\begin{document}

	\begin{abstract}
		In this note we propose the generalization of the notion of a holomorphic contact structure on a manifold (smooth variety) to varieties with rational singularities and prove basic properties of such objects. Natural examples of \textit{singular contact varieties} come from the theory of nilpotent orbits: every projectivization of the closure of a nilpotent orbit in a semisimple Lie algebra satisfies our definition after normalization. We show the existence of stratification \textit{\`a la} Kaledin and that crepant morphism preserve contact structure, analogous to well-known theorems for symplectic singularities. Moreover, as an application of developed tools, we show that any singular projective contact threefold admits a resolution by a contact manifold of the form $\mathbb{P}(TS)$ for some ruled surface $S$.
	\end{abstract}
	\title{Singular contact varieties}
	\section{Introduction}
	\subsection{Background}
	\textit{Contact geometry is an odd-dimensional counterpart of symplectic geometry} is a slogan that is frequently employed when discussing contact structures on manifolds. In the complex algebraic setting, smooth projective varieties equipped with a holomorphic contact structure are researched for their own merit: they are uniruled, so to study them one can employ methods using rational curves and their contractions \cite{KPSW}. Nevertheless, their classification is still open in the prime Fano case and conjecturally (\cite{LeBS94}) the only smooth Fano contact varieties are projectivizations of minimal nilpotent orbits for simple Lie groups. \\
	In the meantime, the seminal paper of Beauville (\cite{Bea00}) marked the birth of the singular symplectic geometry, that has since then become an important field of research, see for example classical, although not very recent survey by Fu \cite{Fu05}. The fundamental examples of (singular) symplectic singularities come from the Lie theory: every coadjoint orbit of a complex semisimple Lie group is equipped with the Kostant-Kirillov symplectic form, and the normalizations of the closures of nilpotent orbits have symplectic singularities. Some properties of general symplectic varieties, like the existence of the canonical stratification can be easily described on those model examples.\\  
	On the other hand, there was little interest in defining and studying contact structures in singular setting: Campana and Flenner proposed a definition of a contact singularity \textit{\`a la} Beauville \cite{CF01} and Fu \cite{Fu06} studied the contact resolutions of projectivized nilpotent orbits. Therefore the purpose of this note it to propose the notion of a \textit{singular contact variety} in such a way that it encompasses the instances mentioned above and allows extending the classical construction of symplectization for smooth contact varieties.
	\subsection{Overview}
	As we have just indicated, we begin by giving our definition of a singular contact variety (Definition \ref{defn.contact}). It has contact singularities in the sense of \cite{CF01}, but our point of view is more global and takes into account the possible twist of the contact form. We motivate our definition by extending the classical construction of symplectization, i.e.~we present a correspondence between singular contact varieties and principal $\mathbb{C}^*$-bundles equipped with a homogeneous symplectic form (Theorem \ref{thm.symplectization}). This allows us to prove an analogue of canonical stratification that for symplectic varieties was obtained by Kaledin \cite{Kal06} (Theorem \ref{thm.contact.stratification}). \\
	Then we prove other properties of singular contact varieties that have analogues in the symplectic world (however our proofs do not rely on the constructed correspondence). Namely, we show that crepant morphisms preserve contact structure (Theorem \ref{thm.crepant.is.contact}) and provide a criterion for the existence of a contact structure on the quotient of a (singular) contact variety (Theorem \ref{thm.quotient}). We use this criterion to present some easy toric examples in the form of quotients of projective spaces (Examples \ref{exa.fav} and \ref{exa.p5}).  \\
	We also observe that a recent result of Cao and H{\"o}ring on integrability of differential forms on singular spaces \cite{CH22} in particular implies that projective singular contact varieties are uniruled, just like in the smooth case (Proposition \ref{prop.not.psef}). This is a crucial property that opens up possibilities for utilizing Mori-theoretic methods in the future work. \\
	Our final result and the application of the developed tools is to show that projective contact threefolds can be resolved by a projective contact manifold of the form $\mathbb{P}(TS)$, where $S$ is a ruled surface (Theorem \ref{thm.threefolds}).
	\subsection{Notation and conventions}
	All objects considered in this note are complex algebraic varieties and by symplectic (contact) structure we always mean a holomorphic one. Likewise, a \textit{manifold} for us is a smooth algebraic variety. For a line bundle $L$ we denote by $L^\bullet$ the total space of its dual with the image of the zero section removed, it is clearly a $\mathbb{C}^*$--principal bundle. Moreover we do not make a distinction between line bundles and Cartier divisors. For the projectivized bundles we follow the Grothendieck convention, i.e.~$\mathbb{P}(\mathcal{E})$ is a space parametrizing one dimensional quotients of $\mathcal{E}$.
	
\section{Definition of contact variety and symplectization}
	\begin{defn}\label{defn.contact}
		Contact variety is an algebraic variety $X$ over $\mathbb{C}$ of odd dimension $2n+1$ with rational singularities and a globally defined line bundle $L$ such that on the smooth locus $X_{sm}$ we have an exact sequence of vector bundles:
		\begin{displaymath}
			0 \rightarrow F \rightarrow TX_{reg} \xrightarrow{\vartheta} L_{|X_{reg}} \rightarrow 0
		\end{displaymath}
		which defines contact structure on $X_{sm}$, i.e. $d\vartheta : \bigwedge^2 F \rightarrow  L_{|X_{sm}}$ is nowhere degenerate. Equivalently one can demand that  $\vartheta \wedge (d \vartheta)^{\wedge n}$ as an element of $H^0(X_{sm}, \Omega^{2n+1}_{|X_{sm}} \otimes L^{n+1}_{|X_{sm}})$ has no zeroes. We will sometimes call $L$ from the definition \textit{contact line bundle}.
	\end{defn}
	It is clear that the singularities of such varieties are contact in the sense of \cite{CF01}. We will note some easy consequences of the given definition, which will be useful later:
	\begin{prop}\label{prop.cartier}
		$-K_X$ is a Cartier divisor and we have $\mathcal{O}(-K_X) = L^{\otimes (n+1)}$ in $\Pic(X)$. Consequently the singularities of $X$ are Gorenstein and canonical.
	\end{prop}
	\begin{proof}
		By an abuse of notation we will use the same symbol $L$ for the contact line bundle and the corresponding Weil divisor. On $X_{sm}$ we have an equality of classes of Weil divisors $(n+1)L_{|X_{sm}} = -K_{X_{sm}}$ given by $\vartheta \wedge (d \vartheta)^{\wedge n}$. By the normality of $X$ we can prolong this equality to whole $X$ by taking unique closures of both divisors. Then, because $L$ corresponds to a line bundle (Cartier divisor), so does the canonical class. Moreover, the singularities of $X$ are rational by definition, so they are in particular Cohen-Macaulay and Gorenstein. Finally, the pair $(X,0)$ is canonical by \cite[Cor. 5.24]{KM92}.
	\end{proof}
	There is a classical construction of symplectization, described for example in \cite[Thm. E.6]{Bu09} that associates to each contact manifold the total space of a principal $\mathbb{C}^*$-bundle equipped with a homogeneous symplectic form. We have defined contact varieties in such a way to preserve this correspondence in the singular setting. To be precise, we can show the following:
	\begin{thm}\label{thm.symplectization}
		Let $X$ be a contact variety with the contact line bundle $L$ and a twisted form $\vartheta$. Then $L^\bullet$ is a variety with symplectic singularities such that the symplectic form $\omega$, induced by $\vartheta$ is homogeneous of weight 1 with respect to the natural $\mathbb{C}^*$-action on $L^\bullet$. \\
		On the other hand, if $\mathbb{P}i: Y \rightarrow Z$ is a principal $\mathbb{C}^*$-bundle having the structure of symplectic variety such that the 2-form is homogeneous of weight 1, then $Z$ is a contact variety.
	\end{thm}
	\begin{proof}
		First recall that by \cite[Theorem 6]{Nam01} a normal variety is symplectic if and only if it has rational Gorenstein singularities and its smooth part admits a holomorphic symplectic form. For the first statement, observe that the singularities of $L^\bullet$ are rational and Gorenstein, as it is a locally trivial $\mathbb{C}^*$-bundle over $X$. Therefore it is enough to provide a homogeneous holomorphic symplectic form on $L^\bullet_{sm}$. But this space is nothing else that the principal bundle over the $X_{sm}$, so one can do it using the standard construction \cite[Section C.5]{Bu09}. \\
		For the second claim, $Z$ is a quotient of a variety having rational singularities by a reductive group, so it also has rational singularities by \cite{Bou87}. Moreover, there exists a globally defined line bundle on $Z$, obtained as the dual of the line bundle associated to $Y$. Analogously as before, $Z_{sm} = \mathbb{P}i (Y_{sm})$, so we construct a twisted form on $Z_{sm}$ using the construction for the smooth varieties \cite[Prop. C.16]{Bu09}.
	\end{proof}
	
	\section{Kaledin's stratification for contact varieties}
	The fundamental application of the correspondence presented above is to provide an analogue of Kaledin's stratification for symplectic varieties \cite{Kal06}. Namely, we have:
	\begin{thm}\label{thm.contact.stratification}
	Let $(X,L,\vartheta)$ be a contact variety. Then there exists a finite stratification $ X = X_0 \supset X_1 \supset ... \supset X_k$ such that:
	\begin{enumerate}
		\item $X_{i+1}$ is the singular part of $X_{i}$.
		\item The normalization of every irreducible component of each stratum is a contact variety.
	\end{enumerate}
\end{thm}
\begin{proof}
	Let $(L^\bullet, \omega)$ be the symplectization of $(X, L, \vartheta)$ and denote by $\mathbb{P}i$ the natural projection. Consider the symplectic stratification of $L^\bullet$ existing by \cite[Thm 2.3]{Kal06}. To prove our theorem it is enough to show that all the strata and their normalizations have a structure of a principal $\mathbb{C}^*$-bundle induced from the one on $L^\bullet$ and that the induced forms are homogeneous of weight $wt(\omega)$. \\
	First, observe that the singular locus of $L^\bullet$ must necessarily be preserved by the $\mathbb{C}^*$ action, so any of its components $Z$ still admits the structure of a principal $\mathbb{C}^*$-bundle. Then, the action can be lifted to the normalization $Z^n$. To see it, consider the diagram:
	\begin{center}
		\begin{tikzcd}
			\mathbb{C}^* \times Z^n \arrow[r, dashed] \arrow[d, "\id \times \eta"] & Z^n \arrow[d, "\eta"] \\
			\mathbb{C}^* \times Z \arrow[r] & Z,
		\end{tikzcd}
	\end{center}
	where $\eta$ is the normalization morphism and the lower horizontal arrow comes from the action. The existence of the dashed arrow comes from the universal property of normalization and it defines the $\mathbb{C}^*$ action on the normalization. Moreover, $Z^n$ also admits the structure of a principal $\mathbb{C}^*$-bundle, as it is equal to the pullback (via the normalization morphism of $\mathbb{P}i(Z)$) of the principal bundle on $\mathbb{P}i(Z)$. \\
	Now we need that the induced forms have the same weight as $\omega$. To that end, observe that in Kaledin's proof the symplectic forms on the strata agree with the globally defined Poisson structure. If we start with the symplectic form on $Y_{sm}$ which is homogeneous of weight $k$, then by \cite[Lemma D.15]{Bu09} the Poisson bracket is homogeneous of weight $-k$. Now, the bracket extends over the singular locus and it is still homogeneous of weight $-k$, so the induced forms on strata, which have to agree with the bracket have weight $k$.
\end{proof}
	\begin{cor}
		The codimensions of the strata are even. Moreover, if the singularities of contact variety are terminal, then $\codim(X_{sing}) \ge 4$.
	\end{cor}
		\section{Quotients}
	In this section we will consider an action of a (finite) group $G$ on a possibly singular contact variety $X$ preserving the contact distribution and our purpose is to determine when the quotient admits a contact structure. To that end, we need to take a closer look at the contact line bundle. In the smooth case, where the contact distribution is defined everywhere, this line bundle is automatically linearized, i.e.~there exists a canonical lift of the action of $G$ to the total space of $L$. Indeed, $L$ is defined as the quotient of $TX$ by $F$ and by the definition of $G$, both of those vector bundles come with a $G$-linearization. In the singular case we need a more subtle argument:
	\begin{prop}
		Let $(X,F,L)$ be a singular contact variety and $G$ a group of automorphisms preserving the contact distribution $F$. Then $L$ is equipped with a canonical $G$-linearization.
	\end{prop}
	\begin{proof}
		Consider the product $G \times X$ with two morphisms to $X$: $\alpha$ corresponding to the action and $\mathbb{P}i_2$ -- projection from the product. Then by \cite[Lemma 2.9]{Bri15} the bundle $L$ is linearized if and only if $\mathbb{P}i_2^* L \simeq \alpha^*L$. On the smooth locus we have a canonical action of $G$ on $L$, as it is a quotient of $TX_{sm}$ by $F$ there. Moreover, $G$ acts on $TX_{sm}$ preserving $F$, so we have the above mentioned isomorphism of pullback bundles on $X_{sm}$. As the singular locus has codimension at least 2 by normality, this isomorphism extends to the whole $X$. Indeed, on any open $U \subset X$ trivializing both bundles such isomorphism is given by a nowhere vanishing regular function $f$ defined on $U \cap X_{sm}$. Any such $f$ can be extended to a nowhere vanishing regular function on the whole $U$ by normality of $X$. The resulting linearization is canonical, as the unique extension of canonically defined linearization on the smooth locus.
	\end{proof}
	To formulate a statement about the quotients of contact varieties, we need to introduce the notion of the descent of a vector bundle.
	\begin{defn}
		We say that a vector bundle $\mathcal{E}$ on a $G$-variety $Y$ with a good quotient $\mathbb{P}i: X \rightarrow X/G$ \textit{descends} if there exists a vector bundle $\mathcal{E}'$ on $X/G$ such that we have a $G$-equivariant isomorphism $\mathcal{E} \simeq \mathbb{P}i^* \mathcal{E}'$.
	\end{defn}
	We are now ready to state and prove the following criterion:
	\begin{thm}\label{thm.quotient}
		Let $(X,F,L, \vartheta)$ be a contact variety of dimension $2n+1$ and $G$ a finite group of contactomorphisms. Then the quotient $X/G$ has the structure of a contact variety $(F',L', \vartheta')$ induced by the one on $X$ if and only if the contact line bundle $L$ descends to the quotient. 
	\end{thm}
	\begin{proof}
		To begin with, we recall the result of Boutot \cite[Corollaire]{Bou87} who showed that any quotient of a variety with rational singularities by a reductive group still has rational singularities. Denote by $Z_g$ the locus of points of $X$ fixed by $g \in G \setminus \{ \id \}$ and let $Z = \bigcup_{g \in G \setminus \{\id \}} Z_g$. Any component of $Z$ has codimension at least 2 as it has to be contact or isotropic.\\
		If $L$ descends to a line bundle $L'$, then consider the set $X_{sm} \setminus Z$. Over it, we can push down $\vartheta$ to the quotient. Then, as the complement of $\mathbb{P}i(X_{sm} \setminus Z)$ in $(X/G)_{sm}$ has codimension at least 2, we can extend the pushdown of $\vartheta$ to a twisted form $\vartheta'$ defined on the whole smooth locus of $X/G$ and nonzero everywhere. Analogously one can push down and extend $\vartheta \wedge d\vartheta^{\wedge n}$ to claim nondegeneracy of $\vartheta'$ wherever it is defined. Consequently, $X/G$ has a canonical divisor that is Cartier, and as the singularities are rational, we are done. \\
		Going the other way around, if $X/G$ is equipped with a compatible contact structure with a line bundle $L'$, then its pullback induces a contact structure on $X$ outside of the ramification locus, which has codimension at least 2, as it coincides with $Z$. As this induced structure coincides with the original one wherever it is defined, it follows that $\mathbb{P}i^* L' \simeq L$, so $L'$ is the descent of $L$.
	\end{proof}
	The question on the existence of the descent was completely answered by Kempf:
	\begin{lem}[Kempf's lemma, {\cite[Th\'eor\`eme 2.3]{DN89}}]
		Let $Y$ be an algebraic variety equipped with an algebraic action of a reductive group $G$ such that there exists a good quotient $\mathbb{P}i: Y \rightarrow Y/G$. Let $\mathcal{E}$ be a $G$-vector bundle. Then $\mathcal{E}$ descends to a vector bundle on the quotient (equal to $(\mathbb{P}i_* \mathcal{E})^G$) if and only if for all closed points $y \in Y$ such that the orbit $G\cdot y$ is closed, the stabilizer $G_y$ of $y$ acts trivially on the fiber $\mathcal{E}_y$.
	\end{lem}
	One of the consequences of the lemma is that the descent, if exists, is unique. Moreover, thanks to it we can claim a corollary that will be useful later on:
	\begin{cor}\label{cor.contact.quotient}
		Let $(X,F, L, \vartheta)$ be a smooth contact manifold, $G$ a finite group of contactomorphisms and $G_x$ a stabilizer of a point $x \in X$. Then the quotient $X/G$ is a contact variety if and only if $\forall_{x \in X} \forall_{g \in G_x} g^* \vartheta = \vartheta$.
	\end{cor}
	\begin{proof}
		First observe that every orbit is closed, as the group $G$ is finite. Let $x \in X$ be a point with a nontrivial stabilizer. Then $T_xX = F_x \oplus L_x$ by the contact exact sequence and $\vartheta$ is just a projection $\mathbb{P}i_2$ onto the second factor, so the stabilizer acts trivially on $L_x$ if and only if $g^* \vartheta = \vartheta$.
	\end{proof}
	\section{Resolutions of singularities}
	Now we turn to study of resolution of singularities of contact varieties. To begin with, we need to discuss the notion of sheaf reflexivity. A coherent sheaf is \textit{reflexive} if it is isomorphic to its second dual. Hartshorne showed (\cite[Prop. 1.6]{Ha80}) that a sheaf $\mathcal{F}$ on a normal and integral scheme $Y$ is reflexive if and only if it is torsion free and satisfies the Barth normality condition, i.e.~for any open $U \subset Y$ and closed $Z \subset U$ of codimension at least 2 the restriction $\mathcal{F}(U) \rightarrow \mathcal{F}(U \setminus Z)$ is bijective, that is sections defined outside of $Z$ posses a unique extension to it. To prove our main theorem, we need the following result concerning reflexivity of differentials:
	\begin{thm}[{\cite[Thm 1.4]{GKKP11}}] \label{thm.reflexivity}
		Let $X$ be a quasi-projective variety with canonical singularities and $g: \widetilde{X} \rightarrow X$ a resolution of singularities. Then the sheaf $g_* \Omega^p_{\widetilde{X}}$ is reflexive for $p \le \dim X$.
	\end{thm}
	As the pushforward sheaf is defined over any open $U \subset X$ by $g_*\Omega^p_{\widetilde{X}}(U) = \Omega^p_{\widetilde{X}}(g^{-1}(U))$, then the reflexivity essentially means that if the codimension of the singular locus is at least 2, then any differential form on the smooth locus of $X$ extends uniquely to the everywhere defined differential form on the resolution. We are now ready to state and prove that crepant morphisms preserve contact structure:
	\begin{thm}\label{thm.crepant.is.contact}
		Let $(X,F,L, \vartheta)$ be a quasi-projective contact variety. Suppose that $f: X' \rightarrow X$ is a birational and crepant morphism such that $f (\Exc(f)) \subset X_{sing}$. Then $X'$ is again a contact variety with the distribution $F'$ such that $f_* F' = F$ on the smooth locus of $X$ and the contact line bundle is $f^*L$. In particular, a terminalization of a contact variety is contact and a crepant resolution of singularities produces a classical contact manifold.
	\end{thm}
	\begin{proof}
		Let $g \colon \widetilde{X} \rightarrow X'$ be a resolution of singularities of $X'$, which composed with $f$ becomes a resolution of singularities of $X$. As $\vartheta$ is defined on the smooth locus of $X$, it can be considered as a section of a sheaf $((fg)_*\Omega^1_{\widetilde{X}}) \otimes L$. This sheaf is reflexive, as it is a product of $(fg)_* \Omega^1_{\widetilde{X}}$ that is reflexive by Theorem \ref{thm.reflexivity} and a line bundle $L$. Moreover, by the projection formula we can identify this product sheaf with  $(fg)_*(\Omega^1_{\widetilde{X}} \otimes (fg)^* L)$. Now, as $X$ is normal, its singular locus has codimension at least 2, therefore $\vartheta$ extends to a (global) section of $\Omega^1_{\widetilde{X}} \otimes (fg)^*L$. As $\widetilde{X}$ is a resolution of singularities of $X'$, we have an isomorphism $\widetilde{X} \setminus \Exc(g) \simeq X_{sm}'$, which allows us to define $\vartheta'$ on the smooth locus of $X'$. \\
		Now recall that $\vartheta \wedge (d\vartheta)^{\wedge n}$ can be extended to the whole $X$ and it is a nowhere vanishing section of $\mathcal{O}(K_X) \otimes L^{\otimes (n+1)}$. We pull back this section to $X'$, where it defines an isomorphism between $\mathcal{O}(-K_{X'})$ and $f^*L^{\otimes (n+1)}$. Moreover, on the intersection of $X_{sm}'$ and any open $U \subset X'$ trivializing $f^*L$ this section agrees with $\vartheta' \wedge (d\vartheta')^{\wedge n}$. \\
		Finally observe that $\vartheta'$ is surjective on $X_{sm}'$. If this were not the case at some point $x \in X_{sm}'$, then $\vartheta' \wedge (d\vartheta')^{\wedge n}$ would be zero at $x$, which is absurd. 
	\end{proof}
	We should also expect that the converse also holds, i.e.~that if the resolution of a contact variety is a contact manifold with compatible contact structure, then the morphism is crepant. In the projective case we know by \cite[Lemma 2.10]{KPSW} that contact manifolds do not admit birational Mori contractions, so the resolution morphism would have to be either crepant or contract some curves that are positive with respect to the canonical divisor. Consequently, one should rule out this last possibility, but the author does not have an argument for that.
	\section{Examples -- projectivized orbits and toric quotients}
	As we have indicated in the introduction, our notion of a contact variety encompasses projectivized orbits:
	\begin{prop}
		For a semisimple group $G$ with a Lie algebra $\mathfrak{g}$ consider a nilpotent orbit $\mathcal{O} \subset \mathfrak{g}^*$ and its projectivization $\mathbb{P}(\mathcal{O}) \subset \mathbb{P}(\mathfrak{g}^*)$. Then the normalization of its closure $\overline{\mathbb{P}(\mathcal{O})}^n$ is a contact variety. 
	\end{prop}
	\begin{proof}
		Recall that the normalization of a nilpotent orbit closure for a semisimple Lie group is a symplectic variety for a Kostant-Kirillov 2-form, as observed in \cite[2.5]{Bea00a}. Moreover, every nilpotent orbit is preserved by a natural $\mathbb{C}^*$-action on $\mathfrak{g}^* \setminus \{ 0\}$ and the symplectic form is homogeneous of weight 1. Therefore, as the normalization commutes with the projectivization, we are in the situation of Theorem \ref{thm.symplectization} and conclude that the quotient by the $\mathbb{C}^*$-action, i.e.~the projectivization admits a contact structure. Note that the contact line bundle comes from the restriction of $\mathcal{O}_{\mathbb{P}(\mathfrak{g}^*)}(1)$.
	\end{proof}
	
	\begin{exa}[Author's favourite example]\label{exa.fav}
		 Start with an affine space with fixed even dimension $\mathbb{C}^{2n+2}$ and a symplectic form: $dx_0 \wedge dx_{n+1} + ... + dx_n \wedge dx_{2n+1}$. Consider a finite group $\widehat{G}$ of symplectomorphisms generated by $\xi_i$ for $i=1,...,n$, where $\xi_i$ acts on a vector by multiplying its $i$-th and $i+n+1$-th coordinate by $(-1)$, i.e.:
		\begin{displaymath}
			\xi_i (x_0,...,x_{2n+1}) = (x_0,...,x_{i-1}, -x_i,x_{i+1},...,x_{i+n}, -x_{i+n+1},x_{i+n+2},...,x_{2n+1}).
		\end{displaymath}
		Note that $\widehat{G}$ is a subgroup of $T^{2n+2}$, the torus giving $\mathbb{C}^{2n+2}$ the structure of a toric variety. Now take the associated projective space $\mathbb{P}^{2n+1}$ equipped with an induced contact form $\vartheta = \sum_{i=0}^{n+1} (x_i dx_{i+n+1} - x_{i+n+1}dx_i)$. Observe that action of $\widehat{G}$ descends to the projective space, it is a subgroup of the torus $T^{2n+1} \subset \mathbb{P}^{2n+1}$ and it preserves the twisted form. Denote the resulting group of contactomorphisms by $G \simeq \mathbb{Z}_2^n$ and consider the quotient $X = \mathbb{P}^{2n+1}/G$, which by Corollary \ref{cor.contact.quotient} is a contact variety. \\
		To describe $X$ explicitly as a toric variety, let $M = \Hom(T^{2n+1}, \mathbb{C}^*)$ be the lattice of characters and $N \simeq \mathbb{Z}^{2n+1}$ the dual lattice spanned by $e_1,...,e_{2n+1}$. Then $\mathbb{P}^{2n+1}$ is described by $\Sigma \subset N_{\mathbb{R}}$. It is a complete fan having $\rho_1 = e_1,...,\rho_{2n+1} = e_{2n+1}, \rho_0 = -e_1 -... - e_{2n+1}$ as rays and such that every subset of rays is a cone. Now, finite subgroups of torus correspond bijectively to finite index sublattices $\iota: M' \hookrightarrow M$, so dually we have an inclusion $N \rightarrow N'$ with a finite cokernel (equal to $G$), so that $N_\mathbb{R} = N'_\mathbb{R}$. Therefore, to obtain the fan $\Sigma'$ of the quotient variety take the image of $\Sigma$ via the adjoint $\mathbb{P}i$ of the inclusion $\iota$. Our choice of generators of $M'$ will be: $w_i = e^*_i + e^*_{i+ n+1}$ and $w_{n+i+1} = e^*_i - e^*_{i+n+1}$ for$n =1,...,n$ and $w_{n+1} = e^*_{n+1}$, so that the matrix of $\iota$ is symmetric and in consequence it also describes $\mathbb{P}i$.\\
		The fan of $X$ is spanned by the following $2n+2$ rays: $\rho'_i = e_i + e_{i+ n+1}$ and $\rho'_{n+i+1} = e_i - e_{i+n+1}$ for $n =1,...,n$, $\rho'_{n+1} = e_{n+1}$ and $\rho'_0 = -2 \cdot (e_1 +... + e_n) - e_{n+1}$ and any subset of rays forms a cone. We denote a cone of the form $\cone(\rho'_i, \rho'_{n+i+1})$ for $i=0,...,n$ (mind the case $i=0$) by $\sigma_i$. Every such cone is singular as its generators cannot be extended to a basis of a whole lattice, and it corresponds to a codimension 2 singular subvariety of $X$, that is the image of $\mathbb{P}^{2n-1} = \{[x_0:...:x_{2n+1}] | x_i = x_{i+n+1} = 0\}$ via the quotient map.\\
		Toric resolution of singularities are provided by particular refinements of the fans. In our case, where every singular cone contains some $\sigma_i$ as a subcone, we first define $n+1$ new rays $\rho'_{E_i} = \frac{1}{2}(\rho'_i +\rho'_{i+n+1})$. Then we divide every $\sigma_i$ onto two cones: $\sigma'_i = \cone(\rho'_i, \rho'_{E_i})$ and $\sigma''_i = \cone(\rho'_{n+i+1}, \rho'_{E_i})$. We do analogously for every cone containing any $\sigma_i$ as a subcone. \\
		In this way we obtain a smooth projective toric variety $\widetilde{X}$ that maps onto $X$ and does not change the smooth locus, i.e.~a resolution of singularities. Moreover, one can compute that the pullback of $K_X$ is the canonical divisor of $\widetilde{X}$, so the resolution is crepant. Now, by \cite[Th\'eor\`eme, p.1]{Dr99} and Theorem~\ref{thm.crepant.is.contact} $\widetilde{X}$ has to be isomorphic to $\mathbb{P}(T(\mathbb{P}^1 \times ...\times \mathbb{P}^1))$. It can also be seen directly, as described in \cite[Prop. 7.3 and (7.6')]{Oda78}. We take the lattice $N'$ and project it by restricting to last $n+1$ coordinates. Then the image of $\Sigma'$ is a product fan for $n+1$ copies of $\mathbb{P}^1$, so $\widetilde{X}$ is equipped with a projection onto $\mathbb{P}^1 \times ... \times \mathbb{P}^1$. The kernel of the lattice projection contains a standard fan of $\mathbb{P}^{n}$, and this is a fiber of this projection. Toric computations show that this bundle is in fact the projectivization of $\mathcal{O}(2,...,2) = T\mathbb{P}^1 \times ... T\mathbb{P}^1$.
	\end{exa}
	The significance of the example above comes from the fact that it allows us to provide a link between $\mathbb{P}^{2n+1}$ and $\mathbb{P}(T(\mathbb{P}^1 \times ... \times \mathbb{P}^1))$, i.e. two disjoint families of contact manifolds via our notion of a contact variety. It is an interesting question whether other contact manifolds can be linked similarly. 
	\begin{exa}\label{exa.p5}
		Now take $X= \mathbb{P}^5$ with the contact and toric structure as described in the previous example and let $\xi$ be the generator of $\mathbb{Z}_2$. Assume that $\mathbb{Z}_2$ acts on $X$ by: $\xi \cdot [x_0:x_1:x_2:x_3:x_4:x_5] = [x_0:-x_1:x_2:x_3:-x_4:x_5]$ and consider the quotient of $X$ by this action. Then the fixed point locus has two components, described by $x_1 = x_4 = 0$ ($\mathbb{P}^3$) and $x_0 =x_2 =x_3= x_5 = 0$ ($\mathbb{P}^1$), which get mapped isomorphically onto two components of the singular locus in the quotient. The (toric) resolution is provided by two disjoint blow-ups centered at those components. One can check that the resulting smooth variety $\widetilde{X}$ has no chance of being contact, as its canonical divisor is not divisible by 3 in the class group. Nevertheless, the partial resolution obtained by blowing up just the bigger component is crepant, so it is another projective contact variety, call it $X'$. One can verify that $\widetilde{X}$ admits a twisted form that maps $T\widetilde{X}$ onto the pullback of the contact line bundle, but the nondegeneracy condition is satisfied on some open subset (equal to the complement of the exceptional divisor). Such varieties are sometimes called \textit{generically contact}.
	\end{exa}
	\section{Existence of rational curves}
	It would be desirable to have a full classification of possible Mori contractions for singular contact varieties, analogous to the one showed in the smooth projective case in \cite[Thm 1.1]{KPSW}. Unfortunately it is currently out of reach, but we can recreate the initial steps. To begin with, Demailly proved some conditions that imply the integrability of a distribution defined as the kernel of the twisted form \cite[Main theorem]{Dem02}. The consequence of this result is that for a projective contact manifold the canonical divisor is not pseudoeffective \cite[Cor. 2]{Dem02}, so in turn such manifold is uniruled by \cite[Thm 2.2]{BDPP13}. The theorem of Demailly was very recently generalized by Cao and H\"oring \cite{CH22}, so that we can claim:
	\begin{prop}\label{prop.not.psef}
		Let $(X,F,L,\vartheta)$ be a projective contact variety. Then $L^*$ is not pseudoeffective. Consequently, $X$ is uniruled and it admits a Mori contraction.
	\end{prop}
	\begin{proof}
		A projective contact variety is a normal compact K\"ahler space with klt singularities and a line bundle $L^*$ is a reflexive sheaf of rank one, so the assumptions of \cite[Thm 1.2]{CH22} are satisfied. In consequence, if $L^*$ were pseudoeffective, then the kernel $F$ of the twisted form $\vartheta \in H^0(X, (\Omega^1_X)^{**} \otimes L)$ would define a foliation, but it is absurd, as $F$ is non-integrable. \\
		As $L^*$ so also $K_X$ are not pseudoeffective, \cite[Thm 2.2]{BDPP13} implies that $X$ admits a moving family of rational curves intersecting negatively with $K_X$, so in particular the Mori cone has a nontrivial $K_X$-negative part.
	\end{proof}
	In the singular case the Cone Theorem \cite[Thm 3.7]{KM98} states that any curve $C$ spanning an extremal ray has bounded intersection with $-K_X$. It can be translated to the condition that $L \cdot C \in \{1,2,3\}$ for the contact line bundle $L$. Unfortunately we cannot further mimic the proof of \cite[Thm 1.1]{KPSW} as for the rational curves on singular varieties we cannot present the bound on dimension of the space of morphisms \cite[Ch. I, Thm 2.16]{Ko96} in terms of cohomology. Consequently, the locus-fiber inequality \cite[Ch. IV, 2.6.1]{Ko96} does not take an easily computable form so we cannot determine the locus of the contraction in this way.
	\section{Projective threefolds}\label{section.threefolds}
	The tools that we have developed so far allow us to claim the following in the case of projective contact threefolds:
	\begin{thm}\label{thm.threefolds}
		Let $(X,L)$ be a projective singular contact threefold.  Then $X$ has a crepant resolution of singularities of the form $\mathbb{P}(TS)$, where $S$ is a ruled surface.

\end{thm}
\begin{proof}
To begin with, we show that $X$ has a crepant resolution of singularities. Indeed, let us denote by $f\colon \widetilde{X} \rightarrow X$ the crepant terminalization of $X$, that exists by \cite[Cor. 1.4.4]{BCHM06} (the assumption are satisfied by Proposition \ref{prop.cartier}). It is a contact variety by Theorem \ref{thm.crepant.is.contact}. The singular locus of a terminal variety has codimension at least 3 by \cite[Lem. 1.3.1]{BS95} but on the other hand by the stratification theorem \ref{thm.contact.stratification} its codimension must be even, therefore it is empty and $\widetilde{X}$ already resolves all singularities and is crepant. As $\widetilde{X}$ is a projective contact manifold with $b_2 (\widetilde{X}) >1$, it must be isomorphic to $\mathbb{P}(TS)$ with the contact line bundle $\mathcal{O}_{\widetilde{X}}(1)$ for some smooth projective surface $S$ by \cite[Thm 1.1]{KPSW} and moreover $f^* L = \mathcal{O}_{\widetilde{X}}(1)$. \\
Now we will show that $S$ has to be a ruled surface. To that end, recall that every fiber of $f$ is covered by rational curves by \cite[Cor. 1.6]{HM07}. Denote by $E$ an irreducible (divisorial) component of the exceptional locus of $f$. Ee have a rational curve $C \subset E$ such that $C \cdot \xi = 0$, where $\xi$ is a class of $\mathcal{O}_{\widetilde{X}}(1)$, as the morphism contracting $C$ is crepant. Denote by $\gamma: \mathbb{P}^1 \rightarrow \mathbb{P}(TS)$ the normalization of $C$ and by $\mathbb{P}i: \mathbb{P}(TS) \rightarrow S$ the natural projection. Consider the relative Euler sequence (\cite[Lem. 2.5]{KPSW}):
\begin{displaymath}
	0 \rightarrow \Omega^1_{\mathbb{P}(TS)/S} \otimes \mathcal{O}_{\widetilde{X}}(1) \rightarrow \mathbb{P}i^* TS \rightarrow \mathcal{O}_{\widetilde{X}}(1) \rightarrow 0
\end{displaymath}
and pull it back via $\gamma$, remembering that every vector bundle on $\mathbb{P}^1$ splits and that $\mathcal{O}_{\widetilde{X}}(1)$ is trivial on $C$. We obtain:
\begin{displaymath}
	0 \rightarrow \mathcal{O}_{\mathbb{P}^1}(a) \rightarrow \mathcal{O}_{\mathbb{P}^1}(2) \oplus \mathcal{O}_{\mathbb{P}^1}(b) \rightarrow \mathcal{O}_{\mathbb{P}^1} \rightarrow 0,
\end{displaymath}
where the known summand in the splitting type of $\mathbb{P}i^* TS$ comes from the fact that $\mathbb{P}i_*(C)$ is a rational curve on $S$.	Now, any map from $\mathcal{O}_{\mathbb{P}^1}(2)$ to $\mathcal{O}_{\mathbb{P}^1}$ is necessarily zero, so we must have an epimorphism $\mathcal{O}_{\mathbb{P}^1}(b) \rightarrow \mathcal{O}_{\mathbb{P}^1}$, but this is only possible if $b =0$. Consequently, we have $\mathbb{P}i_*(C) \cdot K_S = -2$ and $(\mathbb{P}i_*(C))^2 = 0$, so $S$ is a ruled surface by classical MMP for surfaces, see for example \cite[Thm 1.28]{KM98}. Moreover, $E$ dominates $S$, as otherwise it would map to a curve with negative self-intersection.
\end{proof}
	
	

	

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