r})=24+24h^{2,0}(X_{r}) >0.
\end{align}
The Miyaoka--Yau inequality (\ref{eq:Yauineq}) shows then that $X_{r}$ is not of general type.
We claim that it must in fact be of negative Kodaira dimension.
Indeed, otherwise $c_1^3(X_{r})=0$ and so
$$
p_1(X_{r})c_1(X_{r})=c_1^3(X_{r})-2c_1c_2(X_{r})=-48-48 h^{2,0}(X_{r})
$$
would be non-zero and very divisible, whereas $h^{2,0}(X_{r})$ is bounded by $b_2(X_{r})=b_2(X)$.

We have thus shown that $X_{r}$ is a Mori fibre space.
Since there are only finitely many deformation types of Fano threefolds, and since $p_1(X_{r})$ is sufficiently divisible, $X_{r}$ cannot be Fano.

Let us assume that $f:X_{r}\longrightarrow C$ is a Mori fibre space over a curve.
A  general fibre $Q$ of $f$ is a smooth quadric surface and so
$p_1(X_{r}).Q=p_1(Q)=0
$.
On the other hand, the line $[Q]\cdot \mathbb Q\subset H^2(X_{r},\mathbb Q)$ lies in the locus where the cubic form vanishes and so it is, up to at most three possibilities, uniquely determined by (\ref{H*X}).
(This uses $b_2(X_{r})=2$, which follows from $\rho(X_r\slash C)=1$ and $h^{2,0}(X_r)=0$.)
Since $\omega$ is sufficiently general, the restriction of $p_1(X_{r})\in H^2(X_{r},\mathbb Q)^{\vee}$ to any of these lines will be nonzero and so this case cannot happen.

It remains to deal with the case where $f:X_{r}\longrightarrow S$ is a Mori fibre space over a K\"ahler surface $S$.
By Theorem \ref{thm:MMP}, $f$ is an unramified conic bundle.
Since $f^{\ast}H^2(S,\mathbb Z)\subset H^2(X_{r},\mathbb Z)$ is a hyperplane which is contained in the vanishing locus of the cubic form, it follows from (\ref{H*X}) that the subspace $f^{\ast}H^2(S,\mathbb Z)\subset H^2(X_{r},\mathbb Z)$ is determined uniquely up to at most three possibilities.
Therefore, the line in $H^4(X_r,\mathbb Q)$ that is spanned by the class of a fibre of $f$ is also determined up to at most three possibilities.
Since $\omega$ is general, $p_1(X_{r})$ is not contained in any of those lines.
This contradicts Lemma \ref{lem:p1(Mfs)}, which finishes the proof of the theorem.
\end{proof}

\begin{remark}
Any simply connected spin $6$-manifold $M$ with torsion free cohomology can be written as a connected sum $M=M_0\sharp^{b_3/2} (S^3\times S^3)$, with $H^3(M_0,\mathbb Z)=0$;
the diffeomorphism type of $M_0$, called the core of $M$, is uniquely determined by $M$, cf.\ \cite[Corollary 1]{OV95}.
In particular, Theorem \ref{thm:b_3=0:2} applies to the core $M_0$ of any such manifold $M$ as long as $b_2(M)>0$.
\end{remark}

\begin{remark} \label{rem:b_3=0:nongeneraltype}
A K\"ahler threefold $X$ with spin structure and $b_3(X)=0$ cannot be of general type.
Indeed, if it was of general type, then we may by Theorem \ref{thm:MMP} assume that it is minimal and so the Miyaoka--Yau inequality (\ref{eq:Yauineq}) holds.
In particular, $c_1c_2(X)$ is negative, which contradicts Riemann--Roch (\ref{eq:RR}), because $b_1(X)=0$ by Hard Lefschetz.
\end{remark}

In view of  Theorem \ref{thm:b_3=0} and Remark \ref{rem:b_3=0:nongeneraltype}, it is natural to ask for non-K\"ahler manifolds with the homotopy type of a variety of general type.
We were unable to find such examples in the literature.
The next result fills this gap; it applies for instance to arbitrary complete intersection threefolds $X\subset \mathbb P^N$, which might very well be non-spin.

\begin{proposition} \label{prop:b_2=1}
Let $X$ be a simply connected $6$-manifold with $H_2(X,\mathbb Z)\cong\mathbb Z$.
Then there are infinitely many pairwise non-homeomorphic $6$-manifolds with the same oriented homotopy type as $X$, but which are not homeomorphic to a K\"ahler manifold.
\end{proposition}

\begin{proof}
Let $Y$ be a K\"ahler threefold which is homotopy equivalent to $X$.
Since $b_2(Y)=1$, $Y$ is either Fano, Calabi-Yau or $K_Y$ is ample.
Moreover, the ample generator $L$ of $H^2(Y,\mathbb Z)$ is uniquely determined by the topological property $L^3>0$.
If $Y$ is Calabi-Yau, then $p_1(Y)=-2c_2(Y)$ and so
\begin{align} \label{eq:miyaoka}
p_1(Y).L<0
\end{align}
by Miyaoka's inequality \cite[Theorem 1.1]{miyaoka87}.

In order to construct infinitely many pairwise non-homeomorphic manifolds $M_i$ with the homotopy type of $X$, we may now proceed similar to the proof of Theorem \ref{thm:b_3=0}.
By the classification of simply connected $6$-manifolds with torsion free cohomology, see \cite[Section 1]{OV95}, we can choose $p_1(M_i)$ sufficiently divisible and of a sign which violates Miyaoka's inequality (\ref{eq:miyaoka}).
It follows that $M_i$ cannot support a Calabi--Yau structure.
For sufficiently divisible $p_1(M_i)$,
 the cases where the canonical bundles are ample or anti-ample cannot happen either, because the corresponding families are bounded by the boundedness of Fano threefolds and Matsusaka's big theorem together with the Miyaoka--Yau inequality, respectively, cf.\ \cite[Proposition 28]{schreieder-GT}.
\end{proof}

\begin{remark}
Borel's conjecture predicts that a homotopy equivalence between closed manifolds with contractible universal cover is homotopic to a homeomorphism; this is known for large classes of fundamental groups, such as hyperbolic groups, see \cite{bartels-lück}.
Thus the assumption on $\pi_1(X)$
in Theorem \ref{thm:b_3=0:2} and Proposition \ref{prop:b_2=1}
is important.
\end{remark}

By Remark \ref{rem:b_3=0:nongeneraltype}, the following question remains open, but see Corollary \ref{cor:6mfd-not-gentyp} below, where a weaker statement is proven.

\begin{question}
Is there a non-K\"ahler manifold with large $b_2$
that has the oriented homotopy type of a smooth complex projective variety of general type?
\end{question}

As mentioned in the introduction, most topological constraints known for K\"ahler or projective manifolds are only sensitive to the homotopy type, but not to the homeomorphism type.
For instance, Voisin used restrictions on the cohomology algebra to produce K\"ahler manifolds that are not homotopy equivalent to smooth complex projective varieties \cite{voisin-kodaira}, but the following question remained open.

\begin{question} \label{question:voisin}
Is there a K\"ahler manifold which has the oriented homotopy type but not the homeomorphism type of a smooth complex projective variety?
\end{question}

\section{Proof of Theorem \ref{thm:deformations}} \label{sec:gentype}

Theorem \ref{thm:deformations} stated in the introduction follows from the following result.

\begin{theorem}\label{thm:deformations:2}
Let $b\in \mathbb N$.
Then there are only finitely many deformation types of smooth complex projective threefolds $X$ of general type, such that $c_1(X)$ is numerically divisible and $b_i(X)\leq b$ for all $i$.
\end{theorem}

\begin{proof}
Let $X$ be a smooth complex projective threefold of general type such that $c_1(X)$ is numerically divisible.
By Theorems \ref{thm:MMP} and \ref{thm:p>2}, there is some $r\geq 0$ and a smooth complex projective minimal threefold $Y$ of general type such that $X$ is obtained from $Y$ by a finite sequence of $r$ blow-ups along points.
Clearly, any variety that is obtained from $Y$ by a sequence of $r$ blow-ups along points is deformation equivalent to $X$.
Moreover, the number $r$ of blow-ups is bounded from above by $b_2(X)-1$.

By \cite[Theorem 1.2]{cascini-tasin} (or Proposition \ref{prop:gentype} above), there is a constant $c$ which depends only on the Betti numbers of $X$ such that
$$
\operatorname{vol}(Y,K_Y)\leq c .
$$
In order to prove Theorem \ref{thm:deformations:2}, it is therefore enough to prove that there are only finitely many deformation equivalence classes of smooth complex projective minimal threefolds of general type and with volume bounded by $c$.

By \cite{HM06,takayama,tsuji2}, threefolds of  general type and with bounded volume are birationally bounded.
That is, there is a projective morphism of normal quasi-projective schemes $\pi^{bdd}: \mathcal X^{bdd} \longrightarrow B$ such that any smooth complex projective threefold of general type and of volume at most $c$ is birational to a fibre of $\pi^{bdd}$.
By Noetherian induction, replacing $B$ by a disjoint union of locally closed subsets and resolving $\mathcal X^{bdd}$, we get a smooth projective family $\pi^{sm}: \mathcal X^{sm} \longrightarrow B$.
By the deformation invariance of plurigenera \cite{siu}, we may assume that any fibre of $\pi^{sm}$ is of general type.
Up to replacing $B$ by a finite \'etale covering, we can apply the MMP in families \cite[Theorem 12.4.2]{KM92} to $\pi^{sm}$ and obtain a projective family $\pi: \mathcal X \longrightarrow B$ of minimal models of general type, such that any smooth complex projective threefold of general type and with volume bounded by $c$ is birational to a fibre of $\pi$.

In the next step of the proof, we use the family $\pi$ to construct a second family $\pi':\mathcal X'\longrightarrow B'$ with the following property:
\begin{enumerate}
\item[($\ast$)]
if $b\in B_i$ is a very general point  of some component $B_i$ of $B$, then the fibre $X_b=\pi^{-1}(b)$ has the property that any of its minimal models is isomorphic to a fibre of $\pi':\mathcal X'\longrightarrow B'$.
\end{enumerate}

In order to explain our construction, let $B_i$ be a component of $B$ and consider the geometric generic fibre $\mathcal X_{i,\overline {\eta}}$ of $\mathcal X_i\longrightarrow B_i$.
The number of minimal models of $\mathcal X_{i,\overline {\eta}}$ is finite by \cite{KM87} and they are all connected by flops by \cite{kollar89}.
If $\mathcal X_{i,\overline {\eta}}'$ is such a minimal model, then there is a sequence of flops $\mathcal X_{i,\overline {\eta}}\dashrightarrow \mathcal X_{i,\overline {\eta}}'$.
This sequence of flops is defined over some finite extension of the function field $k(B_i)$, hence over the ring of regular functions of some affine variety $B_i'$ which maps finitely onto a Zariski open and dense subset of $B_i$.
In particular, we can spread $\mathcal X_{i,\overline {\eta}}'$  over $B_i'$ to get a family
$$
\mathcal X'_{i}\longrightarrow B_i' ,
$$
whose geometric generic fibre is $\mathcal X_{i,\overline {\eta}}'$.

We define $\pi':\mathcal X'\longrightarrow B'$ to be the disjoint union of all such families $\mathcal X'_{i}\longrightarrow B_i'$ that we can construct in the above way from all the minimal models of all geometric generic fibres $\mathcal X_{i,\overline \eta}$ of $\pi$.
In particular, the number of irreducible components of $B'$ corresponds to the number of minimal models of the geometric generic fibres $\mathcal X_{i,\overline \eta}$ and so $B'$ has finitely many components.
By construction, any minimal model of any $\mathcal X_{i,\overline \eta}$ appears as geometric generic fibre of $\pi'$ over some component of $B'$.
Since the geometric generic fibre is abstractly (as scheme over $\mathbb Z$) isomorphic to any very general fibre, this implies that property ($\ast$)  holds; for convenience of the reader we give some details below.
\begin{lemma}
Property ($\ast$) holds.
\end{lemma}
\begin{proof}
Let $X_b$ be the fibre above a very general point $b\in B_i$; and let $k$ be a finitely generated extension of $\mathbb Q$ over which $B_i$ can be defined.
Since $B_i'$ is affine, $B_i\subset \mathbb A^N$ for some $N$ and so we may write $b=(b_1,\dots ,b_N)$.
We can choose an isomorphism of fields
$$
\sigma:\mathbb C\stackrel{\sim} \longrightarrow \overline{\mathbb C(B_i)} ,
$$
which restricts to an isomorphism between the subfield $k(b_1,\dots ,b_N)$ and the function field of $B_i$ over $k$.
The field automorphism $\sigma$ induces an isomorphism
$$
\varphi: X_b \longrightarrow   \mathcal X_{i,\overline \eta}
$$
of schemes over $\mathbb Z$ (on stalks, this map is not $\mathbb C$-linear but $\sigma^{-1}$-linear).
Any sequence of flops $X_b\dashrightarrow X^+_b$ corresponds via $\varphi$ to a sequence of flops of the geometric generic fibre $\mathcal X_{i, \overline \eta}\dashrightarrow \mathcal X_{i, \overline \eta}^+$.
Moreover, $\mathcal X_{i, \overline \eta}^+$ is a minimal model of $\mathcal X_{i, \overline \eta}$ because it is obtained by a sequence of flops.
By the construction of the family $\pi'$, $\mathcal X_{i, \overline \eta}^+$ is isomorphic to the geometric generic fibre of $\pi'$ over some component $B'_j$ of $B'$:
$$
\mathcal X_{i, \overline \eta}^+\cong \mathcal X'\times_{B'}\overline {\mathbb C(B'_j)} .
$$
By construction of $\pi'$, $B'_j$ maps finitely onto a Zariski open subset of $B_i$, which yields an identification $\overline {\mathbb C(B_i)}= \overline {\mathbb C(B'_j)}$.
Therefore, $\sigma$ induces a ($\sigma$-linear) isomorphism
$$
\psi:\mathcal X_{i, \overline \eta}^+\longrightarrow X'_{b'}
$$
of schemes over $\mathbb Z$, where $X'_{b'}={\pi'}^{-1}(b')$ is a fibre of $\pi'$ for some point $b'\in B_j'$ which maps to $b$ via $B_j'\longrightarrow B_i$.
Moreover, the composition
$$
X_b \stackrel{\varphi}\longrightarrow  \mathcal X_{i,\overline \eta}\xymatrix@1@=15pt{\ar@{-->}[r]&} \mathcal X_{i,\overline \eta}^+ \stackrel{\psi}\longrightarrow  X_{b'}' ,
$$
is a rational map $X_b\dashrightarrow X'_{b'}$ which is $\mathbb C$-linear on stalks.
Therefore, $X_b\dashrightarrow X'_{b'}$ is a sequence of flops of complex projective varieties, which identifies $X_{b'}'$ with $X_b^+$.
This proves that ($\ast$) holds.
\end{proof}

Let us now consider the following set of deformation equivalence classes of smooth fibres $X'_{b'}:={\pi'}^{-1}(b')$ of $\pi'$:
$$
\mathcal S:=\{[X'_{b'}] \mid  b'\in B'\ \text{such that $X'_{b'}$ is smooth}\} .
$$
Since $B'$ has finitely many irreducible components, and since smoothness is an open condition,  $\mathcal S$ is finite.

To conclude, let $X$ be any smooth complex projective minimal threefold of general type and with $\operatorname{vol}(X,K_X)\leq c$.
We claim that the deformation equivalence class $[X]$ belongs to $\mathcal S$, which implies the theorem.
To prove the claim, recall that $\pi:\mathcal X\longrightarrow B$ is a family of complex projective minimal threefolds with the property that any smooth threefold of general type whose volume is bounded by $c$ is birational to a fibre of $\pi$.
In particular, there is a component $B_{i_0}$ and a point $0\in B_{i_0}$ such that the fibre $X_0$ above $0$ is birational to $X$.
Then, since $X$ and $X_0$ are birational minimal models, they are connected by a sequence of flops.
Therefore, by \cite[Theorem 11.10]{KM92}, we can find an analytic neighbourhood $U\subset B_{i_0}$ of $0 \in B_{i_0}$, such that the base change $\mathcal X_U \longrightarrow U$ admits a sequence of flops to get a family $\mathcal X^+_U \longrightarrow U$ whose central fibre is isomorphic to $X$.
Since $X$ is projective, we may by \cite[Theorem 12.2.10]{KM92} assume that all fibres of $\mathcal X^+_U \longrightarrow U$ are projective.
Moreover, a very general fibre $X^+_t$ of $\mathcal X^+_U \longrightarrow U$ is connected to a very general fibre of $\mathcal X_U \longrightarrow U$ by a sequence of flops.
Therefore, $X^+_t$ is a minimal model of a very general fibre of $\mathcal X_{i_0} \longrightarrow B_{i_0}$.
Thus, by $( \ast )$, $X^+_t$ is isomorphic to a fibre of $\pi'$.
Since $X$ is smooth, $X^+_t$ is smooth, which implies
$
[X]=[X^+_t] \in \mathcal S .
$
This finishes the proof of Theorem \ref{thm:deformations:2}.
\end{proof}

\begin{remark} \label{rem:newpaper}
Generalizing the above argument, we have shown in a joint work with Martinelli  \cite{MST16} that in arbitrary dimension, minimal models of general type and bounded volume form a bounded family.
\end{remark}

\begin{remark} \label{rem:b2=2:unbounded}
The $\mathbb P^1$-bundle $X_n:=\mathbb P(\mathcal O_{\mathbb P^2}(2n+1)\oplus \mathcal O_{\mathbb P^2})$ over $\mathbb P^2$ is a simply connected spin threefold with torsion free cohomology, $b_2(X_n)=2$ and $b_3(X_n)=0$.
However, $c_1^3(X_n)$ is unbounded for $n\to \infty$, and so the $X_n$'s do not belong to a finite number of deformation types.
Therefore, the general type assumption is needed in Theorem \ref{thm:deformations}.
The next example shows that the spin assumption is also necessary.
\end{remark}

\begin{example} \label{ex:deformations:gentype}
There are infinitely many deformation types of smooth projective threefolds of general type and with bounded Betti numbers.
Indeed, we start with a threefold $Y$ of general type which contains a smooth quadric surface $Q$.
Then, $Q$ contains smooth rational curves $C_i\subset Q$ such that the degree of the normal bundle of $C_i$ in $Q$ is unbounded.
The blow-up $X_i:=Bl_{C_i}Y$ satisfies
$
K_{X_i}^3=K_Y^3+2\deg(N_{C_i\slash X})+6
$,
see \cite[Proposition 4.8]{cascini-tasin} and \cite[Proposition 14]{OV95}.
Hence, $c_1^3(X_i)$ is unbounded in $i$, although the Betti numbers of $X_i$ do not depend on $i$.
\end{example}

Using the classification of simply connected spin $6$-manifolds whose cohomology is not necessarily torsion free, see \cite{zhubr}, Theorem \ref{thm:deformations} implies easily the following; the details are analogous (but simpler) to the proof of Theorem \ref{thm:b_3=0} and so we leave them out.

\begin{corollary} \label{cor:6mfd-not-gentyp}
Let $X$ be any simply connected
K\"ahler threefold with spin structure.
Then there are infinitely many pairwise non-homeomorphic simply connected closed spin $6$-manifolds $M_i$, that have the same oriented homotopy type as $X$,
but which are not homeomorphic to any smooth complex projective variety of general type.
\end{corollary}

Corollary \ref{cor:6mfd-not-gentyp} has the following amusing consequence: either there is a large supply of examples of homotopy equivalent K\"ahler manifolds with different Kodaira dimensions, or one can produce many more smooth manifolds which do not carry a K\"ahler structure although they have the homotopy type of a smooth complex projective variety of general type.
The known examples of homotopy equivalent K\"ahler manifolds with different Kodaira dimensions are rather scarce and come usually from particular complex surfaces, see for instance \cite{kollar}; it seems unlikely that they form a really big class.

\section{Mori fibre spaces}

In this section, we analyse the Chern numbers of  smooth Mori fibre spaces with spin structure  in dimension three.
This will be used in the proof of Theorem \ref{thm:w2=0} in Section \ref{sec:bounded}.

\subsection{Mori fibre spaces over curves} \label{sec:Mfs:curves}

In this subsection we prove the following result.

\begin{proposition} \label{prop:MFS:curves}
Let $(X_i)_{i\geq 0}$ be a sequence of compact K\"ahler manifolds of dimension $n$, admitting the structure of a Mori fibre space $X_i\longrightarrow C_i$ over a curve $C_i$.
Suppose that
\begin{enumerate}
\item the Betti numbers of the $X_i$'s are bounded;
\item there is an isomorphism $H^{2\ast}_{tf}(X_i,\mathbb Z)\stackrel{\sim} \longrightarrow H^{2\ast}_{tf}(X_0,\mathbb Z)$ between the even torsion free cohomology algebras, which respects the Pontryagin classes.
\end{enumerate}
Then the images of the Chern classes $c_1(X_i)$ and $c_2(X_i)$ in $H^{2\ast}_{tf}(X_0,\mathbb Z)$ are bounded.
In particular, for all $0\leq k\leq n/2$, the sequence of Chern numbers $c_1^{n-2k}c_2^k(X_i)$ is bounded.
\end{proposition}

\begin{proof}
Let $F_i$ denote the general fiber of $X_i\longrightarrow C_i$.
Then
$$
H^2(X_0,\mathbb Q)=x\cdot \mathbb Q\oplus y\cdot \mathbb Q,
$$
where $y=c_1(X_0)$, and $x\in H^2_{tf}(X_0,\mathbb Z)$ is primitive such that $[F_0]$ is a positive multiple of $x$.
The ring structure satisfies $x^2=0$, $y^{n-1}x=c_1^{n-1}(F_0)$ and $y^n=c_1^n(X_0)$.
From now on we will use the isomorphism $H^2(X_i,\mathbb Q)\cong H^2(X_0,\mathbb Q)$ to think about $x$ and $y$ as basis elements of $H^2(X_i,\mathbb Q)$.

We can write
$
c_1(X_i) = a_i\cdot x+ b_i\cdot y
$,
for some $a_i,b_i \in \mathbb Q$.
Next, since $[F_i]^2=0$,  $[F_i]=\lambda_i \cdot x $, for some $\lambda_i\in \mathbb Z$.\footnote{In fact, $\lambda_i=\pm 1$, because $X_i\longrightarrow C_i$ has a rational section by Grabber--Harris--Starr's theorem, but we do not use this here.}
Moreover, since $F_i\subset X_i$ has trivial normal bundle,
$
c(X_i)|_{F_i}=c(F_i)
$.
This implies
$$
c_1^{n-1}(F_i)=c_1(X_i)^{n-1}[F_i]= (a_i\cdot x+ b_i\cdot y)^{n-1} \lambda_i \cdot x=b_i^{n-1}\lambda_i \cdot y^{n-1}x.
$$
Since $F_i$ is Fano, $b_i$ and $\lambda_i$ are both nonzero and bounded, because Fano varieties of a fixed dimension form a bounded family.

What follows is inspired by \cite[Theorem 4.2.3]{kollar}.
We have
$$
\chi(X_i,\mathcal O_{X_i})= \sum_{s\geq 0} \frac{1}{2^{n+2s}(n-2s)!}c_1(X_i)^{n-2s} A_s(p_1,...,p_s),
$$
where $A_s(p_1,...,p_s)$ is a polynomial in the Pontryagin classes.
Since $b_i$ is bounded and $x^2=0$, the above expression is a linear polynomial in $a_i$ with bounded coefficients,
$$
\chi(X_i,\mathcal O_{X_i})=\mu_1(i)\cdot a_i + \mu_0(i) .
$$
Here,
\begin{align*}
\mu_1 (i)&= \sum_{s\geq 0} \frac{1}{2^{n+2s}(n-2s)!}(n-2s)x\cdot (b_iy)^{n-1-2s} A_s(p_1,...,p_s)\\
&=\frac{1}{\lambda_i} [F_i] \cdot  \sum_{s\geq 0} \frac{1}{2^{n+2s}(n-1-2s)!} (b_iy)^{n-1-2s} A_s(p_1,...,p_s)\\
&=\frac{1}{2\lambda_i} [F_i] \cdot  \sum_{s\geq 0} \frac{1}{2^{n-1+2s}(n-1-2s)!} c_1(X_i)^{n-1-2s} A_s(p_1,...,p_s)\\
&=\frac{1}{2\lambda_i}\chi(F_i,\mathcal O_{F_i}) \\
&= \frac{1}{2\lambda_i} ,
\end{align*}
where we used that the Chern and Pontryagin classes of $X_i$ restrict to those of $F_i$.
The above computation shows $\mu_1(i)\neq 0$ for all $i$.

Since the Betti numbers of $X_i$ are bounded, and $X_i$ is K\"ahler, $\chi(X_i,\mathcal O_{X_i})$ attains only finitely many values.
Since $\mu_0(i)$ and $\mu_1(i)\neq 0$ are bounded, $a_i$ is bounded.
This proves that $c_1(X_i)$ is bounded.
The boundedess of $c_2(X_i)$ follows from $p_1(X_i)=c^2_1(X_i)-2c_2(X_i)$.
This proves the proposition.
\end{proof}

\begin{remark} \label{rem:MFS:curves}
The above proof shows that for $n=3$, one can replace condition (2) in Proposition \ref{prop:MFS:curves} by the following slightly weaker assumption:
for each $i$ there is an isomorphism $H^2_{tf}(X_i,\mathbb Z)\longrightarrow H^2_{tf}(X_0,\mathbb Z)$ which respects the trilinear forms, given by cup products, and the linear forms, given by the first Pontryagin classes.
\end{remark}

\subsection{Unramified conic bundles over surfaces} \label{sec:Mfs:surface}

The aim of this subsection is to prove

\begin{proposition} \label{prop:MFS:surfaces}
Let $(X_i)_{i\geq 0}$ be a sequence of smooth K\"ahler threefolds with the structure of an unramified conic bundle $f_i:X_i\longrightarrow S_i$  over a smooth K\"ahler surface $S_i$.
Suppose that
\begin{enumerate}
\item the Betti numbers of the $X_i$'s are bounded;
\item for each $i$, there is an isomorphism $H^2_{tf}(X_i,\mathbb Z)\cong H^2_{tf}(X_0,\mathbb Z)$ which respects the trilinear forms given by cup products.
\end{enumerate}
Then the sequence of Chern numbers $c_1^3(X_i)$ is bounded.
\end{proposition}

We need the following lemma, which is well--known (at least) in the projective case.

\begin{lemma} \label{lem:Spin:Mfs}
Let $X$ be a smooth K\"ahler threefold admitting the structure of an unramified conic bundle $f:X\longrightarrow S$ over a smooth K\"ahler surface $S$.
Then we have the following numerical equivalence on $S$:
$$
f_{\ast}K_X^2\equiv -4K_S .
$$
\end{lemma}

\begin{proof}
Note first that the N\'eron--Severi group $\operatorname{NS}(S)$ is generated by smooth curves $C\subset S$; this is clear if $S$ is projective and it follows easily from the classification of surfaces if $S$ is a non-projective K\"ahler surface, see \cite{barth-etal}.
It thus suffices to compare the intersection numbers of $f_{\ast}K_X^2$ and $-4K_S$ with a smooth curve $C\subset S$.
This follows from an easy computation, where one uses that  $R:=f^{-1}(C)$  is a minimal ruled surface with normal bundle $f^\ast \mathcal O_S(C)|_R$, cf.\ \cite{mori-mukai} and \cite[Proposition 7.1.8]{iskovski}.

The following alternative (and slightly more general) argument was suggested to us by D.\ Kotschick.
We have $TX=f^\ast TS\oplus Tf$, where $Tf=\ker(f_\ast: TX\longrightarrow f^\ast TS)$ is the tangent bundle along the fibres of $f$.
Hence,
$$
c_1^2(X)=f^\ast c_1^2(S)+2c_1(Tf)f^\ast c_1(S) +c_1^2(Tf) .
$$
As in Lemma \ref{lem:p1(Mfs)}, $Tf$ is stably isomorphic to $f^\ast E$ for a real rank three vector bundle $E$ on $S$.
Since $Tf$ is a complex line bundle, we deduce $c_1^2(Tf)=p_1(Tf)\in f^\ast H^4(S,\mathbb Z)$ and so $f_\ast c_1^2(Tf)=0$.
Hence,
$$
f_\ast c_1^2(X)
f_\ast(2c_1(Tf)f^\ast c_1(S))=4c_1(S) ,
$$
because $c_1(Tf)$ restricts to $c_1(\mathbb P^1)=2$ on each fibre.
\end{proof}

\begin{proof}[Proof of Proposition \ref{prop:MFS:surfaces}]
Using the isomorphism $H^2_{tf}(X_i,\mathbb Z)\cong H^2_{tf}(X_0,\mathbb Z)$ which respects the trilinear forms given by cup products, we identify degree two cohomology classes of $X_i$ with those of $X_0$.
Using Poincar\'e duality, we further identify classes of $H^4_{tf}(X_i,\mathbb Z)$ with linear forms on $H^2_{tf}(X_i,\mathbb Z)\cong H^2_{tf}(X_0,\mathbb Z)$.

The codimension one linear subspace ${f_i}^*\mathbb P(H^2(S_i,\mathbb C))$ of $\mathbb P(H^2(X_0,\mathbb C))$ is contained in the cubic hypersurface $\{\alpha\mid \alpha^3=0\}$.
Passing to a suitable subsequence we can therefore assume that
$$
f_i^*H^2(S_i,\mathbb C)\subset  H^2(X_0,\mathbb C)
$$
does not depend on $i$.
Let $\ell_i \in H^4_{tf}(X_0,\mathbb Z)$ be the class of a fiber of $f_i$.
The action of this class on $H^2(X_0,\mathbb Q)$ has kernel $f_i^*H^2(S_i,\mathbb Q)$, and so $\ell_i\cdot \mathbb Q$ is independent of $i$.
Since $\ell_i$ is an integral class with $K_{X_i}.\ell_i=-2$, we may after possibly passing to another subsequence thus assume that $\ell_i=\ell$ does not depend on $i$.

For any class $y\in H^2(X_0,\mathbb Q)$ which does not lie in ${f_i}^*H^2(S_i,\mathbb Q)$, we have
$$
H^2(X_0, \mathbb Q)={f_i}^*H^2(S_i,\mathbb Q) \oplus y\cdot \mathbb Q \ \ \text{and}\ \ H^4(X_0, \mathbb Q)={f_i}^*H^2(S_i,\mathbb Q)\cdot y \oplus \ell\cdot \mathbb Q .
$$
In particular, $y^2=u y + \lambda \ell$ for some $\lambda \in \mathbb Q$ and $u \in {f_i}^*H^2(S_i,\mathbb Q)$.
Replacing $y$ by a suitable multiple of $y-\frac{1}{2}u$, we may thus assume that
$$
y.\ell=-2\ \ \text{and}\ \ y^2 \in {f_i}^*H^4(S_i,\mathbb Q)=\ell\cdot \mathbb Q .
$$
Using this class, we have
$$
K_{X_i}=y+f_i^* z_i ,
$$
for some $z_i \in H^2(S_i,\mathbb Q)$.
We claim that $z_i^2\in H^4(S_i,\mathbb Q)\cong \mathbb Q$ is a bounded sequence, which implies the theorem, because
$$
K_{X_i}^3=y^3+3f_i^* z_i^2\cdot y=y^3-6 z_i^2 ,
$$
and $y^3$ does not depend on $i$.

By Lemma \ref{lem:Spin:Mfs}, we have a numerical equivalence
$$
{f_i}_*K_{X_i}^2 \equiv -4z_i \equiv -4K_{S_i}.
$$
In particular, $z_i^2=K_{S_i}^2$ is bounded by the Betti numbers of $X_i$, as we want.
This finishes the proof of Proposition \ref{prop:MFS:surfaces}.
\end{proof}

\subsection{Chern numbers of Mori fibre spaces with divisible canonical class}

The results of the previous subsections imply the following result.

\begin{corollary} \label{cor:Mfs:Kaehler}
Let $X$ be a smooth K\"ahler threefold which admits the structure of a Mori fibre space $f:X\longrightarrow B$.
If $c_1(X)$ is numerically divisible, then the Chern numbers of $X$ are determined up to finite ambiguity by the following invariants:
\begin{enumerate}
\item the Betti numbers of $X$;
\item the triple $(H^{2}_{tf}(X,\mathbb Z),F_X,p_1(X))$, where $F_X$ denotes the cubic form on  $H^{2}_{tf}(X,\mathbb Z)$, given by cup product, and $p_1(X)$ denotes the linear form given by the first Pontryagin class.
\end{enumerate}
\end{corollary}

\begin{proof}
The case where $B$ is a point follows from the boundedness of Fano threefolds \cite{iskovski}.
Next, note that  $F_X$ determines the trilinear form on $H^{2}_{tf}(X,\mathbb Z)$, given by cup product.
If $B$ is a curve, then the assertion follows therefore from Proposition \ref{prop:MFS:curves} and Remark \ref{rem:MFS:curves}, and if $B$ is a surface, we conclude via Theorems  \ref{thm:MMP} and \ref{thm:p>2}, and Proposition \ref{prop:MFS:surfaces}.
\end{proof}

\section{Proof of Theorem \ref{thm:w2=0}} \label{sec:bounded}

In the proof of Theorem \ref{thm:w2=0}, we will use the following result from \cite{cascini-tasin}.

\begin{lemma}[\cite{cascini-tasin}] \label{lem:blowup-point}
Let $Y$ be a smooth complex projective threefold and let $f:X\longrightarrow Y$ be the blow-up of a point of $Y$.
If $E\subset X$ denotes the exceptional divisor of $f$, then
\begin{enumerate}
\item the class $[E]\in H^2_{tf}(X,\mathbb Z)$ is determined up to finite ambiguity by the cubic form on $H^2_{tf}(X,\mathbb Z)$.
\item the class $[E]\in H^2_{tf}(X,\mathbb Z)$ determines uniquely the subspace $f^{\ast}H^2_{tf}(Y,\mathbb Z)\subset H^2_{tf}(X,\mathbb Z)$.
\end{enumerate}
\end{lemma}
\begin{proof}
The first assertion follows from \cite[Proposition 13]{OV95} and \cite[Proposition 3.3]{cascini-tasin}.
The second assertion follows from $H^2_{tf}(X,\mathbb Z)= f^{\ast}H^2_{tf}(Y,\mathbb Z) \oplus [E]\cdot \mathbb Z$, which shows that the cup product map
$$
\cup[E]:H^2_{tf}(X,\mathbb Z)\longrightarrow H^4_{tf}(X,\mathbb Z)
$$
has kernel  $f^{\ast}H^2_{tf}(Y,\mathbb Z)$.
\end{proof}

The following theorem implies Theorem \ref{thm:w2=0} from the introduction.

\begin{theorem} \label{thm:w2=0:2}
Let $X$ be a smooth K\"ahler threefold such that $c_1(X)$ is numerically divisible.
Then the Chern numbers of $X$ are determined up to finite ambiguity by the following invariants:
\begin{enumerate}
\item the Betti numbers of $X$; \label{item:thmw2=0:1}
\item the cubic form $F_X\in S^3H^{2}_{tf}(X,\mathbb Z)^{\vee}$, given by cup product;
\item the linear form, given by the first Pontryagin class $p_1(X)\in H^2_{tf}(X,\mathbb Z)^{\vee}$. \label{item:thmw2=0:3}
\end{enumerate}
\end{theorem}

\begin{proof}
Let $X$ be a smooth complex projective threefold such that $c_1(X)$ is numerically divisible.
By Theorems \ref{thm:MMP} and \ref{thm:p>2}, there is a sequence of blow-downs to points in smooth loci
$$
X=Y_r\longrightarrow Y_{r-1}\longrightarrow \dots \longrightarrow Y_1\longrightarrow Y_0=Y,
$$
such that $Y$ is either a smooth minimal model or a smooth Mori fiber space.
Moreover, $c_1(Y)$ is numerically divisible and the number of such blow-downs is bounded by $r\leq \rho(X)-1$, where $\rho(X)$ denotes the Picard number of $X$.

Let $E_i\subset Y_{i}$ be the exceptional divisor of $Y_{i}\longrightarrow Y_{i-1}$ and let $f_i:X\longrightarrow Y_i$ denote the natural map.
By Lemma \ref{lem:blowup-point}, the class $[E_{r}]\in H^2_{tf}(X,\mathbb Z)$ is determined up to finite ambiguity by the cubic form on $H^2_{tf}(X,\mathbb Z)$, and it determines the subspace $f_{r-1}^{\ast}H^2_{tf}(Y_{r-1},\mathbb Z)\subset H^2_{tf}(X,\mathbb Z)$ uniquely.
Repeating this argument $r$ times, we conclude that for all $i$, the classes $f_i^{\ast}[E_i]$ as well as the subspaces $f_i^{\ast}H^2_{tf}(Y_i,\mathbb Z)\subset H^2_{tf}(X,\mathbb Z)$ are up to finite ambiguity determined by the cubic form on $H^2_{tf}(X,\mathbb Z)$.

In particular, the isomorphism type of $H^2_{tf}(Y,\mathbb Z)$ together with the cubic form $F_Y$ given by cup product is up to finite ambiguity determined by the cubic form on $H^2_{tf}(X,\mathbb Z)$.
We aim to show that $p_1(Y)$ is determined up to finite ambiguity as well.
In order to see this, we note
$$
H^2_{tf}(X,\mathbb Z)=f^{\ast}H^2_{tf}(Y,\mathbb Z) \oplus \bigoplus_{i=1}^r f_i^{\ast}[E_i]\cdot \mathbb Z .
$$
Moreover,
$$
c_1(X)=f^{\ast}c_1(Y)-2\sum_{i=1}^r  f_i^{\ast}[E_i] \ \ \text{and}\ \ c_2(X)=f^{\ast} c_2(Y) .
$$
Hence,
\begin{align*}
p_1(X) &= f^{\ast} p_1(Y) +4 \sum_{i=1}^r f_i^{\ast}[E_i]^2 ,
\end{align*}
which proves the claim.

We have thus proven that the triple $(H^2_{tf}(X,\mathbb Z),F_X,p_1(X))$ determines up to finite ambiguity the isomorphism type of $(H^2_{tf}(Y,\mathbb Z),F_Y,p_1(Y))$.
Applying Proposition \ref{prop:gentype} and Corollary \ref{cor:Mfs:Kaehler}, we can therefore bound $c_1^3(Y)$ in terms of the Betti numbers of $X$ and the isomorphism type of $(H^2_{tf}(X,\mathbb Z),F_X,p_1(X))$.
Since $[E_i]^3=1$,
\begin{align} \label{eq:c_1^3(X)}
c_1^3(X)=c_1^3(Y)-8r ,
\end{align}
with $0\leq r \leq \rho(X)-1$.
Hence, $c_1^3(X)$ is also bounded in terms of the Betti numbers of $X$ and the isomorphism type  of $(H^2_{tf}(X,\mathbb Z),F_X,p_1(X))$.
This proves the theorem.
\end{proof}

\subsection{Explicit bounds for $c_1^3$ of non-uniruled K\"ahler threefolds}

Using suitable examples of $\mathbb P^1$-bundles over surfaces, Kotschick showed that for uniruled spin threefolds $X$, the Betti numbers do not bound $c_1^3(X)$, see (proof of) \cite[Theorem 4]{kotschick-advances}.
In contrast to that result, the following corollary of the proof of Theorem \ref{thm:w2=0} shows that such a boundedness result is true if we restrict to non-uniruled spin threefolds.

\begin{corollary} \label{cor:ineq}
Let $X$ be a smooth K\"ahler threefold with spin structure.
If $X$ is not uniruled, then
$$
0\geq c_1^3(X)\geq \min \left(64\chi(X,\mathcal O_{X}) -8b_2(X)+8,-8b_2(X)+8\right) .
$$
\end{corollary}

\begin{proof}[Proof of Corollary \ref{cor:ineq}]
We use the notation from the proof of Theorem \ref{thm:w2=0}.
If $X$ is not of general type, then, by \cite[Corollary 1.4]{HP16}, the Kodaira dimension of $X$ satisfies $\operatorname{kod}(X)\in \{0,1,2\}$.
Hence, $Y$ is a minimal model which is not of general type and so $c_1^3(Y)=0$, see for instance \cite{CHP16}.
It then follows from (\ref{eq:c_1^3(X)}) that $c_1^3(X)=-8r$ for some $0\leq r \leq \rho(X)-1$.

Conversely, if $X$ is of general type, then $Y$ is a minimal model of general type and so the Miyaoka--Yau inequality yields $c_1^3(Y)\geq\frac{8}{3}c_1c_2(Y) $, see \cite{tsuji,zhang}.
Again by (\ref{eq:c_1^3(X)}),
\begin{align*}
c_1^3(X)=c_1^3(Y)-8r \geq \frac{8}{3}c_1c_2(Y) -8r .
\end{align*}
The corollary follows therefore from $c_1c_2(X)=c_1c_2(Y)$ and $0\leq r\leq b_2(X)-1$.
\end{proof}

\begin{remark} \label{rem:thm:ineq:2}
Corollary \ref{cor:ineq} shows that non-uniruled K\"ahler threefolds $X$ with spin structure satisfy $K_X^3\geq 0$.
This fails without the spin assumption.
For instance, if $C$ and $E$ denote smooth projective curves of genus $g$ and $1$, then $X=Bl_{\Delta_C\times 0}(C\times C\times E)$ satisfies $K_X^3=-4g+10$, which is negative for $g\geq 3$.
 \end{remark}

\section{Examples of unbounded Chern classes and deformation types} \label{sec:unbounded}

By Theorem \ref{thm:deformations}, the deformation types of K\"ahler structures of general type on a given spin $6$-manifold are bounded;
by Theorem \ref{thm:w2=0}, the same holds for the Chern numbers of arbitrary K\"ahler structures.
In this section, we show that both results are sharp.

\begin{proposition} \label{prop:dolgachev}
There is a simply connected spin $6$-manifold $M$,
which  admits a sequence of K\"ahler structures $X_i$, such that $c_1(X_i)\in H^2(M,\mathbb Z)$ is unbounded.
\end{proposition}

\begin{proof}
Let $q\geq 3$ be an odd integer.
As in \cite{schreieder-tasin}, we consider the Dolgachev surface $S_q$, which is the elliptic surface, obtained from a general pencil of plane cubic curves $S\longrightarrow \mathbb P^1$ by a logarithmic transformation of order $2$ and $q$ at two smooth fibres.
There is an h-cobordism $W_q$ between $S_q$ and $S_3$ which induces isomorphisms $H^2(S_q,\mathbb Z)\cong H^2(S_3,\mathbb Z)$.

We choose a cohomology class $\omega\in H^2(S_3,\mathbb Z)$  with $\omega^2\neq 0$ and such that the mod $2$ reduction of $c_1(S_3)+\omega$ vanishes.
Since $S_q$ and $S_3$ are h-cobordant, $w_2(S_q)\in H^2(S_3,\mathbb Z \slash 2\mathbb Z)$ does not depend on $q$ and so the mod $2$ reduction of $c_1(S_q)+\omega$ vanishes for all $q$.
Since $h^{2,0}(S_q)=0$, there is a line bundle $L_q\in \operatorname{Pic}(S_q)$ with $c_1(L_q)=\omega$, see \cite[Section 2]{schreieder-tasin}.

Then the $\mathbb P^1$-bundle $X_q:=\mathbb P(L\oplus \mathcal O_{S_q})$ over $S_q$ is spin for all $q$.
Moreover, since $S_q$ is h-cobordant to $S_3$ and since $L\oplus \mathcal O_{S_q}$ extends as a complex vector bundle over the corresponding h-cobordism, it follows from the h-cobordism theorem that $X_q$ is diffeomorphic to $X_3$ for all $q$, cf. \cite{kotschick-JTOP,kotschick-advances,schreieder-tasin}.

From now on we think of $X_q$ as a complex projective structure on a fixed spin $6$-manifold $M$.
If $c_1(X_{q})\in H^2(M,\mathbb Z)$ is bounded, then, by Proposition \ref{prop:almostcx}, the almost complex structures underlying the $X_q$'s belong to finitely many homotopy classes.
Let us therefore assume that the almost complex structures which underly $X_{q_1}$ and $X_{q_2}$ are homotopic.
Then there is an isomorphism of cohomology algebras,
$$
\phi:H^\ast(X_{q_1},\mathbb Z)\longrightarrow H^\ast(X_{q_2},\mathbb Z) ,
$$
which respects the Chern classes.
We use the following isomorphisms
$$
H^2(X_{q_1},\mathbb Z)\cong H^2(S_3,\mathbb Z)\oplus y \mathbb Z\ \ \text{and}\ \ H^2(X_{q_2},\mathbb Z)\cong H^2(S_3,\mathbb Z)\oplus y\mathbb Z ,
$$
to identify the restriction of $\phi$ to degree two cohomology classes with an endomorphism of $H^2(S_3,\mathbb Z)\oplus y\mathbb Z$.
Using this identification, the ring structure is determined by $y^2=-\omega\cdot y$.
Since $\{\alpha\in H^2(S_3,\mathbb Z)\mid \alpha^2=\omega \alpha\}$ is an irreducible quadric, one easily proves that the elements of zero cube in $H^2(X_{q_i},\mathbb C)$ are given by the union of $H^2(S_3,\mathbb C)$ with an irreducible quadric.
In particular,
$
\phi(H^2(S_3,\mathbb Z))=H^2(S_3,\mathbb Z) $.
Since $\phi$ respects the ring structures, we deduce
\begin{align} \label{eq:phi:H^4}
\phi(H^4(S_{q_1}, \mathbb Z))=H^4(S_{q_2},\mathbb Z) ,
\end{align}
where we identify $H^4(S_{q},\mathbb Z)$ via pullback with a subgroup of $H^4(X_q,\mathbb Z)$.
By assumptions, $\phi$ respects the second Chern class
$
c_2(X_{q_i})=c_2(S_{q_i})+ c_1(S_{q_i})(2y+\omega)
$.
Since the Euler number of $S_q$ is independent of $q$, it follows from (\ref{eq:phi:H^4}) that
$$
\phi(c_1(S_{q_1})(2y+\omega))=c_1(S_{q_2})(2y+\omega).
$$
Multiplying this class with $y$ shows that it is non-zero.
Therefore, for $q_1$ fixed, $q_2$ is bounded, because $c_1(S_{q})$ is nonzero and divisible by $(q-2)$, see Proposition 3.7 in \cite[Section I.3]{friedman-morgan}.
This proves the proposition.
\end{proof}

\section*{Acknowledgments}
The first author is member of the SFB/TR 45.
During parts of this project, the second author was supported by the DFG Emmy Noether-Nachwuchsgruppe ``Gute Strukturen in der h\"oherdimensionalen birationalen Geometrie'' and thereby also member of the SFB/TR 45.
We are very grateful to the referee for carefully reading a previous version of the paper.
We thank D.\ Kotschick for detailed comments and P.\ Cascini, M.\ Land, E.\ Sernesi, R.\ Svaldi and B.\ Totaro for conversations.

redpajama_set_name: RedPajamaArXiv


Any experienced English teacher knows the drill: on the dreaded due date, students bring printed copies of their essays to class, where we collect them, take them home, jot inscrutable comments in the margins, bring them back to class, return them, and then watch students promptly toss them in the recycling bin on the way out of the room. The whole cycle borders on farce. Students pretend to spend many hours writing their papers, teachers pretend to spend many hours grading them, and we all pretend like repeating this process over and over again leads to something we in education like to call "student growth." But teachers can finally put an end to this exercise in futility, thanks to an unlikely hero sometimes condemned for its unrelenting pursuit of profit at the expense of the public good…Google.
Ever since I made my students aware that I could easily spot cursory revisions, I've noticed substantial improvement in the quality of their writing.
Google's online word processor, Google Docs, has entirely transformed the way teachers can support students through the writing process, from the very first brainstorming activity to the feedback provided to students on their final drafts.
To illustrate, I'll use my experience helping students write essays about the most beloved and/or despised staple of high school American literature courses, "The Great Gatsby." Let's start with a brainstorming session. At the beginning of a class, I ask my students to enter a response to the following prompt into a shared Google Doc: "When I think of 'The Great Gatsby,' one word that comes to mind is…." Five minutes later, the Google Doc projected on a large screen at the front of the room is filled with origins of essay topics: "disillusionment," "American Dream," "obsession," and "self-deception," to name a consistent few. Especially for weaker students, the collaborative nature of this exercise exposes them to ideas in a manner that the old practice of independent journaling cannot. This method also gives a voice to the introverted students in class whose often brilliant ideas now receive the attention they deserve.
Onto the composition process, where Google Doc's sharing function changes the nature of communication between teacher and student. Because each of my students writes a draft of their essay using a Google Doc that is shared with me, I can immediately reply to questions that students pose in the margins of their document, rather than waiting until students bring a printed copy to class a few days later. I can also intervene if I notice that a student begins an essay with something like, "In 'The Great Gatsby,' the plot is crucial to the novel, as it both impacts the beginning of the story and leads to its denooment[sic]." No need to make this poor kid spend hours writing a draft that will read like a Harvard Lampoon spoof. Sharing documents also increases student accountability for meeting deadlines, an advantage not to be dismissed, especially for younger students still learning time management skills.
Perhaps most dramatically transformed by the use of Google Docs is the revision process. An often unknown feature is something called a "revision history." With one click of a button, the revision history allows me to bring up a literally minute-by-minute account of all of the changes that were made to an essay. Because most students' perceptions of the revision process entail changing a few appearances of the word "but" to "however" and substituting "pestiferous" for "bad," this feature is invaluable as a way to help students self-reflect upon the evolution of their essay. Ever since I made my students aware that I could easily spot cursory revisions, I've noticed substantial improvement in the quality of their writing.
At the end of the writing cycle is the part when teachers do their best to provide feedback that will resonate with students and here again, Google Docs is a game changer. A Google Doc app called Kaizena allows me to provide oral commentary with my laptop's built-in microphone that students can listen to the very instant that I am done recording. Each one of my spoken comments corresponds with a highlighted phrase or section of the essay on the Google Doc, conveying feedback in a manner that makes my former method of scribbling opaque phrases like "needs more work here!" in the margins look totally antiquated. Using the recorder enables me to increase both the quantity and quality of my comments, and most importantly, my students report that they find this technique much more beneficial. They explain that hearing the respectful tone of a teacher's voice makes them less threatened when delivered constructive criticism. One student remarked that it felt like I was sitting right next to him in his room as he listened to me explain the strengths and weaknesses of his essay. A little creepy, yes, but the culmination of this process, quite revolutionary in my mind, sure beats a crumpled-up essay on its way to the recycling center.
This program aired on November 12, 2013. The audio for this program is not available.


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text:
Saudi Arabia documents by Year: 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000
CPJ calls on UN to investigate murder of journalist Jamal Khashoggi in Saudi consulate
CPJ calls on U.N. Secretary General António Guterres to request that the United Nations launch an investigation into the circumstances surrounding the murder of journalist Jamal Khashoggi at the Saudi Arabian consulate in Istanbul, Turkey.
Senate Foreign Relations Committee must keep up pressure over Khashoggi
CPJ writes to the leaders of the U.S. Senate Foreign Relations Committee, requesting that they ensure the Trump administration conducts a quick and thorough investigation into Jamal Khashoggi's killing, as required by the Magnitsky Act, and that they consider holding independent hearings on Saudi Arabia.
Saudi attorney general says Khashoggi was killed during fight in consulate
New York, October 19, 2018--Saudi state media today reported that the country's attorney general has confirmed prominent journalist and Washington Post columnist Jamal Khashoggi died in the country's Turkey consulate, The Associated Press reported. A statement by the attorney general said that Khashoggi was killed after a fight inside...
CPJ calls on Saudi Arabia to account for Jamal Khashoggi's whereabouts
New York, October 6, 2018--The Committee to Protect Journalists today called on Saudi Arabia to immediately account for the whereabouts of prominent Saudi journalist Jamal Khashoggi, who has not been seen since entering the Saudi consulate in Istanbul on October 2. Multiple news outlets reported today that Turkish authorities,...
CPJ expresses concern for Saudi journalist Jamal Khashoggi
October 3, 2018 11:17 AM ET
Beirut, October 3, 2018--The Committee to Protect Journalists is deeply concerned about Jamal Khashoggi, a Saudi journalist who has been living in self-imposed exile in the U.S. since 2017, and urges Saudi authorities to immediately disclose his whereabouts. Khashoggi, a columnist for The Washington Post and a former editor-in-chief...
CPJ Safety Advisory: Pegasus spyware used to target journalists, civil society
[EDITOR'S NOTE: See CPJ's updated safety advisory here https://cpj.org/2019/11/cpj-safety-advisory-journalist-targets-of-pegasus-.php.] In a report published on September 18, Citizen Lab said it had detected Pegasus, a spyware created for mobile devices, in over 45 countries. Pegasus, which transforms a cellphone into a mobile surveillance station, could have been deployed against a...
Saudi blogger Eman Al Nafjan detained
New York, May 22, 2018--The Committee to Protect Journalists today expressed concern over the detention of Eman Al Nafjan, a Saudi blogger who writes about women's issues, and urges the country's authorities to confirm her whereabouts and release her immediately....
Saudi journalist jailed for five years for insulting royal court
February 8, 2018 3:26 PM ET
New York, February 8, 2018--A specialized criminal court in Saudi Arabia sentenced al-Watan columnist Saleh al-Shehi to five years in prison for "insulting the royal court," and imposed a five-year travel ban that will go into effect on his release, according to a report published today in the privately-owned...
Saudi Arabian authorities arrest local journalist following critical commentary
January 5, 2018 3:03 PM ET
New York, January 5, 2018 -- Saudi Arabian authorities must immediately release Saleh al-Shehi from detention and stop arresting journalists who criticize the country's government, the Committee to Protect Journalists said today. Saudi security forces arrested al-Shehi, a columnist for the Saudi Arabian daily al-Watan, on January 3, according...
« 2017 | 2019 »


text:
Amy Schumer & Jennifer Lawrence Danced to "Uptown Girl" at Billy Joel's Chicago Concert
by Abby Schreiber
Last night, Billy Joel proved Taylor Swift isn't the only pop star who can get some A-list surprise guests onstage. During his concert at Chicago's Wrigley Field, he started in on his mega-hit "Uptown Girl" when, in a nod to one of the best Trainwreck scenes, Amy Schumer hopped onstage and began dancing around to the song only to be joined moments later by her future co-star, Jennifer Lawrence. The twosome, along with some other guests, started doing Rockettes-style high kicks and even danced atop Joel's piano. The whole thing was a damn delight. Watch it above.
New Rina Sawayama Will Make You Feel Like That Bitch
Dorian Electra Wants You to Write Their Next Banger
A Closer Look at Every Single Judy Blame Reference From Dior Men's Fall 2020


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MIAMI (CBSMiami) – A Missing Child Alert has been canceled for a two-year-old Bonita Springs boy.
Brodi Zuniga was reported missing late Thursday.
When the Florida Department of Law Enforcement issued the Missing Child Alert they said he may be in the company of 36-year-old Sergio Zuniga and they may be traveling in a dark blue 2005 Nissan Altima.
Just before 11:30 a.m. Friday, they announced that the child had been found and he was safe.


text:
Der Wintersportverein Sterzing / Vipiteno - Wipptal Broncos ist ein italienischer Eishockeyverein aus Sterzing, der momentan in der grenzübergreifenden Alpen Hockey Liga spielt und zuvor Mitglied der italienischen Serie A war. Die Weihenstephan Arena hat eine Kapazität von ca. 1700 Zuschauern.

Geschichte
Der Wintersportverein Sterzing wurde im Jahre 1948 gegründet. Mitte der 1990er-Jahre trennte man sich von den anderen Wintersportsektionen. Die Mannschaft spielte von 1997 bis 2002 in der Serie A. Der Verein spielte auch in der Alpenliga mit. In den Spielzeiten 2002/03 und 2003/04 schickte der Verein aus wirtschaftlichen Gründen keine Seniorenmannschaft in die Meisterschaft. Ab Herbst 2004 spielte Sterzing wieder in der Serie A2. Im Sommer 2010 versuchte der Verein WSV Sterzing Broncos den Aufstieg über den grünen Tisch in die erste italienische Liga (Serie A1). Geplant war dazu ein Abkommen mit dem EV Bozen 84, da der Verband den Broncos zunächst die Auflage erteilt hatte, einen Verein als Farmteam aus der Serie A2 zu finden. Der italienische Eishockeyverband (FISG) lehnte die Anfrage des Vereins schließlich am 9. Juli ab.

2011 stiegen die Broncos mit dem Meistertitel der Serie A2 in die Serie A auf, mussten aber bereits 2012 nach einer Saison im Oberhaus wieder in die Serie A2 zurück.

In der Saison 2016/17 gehörte der Verein zu den Gründungsmitgliedern der grenzübergreifenden Alps Hockey League.

Trainer
Trainerliste:

Bekannte ehemalige Spieler
 Tomáš Kucharčík
  Jan Alston
  Emanuel Viveiros
  Agris Saviels
  Clayton Beddoes
  Rod Hinks
  Jordan Owens
  Claude Boivin
  Oļegs Sorokins

WSV Broncos Wipptal C
Die Wipptal Broncos C sind die zweite Mannschaft der Broncos.

Weblinks
 Website des Wipptal Broncos

Einzelnachweise

Sterzing, HC
Sportverein in Südtirol

Club der Alps Hockey League

redpajama_set_name: RedPajamaWikipedia


Comfort foods are typically energy-dense, high fat and sweet, such as chocolate, ice cream, and French fries. They give distinctive pleasure or make us temporarily feel better. The followings describe 5 factors that can contribute to comfort eating.
1. Feel good. Eating food high in fat, sugar or salt activates the brain's reward system. For example, chocolate has a strong effect on mood, generally increasing pleasant feelings and reducing tension. Highly palatable foods activate the same brain regions of reward and pleasure that are active in drug addiction. Studies of obese individuals show that brain areas associated with drug reward are activated by anticipation and receipt of highly palatable foods (Avena et al., 2012).
2. Comfort food as self-medication. There seems to be a consistent connection between negative emotions and unhealthy foods. A phenomenon called emotional eating. In a bad mood, people are drawn to unhealthy foods (eating sugary and fatty) as a coping mechanism. In contrast, people in a positive mood tend to choose healthy foods (Gardner et al., 2014). However, a study (Mann, 2015) found that although the comfort foods provided some relief from negative moods, but so did the other foods and even receiving no foods at all. Furthermore, comfort foods can put you in bad a mood (Hendy, 2012). In fact, the link between consumption of comfort foods and increased negative moods (due to perhaps crash after a high-carb meal or guilt) lasted over a period of two days.
3. The need to belong. We tend to associate certain foods with members of our family, social gatherings, and people taking care of us (Thanksgiving holidays with family). So when we feel lonely we crave for these foods to give us comfort and security. That is, comfort food's power may lie primarily in the associations it calls to mind (memories of secure attachment). For example, for a college student away from home for a first time, comfort foods may serve as a reminder of family or other relational in times of stress or isolation. Evidence showed that people with positive family relationships were more likely than others to reach for comforting foods on the days that they felt lonely (Jordan et a., 2015).
4. Nostalgic eating. There is a strong link between scents and emotional memory. Smell of foods can evoke vivid and detailed emotional memories of our past (Reid, et al., 2014). Our learning history predisposes us to enjoy certain foods. For example, the scent of something deep-fried corn dogs may remind one's childhood at the state fair during summers. Because odor-evoked memories tend to be positive, the smell improves the mood and produces feelings of social connectedness.
5. Special occasion. We tend to go for special, often unhealthy, foods on celebratory occasions, like birthdays or Thanksgiving. The excuse reflects an underlying dilemma. From a momentary perspective, the indulgence is the best choice; but from a long-term perspective, healthy eating is the best choice. The ideal solution is to somehow do both. This is impossible, except in one situation. If the situation can be framed as the "last time," then the dilemma disappears, since the person can say to herself that a new and better life will begin tomorrow.
Avena, N. M., Gold, J. A., Kroll, C., & Gold, M. S. (2012). Further developments in the neurobiology of food and addiction: Update on the state of the science. Nutrition (Burbank, Los Angeles County, Calif.), 28(4), 341-343.
Gardner, Meryl P., et al. (2014), Better moods for better eating?: How mood influences food choice. Journal of Consumer Psychology. Vol 24, issue 3, pages 320-335.
Hendy H (2012), Which comes first in food–mood relationships, foods or moods? Volume 58, Issue 2, Pages 771–775.
Jordan D. Troisi, Shira Gabriel, Jaye L. Derrick, Alyssa Geisler. Threatened belonging and preference for comfort food among the securely attached. Appetite, 2015; 90: 58.
Mann, Tracy (2015). Secrets from the eating lab: the science weight loss, the myth of willpower, and why you should never diet again. Harper Wave.
Reid, C.A.; Green, J.D.; Wildschut, T.; Sedikides, C. Scent-evoked nostalgia (2015). Memory, 23, 157–166.
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