A complete derived invariant and silting theory for graded gentle algebras

Haibo Jin
Department of Mathematics, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany

Sibylle Schroll
Department of Mathematics, Universität zu Köln, Weyertal 86-90, 50931 Köln, Germany and
Institutt for matematiske fag, NTNU, N-7491 Trondheim, Norway

Zhengfang Wang
Institut für Algebra und Zahlentheorie, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

The first and second author were partially supported by  the DFG through the project SFB/TRR 191 Symplectic Structures
in Geometry, Algebra and Dynamics (Projektnummer 281071066–TRR 191).  The third-named author is supported by the DFG grant (WA 5157/1-1).

On behalf of all authors, the corresponding author states that there is no conflict of interest.

We show that among the derived equivalent classes of homologically smooth and proper graded gentle algebras there is only one class whose perfect derived category does not admit silting objects. This allows us to construct a family of examples where a pre-silting object cannot be completed into a silting object.

As another application we confirm a conjecture by Lekili and Polishchuk that the geometric invariants which they construct for homologically smooth and proper graded gentle algebras are a complete derived invariant.

§ INTRODUCTION

Since their introduction in [3], gentle algebras have been widely studied and occur in many areas of mathematics. For instance, it is shown in [14] that (graded) gentle algebras naturally appear as the endomorphism algebras of formal generators in  partially wrapped Fukaya categories associated to graded marked surfaces.
Later, an inverse construction is given in [23, 27], namely to any graded gentle algebra one may associate  a surface model, i.e. a graded marked surface together with a graded admissible dissection (or formal generator). From the perspective of representation theory, the surface models provide a convenient and geometric approach to study derived categories of graded gentle algebras. For instance, the indecomposable objects, morphisms and algebraic operations (e.g. mapping cones or mutations) between these objects correspond to some natural geometric objects and operations in the associated  surface models, see [12, 14, 15, 27, 30].

In this paper we apply the explicit geometric invariants of graded gentle algebras studied in [23] to give a complete classification of the graded gentle algebras whose perfect derived category admits silting objects and we give two applications of this result. Namely, we give a family of examples of pre-silting
objects that cannot be completed to silting objects, that is they are not partial silting objects. As a second application we give a positive answer to a conjecture by Lekili and Polishchuk <cit.> giving a complete derived invariant for graded gentle algebras in terms of their surface models.
For the ungraded case a complete derived invariant is given in [7, 26].

However, we point out that the difficulty in passing from the ungraded case to the graded case is that in the graded case there might not be a silting object in the perfect derived category whereas in the ungraded case the algebra itself is always silting (even tilting). It is important to note that silting objects give rise to graded admissible dissections in the surface models (see [27] and Proposition <ref>), which establishes a direct link between derived equivalences and the surface models of the corresponding graded gentle algebras.

In the following we state our three main results more precisely.

§.§ Existence of silting objects

The notion of silting object, introduced in [22] as a generalization of a tilting object, plays an important role in the study of the representation theory of algebras. It is one of the standard tools for studying triangulated categories and it enables us to control equivalences of triangulated categories (see, for example, [19, 20]). Silting theory has been shown to  have deep connections with Bridgeland stability conditions, cluster theory, torsion theory, and co-$t$-structures, see, for example,  [4, 6, 11, 16, 18, 29, 25].

Recall that an object $P$ in a triangulated category $\T$ is called pre-silting if

\Hom_{\T}(P, P[i])=0,\quad \text{for each $i> 0$},

and it is called silting if, moreover, $\T$ coincides with the thick subcategory  generated by $P$.   Silting objects appear naturally in many triangulated categories. For instance,  for a non-positively graded dg algebra $A$ (non-positive dg algebra), the perfect derived category $\per(A)$ admits $A$ itself as a silting object. More generally,  if an algebraic, idempotent complete, triangulated category $\T$ admits a silting object then $\T$ is triangle equivalent to $\per(A)$ for some non-positive dg algebra $A$ [19, 21]. On the other hand, it is difficult to determine whether or not $\per(A)$ admits a (pre-)silting object if $A$ is not non-positive, see  Remark <ref>.

The first result of this paper is to provide a comprehensive answer to the existence of silting objects in $\per(A)$ for a homologically smooth, proper graded gentle algebra $A$. To be more specific,  we show the following result.

[Theorem <ref>]
Let $A=\Bbbk Q/I$ be a homologically smooth and proper graded gentle algebra. Then $\per(A)$ admits a silting object if and only if $A$ is not isomorphic to a graded gentle algebra given by the graded quiver

\begin{align*}
\xymatrix{ & 1 \ar@<1pc>[r]^-{\za} \ar@<-1pc>[r]^-{\zg}& 2 \ar[l]_-{\zb}}
\end{align*}

with relations $\{\za\zb, \zb\zg  \}$ and satisfying  $|\za|+|\zb|=1$ and $|\zb|+|\zg|=1$.

The proof of Theorem <ref> uses the existence results of silting objects obtained in [10]. More precisely, it was shown in [10] that the only cases in which  $\per(A)$ might not admit  silting objects are  when  $A$ is derived equivalent to a graded gentle algebra of the form $A^{(n)}$ satisfying certain degree conditions, see Theorem <cit.> and Definition <ref> for more details. In order to give a complete answer for the existence of silting objects, it thus remaines to consider the class of graded gentle algebras of the form $A^{(n)}$.

Our approach to Theorem <ref> is based on the explicit description of  the surface models and the geometric invariants of the graded gentle algebras of  the form $A^{(n)}$, see Section <ref>.  In particular, the calculation of the explicit invariants characterising the homotopy class of the line field of the surface model given in [23] enables us to recognise the derived equivalence classes of the graded gentle algebras of type $A^{(n)}$. This plays a key role in all our results.

§.§ Examples of pre-silting objects which are  not partial silting

A pre-silting object $P$ is called partial silting if there exists $Q$ such that $P\op Q$ is silting. The question whether any pre-silting object in $\per(\Lambda)$, for  an (ungraded) finite-dimensional algebra $\Lambda$,  is partial silting has been of great interest.
It has been known to be true  for  particular classes of algebras, such as piecewise hereditary algebras [4, 9] and representation-finite symmetric algebras [5, 1] . Recently, [24] has given  a negative answer in the general case by giving a pre-silting object in the perfect derived category of a  finite dimensional gentle algebra which is not partial silting.

Combining  Theorem <ref>, with the complete description of the derived equivalence class of the algebras of type $A^{(n)}$ given  in Section <ref> and with the silting reduction of graded gentle algebras considered  in [10],
we give a whole family of examples that have pre-silting objects which are not partial silting in the setting of  finite dimensional algebras.
[Theorem <ref>]

Let $A$ be a zero-graded (ungraded)    gentle algebra of the form $A^{(n)}$, for $n >1$. Then  there exists a pre-silting object in $\per(A)$ which is not partial silting.

We remark  that  for the graded case, one can easily construct a graded gentle algebra $A$ (or more generally, a graded quadratic monomial algebra) along with an idempotent $e$ such that $eA$ is pre-silting in $\per(A)$ but not partial silting,  see Corollary <ref> and Example <ref>.

Furthermore,  we obtain  conditions under which a pre-silting object is partial silting, namely,  we show in Proposition <ref> that for a homologically smooth and proper graded gentle algebra any almost complete pre-silting object is partial silting.

§.§ A complete derived invariant of graded gentle algebras

The  second application of Theorem <ref> is to give a complete derived invariant for graded gentle algebras.
This invariant is given in terms of their geometric surface models. By the equivalences in [14] and [23] this  then also gives a derived invariant for partially wrapped Fukaya categories of graded surfaces with stops.

Classifying algebras up to derived equivalence is a difficult task. Even for small classes of algebras that are closed under derived equivalence, this is difficult and often additional information such as geometric invariants is necessary such as in the case of ungraded gentle algebras and Brauer graph algebras, see for example [7, 26, 28].  More precisely, in the case of ungraded gentle algebras, using the surface model of their derived category described in [27], a complete derived invariant is described in [7, 26]. In the case of graded gentle algebras while partial results exist [14, 23], the construction of a complete derived invariant is still open. More precisely, building on [14], for general graded gentle algebras, a sufficient condition for two algebras to be derived equivalent is given in [23] in terms of winding numbers of curves. However, the converse is still open and is conjectured to be true in <cit.>. We show that the conjecture holds by showing the following results.

[Theorem <ref>]

Let $A$ and $B$ be two homologically smooth and proper graded gentle algebras with associated surfaces models  $(S_{A}, M_{A}, \zD_{A}, \eta_{A})$ and
$(S_{B}, M_{B}, \zD_{B}, \eta_{B})$,  respectively. Then $A$ and $B$ are derived equivalent if and only if there exists an orientation preserving homeomorphism $\varphi: S_{A}\ra S_{B}$ such that $\varphi(M_{A})=M_{B}$  and $\varphi_{*}(\eta_{A})$ is homotopic to $\eta_{B}$.

As a consequence, we show that the geometric invariants of the surface models of graded gentle algebras considered in [23] are a complete derived invariant. More precisely, we have the following result, where we refer to Subsection  <ref> for more details and definitions.
[Corollary <ref>]
Let $A$ and $B$ be two homologically smooth and proper graded gentle algebras with associated surfaces models  $(S_{A}, M_{A}, \zD_{A}, \eta_{A})$ and $(S_{B}, M_{B}, \zD_{B}, \eta_{B})$,  respectively. Then $A$ and $B$ are derived equivalent if and only if $S_A$ and $S_B$ have the same number $b$ of boundary components and there exists a numbering of these boundary components such that for each $i=1,\dots, b$, one has

\[\#(M_{A}\cap\partial_{i} S_{A})= \#(M_{B}\cap\partial_{i} S_{B}),\]

\[w_{\eta_{A}}(\partial_{i} S_{A})=w_{\eta_{B}}(\partial_{i} S_{B}),  \]

and in addition,

* if $g(S_{A})=g(S_{B})=1$, then $\w{\mathcal A}(\eta_{A})=\w{\mathcal A}(\eta_{B})$;

* if $g(S_{A})=g(S_{B}) >1$, then $\zs(\eta_{A})=\zs(\eta_{B})$ and $\mathcal A(\eta_{A})=\mathcal A (\eta_{B})$ whenever the latter two invariants are defined.

We note that the invariants  $\sigma, \w{\mathcal A}, \mathcal A$ defined in Subsection <ref> are expressed in terms of the winding numbers of some non-separating simple closed curves.

§ CONVENTIONS

In this paper, all algebras will be assumed to be over a base field $\Bbbk$.
Arrows in a quiver are composed from left to right as follows: for arrows $a$ and $b$ we write $ab$ for the path from the source of $a$ to the target of $b$. However, we adopt the convention that maps are composed from right to left, that is if $f: X \to Y$ and $g: Y \to Z$ then $gf : X \to Z$.

For a dg algebra $A$,  we denote by $\per(A)$ the perfect derived category of $A$, which is the smallest triangulated full subcategory of the derived category $\D(A)$ generated by $A$ and which is closed under direct summands. We denote by $\D^{\bb}(A)$ the full subcategory of $\D(A)$ consisting of the objects $M$ whose total cohomology is finite-dimensional (that is, $\dim_{k}\bop_{n}\h^{n}(M)<\infty$).  If $\per (A)\subset \D^{\bb}(A)$, we say $A$ is proper. Note that a proper graded algebra (a dg algebra with trivial differential) is finite dimensional. If $A\in \per (A\ot_{k}A^{\rm op})$, we say $A$ is homologically smooth.
Note that if $A$ is homologically smooth and proper, we have $\per(A)=\D^{\rm b}(A)$. All (differential graded) modules considered here are right modules.

Let $\T$ be a triangulated $\Bbbk$-category and let $P\in \T$. We denote by $\thick(P)$ the smallest full  triangulated  subcategory of $\T$  containing  $P$  and closed under direct summands.

§ SURFACE MODELS AND GEOMETRIC INVARIANTS

In this section we recall the surface models and geometric invariants of graded gentle algebras based on [14, 23, 27, 7].

§.§ Graded gentle algebras and their surface model

In this subsection we briefly recall some results on the surface model for a graded gentle algebra.

We begin by recalling the definition of a graded gentle algebra.

A   graded gentle algebra  is a graded algebra of the form  $A= \Bbbk Q/I$  where $Q=(Q_0,Q_1)$ is a finite graded  quiver (i.e. each arrow is assigned to a degree) and $I$ an ideal  of $\Bbbk Q$ generated by paths of length two such that

* each vertex in $Q_0$ is the source of at most two arrows and the target of at most two arrows;

* for each arrow $\za$ in $Q_1$, there is at most one arrow $\zb$ such that  $0 \neq \za\zb\in I$ and at most one arrow $\zg$ such that  $0 \neq \zg\za\in I$. Furthermore,   there is at most one arrow $\zb'$ such that $\za\zb'\notin I$ and at most one arrow $\zg'$ such that $\zg'\za\notin I$.

A graded marked surface is a triple $(S,M, \eta)$ where

* $S$ is a compact oriented surface with nonempty boundary  $\partial S=\sqcup_{i=1}^{b}\partial_i S$ where each $\partial_i S$ is a connected boundary component;

* $M$ is a finite set of marked points  on $\partial S$ such that each $\partial_i S$ has at least one marked point.

* $\eta$ is a line field on $S$, i.e. a section of the projectivized tangent bundle $\mathbb{P}(T S)$. For our purpose $\eta$ is usually considered up to homotopy.

Throughout the paper, we fix the orientation of the surface, such that when drawing surfaces in the plane, the orientation of the surface becomes the clockwise orientation of the plane.

All curves in the surface $S$ are considered up to homotopy where in the case of  a curve that is not closed (like in Definition <ref> below) the homotopy is up to  fixing the endpoints. Furthermore, all  intersections of curves are  to be transversal.

Let $(S,M, \eta)$ be a graded marked surface.
An admissible arc is a non-contractible curve with endpoints in $M$.

Let $(S,M, \eta)$ be a marked surface.

* A graded admissible dissection $\zD$ of $ (S,M)$ is a collection of pairwise non-intersecting (in the interior of $S$)  admissible arcs  on $S$ such that these arcs cut $S$ into polygons each of which has exactly one boundary segment, see Figure <ref>.    We may assume that  $\eta$ is transverse to each of the arcs in $\Delta$ (this is always possible up to homotopy).

* Let $\ell_i$ and $\ell_j$ be two arcs in $\zD$ which share a common endpoint $q \in M$. An oriented angle $\alpha$ from $\ell_i$ to $\ell_j$ at $q$ is an embedded path $u_\alpha$ in a small neighbourhood of $q$ from $\ell_i$ and $\ell_j$ such that, when following the path, the point $q$ is on the right. We call an oriented angle $\alpha$ minimal if the corresponding path $u_\alpha$ does not cross any other arc in $\Delta$.

* The degree of an oriented angle $\alpha$ is defined as the winding number $w_{\eta}(u_\alpha)$ of the corresponding path $u_\alpha$. Note that $u_\alpha$ may be chosen, up to homotopy, such that $\eta$ is transverse to the endpoints of $u_{\alpha}$ and thus $w_{\eta}(u_\alpha)$ is well-defined.

From a graded admissible dissection $\zD$ of a graded marked surface $(S, M, \eta)$, we may construct a graded gentle algebra  $A(\zD)=\Bbbk Q(\zD)/I(\zD)$ where

* the vertices of $Q(\zD)$ are in one-to-one correspondence with the arcs in $\zD$;

* each minimal oriented angle from arcs $\ell_i$ to $\ell_j$ gives rise to an arrow $\za$ from the corresponding vertices $\ell_i$ to $\ell_j$ whose degree equals the degree of the oriented angle.

* The ideal $I(\zD)$ is generated by the length two paths of the form $\za\zb$  for all minimal angles $\za:\ell_i\rightarrow\ell_j$ and  $\zb:\ell_j\rightarrow\ell_k$  intersecting $\ell_j$ at the different ends of $\ell_j$. See Figure <ref> for all the possibilities to have relations.

Since we assume that every connected boundary component has at least one marked point, the associated graded gentle algebra is homologically smooth. Furthermore, since all marked points are on the boundary, the graded gentle algebra is proper.  See, for example, [14, 27].

[pattern=north west lines] (1,-.3) arc[start angle=0, end angle = 180, x radius = 1, y radius = .3];
[pattern=north west lines] (1,-.3) arc[start angle=0, end angle = 180, x radius = 1, y radius = .3];
[green, thick] (0,0) to (60:1.5);
[bend left, ->] (162:.6) to (60:.6);
at (46:.8) $\l_k$;
at (112:.7) $\zb$;

[green, thick] (0, 0) to (120:1.5);
[green, thick] (0, 0) ..controls (1.5,.5).. (3,0);
[bend left, ->] (120:.6) to (18.5:.6);
at (130:1) $\ell_i$;
at (1.5,0.1) $\ell_j$;
at (70:.7) $\za$;

[pattern=north west lines] (1,-.3) arc[start angle=0, end angle = 180, x radius = 1, y radius = .3];
[green, thick] (0, 0) to (140:2);
[green, thick] (0, 0) to (40:2);
[green, thick] (0, 0) to (110:1.5);
[green, thick] (0, 0) to (70:1.5);
[green, dashed] (70:1.5) ..controls (70:2) and (110:2).. (110:1.5);
[bend left, ->] (140:.6) to (110:.6);
[bend left, ->] (70:.6) to (40:.6);

at (150:1) $\ell_i$;
at (30:1) $\ell_k$;
at (120:1.3) $\ell_j$;
at (60:1.3) $\ell_j$;
at (130:.8) $\za$;
at (50:.8) $\zb$;

[pattern=north west lines] (1,-.3) arc[start angle=0, end angle = 180, x radius = 1, y radius = .3];
[green, thick] (0, 0) to (140:2);
[green, thick] (0, 0) to (40:2);
[green, thick] (0, 0) to (110:1);
[green, thick] (0, 0) to (70:1);
[green, dashed] (140:2) ..controls (120:2.5) and (60:2.5).. (40:2);
[bend left, ->] (140:.6) to (110:.6);
[bend left, ->] (70:.6) to (40:.6);

at (150:1) $\ell_j$;
at (30:1) $\ell_j$;
at (113:1.3) $\ell_k$;
at (67:1.3) $\ell_i$;
at (130:.8) $\zb$;
at (50:.8) $\za$;

The local pictures near the marked points for relations in gentle algebras. In the first figure, the two boundary components are not necessarily pairwise distinct. The last two figures may be viewed as a special case of the first one, where the two marked points coincide.

Let $A$ be a graded gentle algebra. We say that $(S, M, \eta, \zD)$ is a surface model of $A$ if $A(\zD)$ is isomorphic to $A$ as graded algebras.

Let $A$ and $A'$ be two graded gentle algebras. Let $(S, M, \eta, \zD)$ (resp. $(S', M', \eta', \zD')$) be a  surface model of $A$ (resp. $A'$). We say that the two surface models are isomorphic if there is an orientation preserving homeomorphism $\varphi\colon S \to S'$ such that $\varphi(M) = M'$ and $\varphi_*(\eta)$ is homotopic to $\eta'$. (Note that here we do not require that $\varphi_*(\Delta)$ is related to  $\Delta'$, compare with Proposition <ref> below.)

By [23, 27] any graded gentle algebra admits at least one surface model. In particular, the surface may be obtained by thickening the ribbon graph associated to the (ungraded) gentle algebra, see <cit.>.

Throughout this paper, we only consider homologically smooth and proper graded gentle algebras as in [23], where the explicit  geometric invariants are only given for this class of algebras.

§.§ Combinatorial geometric invariants of a line field

In this subsection, we recall from [23] the combinatorial invariants of a line field in a graded marked surface. For details see <cit.>.

Let $S$ be a compact oriented surface with boundary  $\partial S=\sqcup_{i=1}^{b}\partial_i S$. Let $\overline S$ be the closed surface obtained from $S$ by filling in each boundary component. The Poincaré duality for $\overline S$ induces a nondegenerate skew-symmetric pairing $\langle\cdot,\cdot\rangle$ on $\mathrm{H}_1(\overline S)\simeq \mathbb Z^{2g}$.   Fix a collection of simple closed curves $s_1, \dotsc, s_g, t_1, \dotsc, t_g$ on $S$ such that their homotopy classes form a symplectic  basis of  $\mathrm{H}_1(\overline S)$ such that $\langle s_i, s_j \rangle = 0 = \langle t_i, t_j \rangle$ and $\langle s_i, t_j \rangle = \delta_{i,j}$ for $1\leq i, j \leq g$.

For any line field $\eta$ the following sequence of integers

\begin{align}\label{align:allthewindingnumbers}
W_\eta:= \{w_{\eta}(\partial_1 S),\dotsb, w_{\eta}(\partial_d S), w_{\eta} (s_1), \dotsb, w_{\eta}(s_g), w_{\eta}(t_1),\dotsb, w_{\eta}(t_g)\}
\end{align}

contains enough information to recover the homotopy class of $\eta$.  For each line field $\eta$ we first define a $\mathbb Z/2\mathbb Z$-valued invariant $\sigma(\eta)$,  which indicates whether the line field comes from a vector field (i.e. $\sigma(\eta) = 0$)

\begin{align}\label{align:sigmainvariant}
\sigma(\eta) = \begin{cases}
0 & \text{if  all the integers in $W_\eta$ are even}\\
1 & \text{otherwise.}
\end{cases}
\end{align}

Depending on the genus $g(S)$ and the parity of $\sigma(\eta)$, we need more invariants to determine the homotopy class of $\eta$.

* When $g(S) = 0$, two line fields $\eta_1$ and $\eta_2$ on $S$ are homotopic if and only if $w_{\eta_1}( \partial_i S) = w_{\eta_2}(\partial_iS)$ for each $1\leq i \leq b$;

* When $g(S)=1$, we define

\begin{align}\label{align:wA}
\w{\mathcal A}(\eta):=\gcd(\{ w_{\eta}(\partial_{1}S)+2, \dots, w_{\eta}(\partial_{d}S)+2, w_{\eta}(s_1), w_{\eta}(t_1)\}).
\end{align}

Then $\eta_1$ and $\eta_2$ are homotopic if and only if $w_{\eta_1}( \partial_i S) = w_{\eta_2}(\partial_iS)$ for $1\leq i \leq b$ and $\widetilde A(\eta_1) = \widetilde A(\eta_2)$. We note, in particular, that $\gcd(m_1, \dotsc, m_k) = 0$ if and only if $m_i = 0$ for each $1\leq i \leq k$.

* When $g(S)> 1$, the line fields $\eta_1$ and $\eta_2$ on $S$ are homotopic if and only if $w_{\eta_1}( \partial_i S) = w_{\eta_2}(\partial_iS)$ for each $1\leq i \leq b$ and in addition any one of the following three conditions, which we will refer to as conditions ($\star$), occurs:

* $\sigma(\eta_1) = \sigma(\eta_2) =1$;

* $\sigma(\eta_1) = \sigma(\eta_2) = 0$ and there exists $1\leq i \leq b$ such that $w_{\eta_1}(\partial_i S) = w_{\eta_2}(\partial_i S) = 0 \ \mod 4$;

* $\sigma(\eta_1) = \sigma(\eta_2) = 0$, for each $1\leq i \leq b$ we have $w_{\eta_1}(\partial_i S) = w_{\eta_2}(\partial_i S) = 2 \ \mod 4$, and the Arf invariants for $\eta_1$ and $\eta_2$ coincide

\[
\mathcal A(\eta_1) = \mathcal A(\eta_2)
\]

where the Arf invariant is defined as follows

\begin{align}\label{align:Arfinvariant}
\mathcal A(\eta_1) := \sum_{i=1}^g \left (\frac{1}{2} w_{\eta_1}(s_i) +1\right) \left(\frac{1}{2} w_{\eta_1}(t_i) +1\right) \ \mod 2.
\end{align}

Let $(S_{1}, M_{1}, \eta_{1})$ and $(S_{2}, M_{2}, \eta_{2})$ be two graded surfaces  with the same number $b$ of boundary components. Then there exists an orientation preserving homeomorphism $\varphi: S_{1}\ra S_{2}$ such that $\varphi(M_{1})=M_{2}$ and $\varphi_{*}(\eta_{1})$ is homotopic to $\eta_{2}$ if and only if
there exists  a numbering of the boundary components of $S_{1}$ and $S_{2}$ such that for each $i=1,\dots, b$, one has

\[\#(M_{1}\cap\partial_{i} S_{1})= \#(M_{2}\cap\partial_{i} S_{2}),\]

\[w_{\eta_{1}}(\partial_{i} S_{1})=w_{\eta_{2}}(\partial_{i} S_{2}),  \]

and in addition,

* if $g(S_{1})=g(S_{2})=1$, then $\w{\mathcal A}(\eta_{1})=\w{\mathcal A}(\eta_{2})$;

* if $g(S_{1})=g(S_{2})>1$, then one of the three conditions ($\star$) as above occurs.

Note that the invariants $\sigma, \w{\mathcal A}$ and $\mathcal A$ are independent of the choice of the symplectic basis of $\mathrm H_1(\overline S)$ given by the simple closed curves $s_i, t_i$, see  <cit.> and [23]. We also note that if the  sequences $W_{\eta_1}$ and $W_{\eta_2}$ in (<ref>) associated to two line fields $\eta_1$ and $\eta_2$ in a surface $S$ coincide then by Theorem <ref> above $\eta_1$ and $\eta_2$ are homotopic. But the converse does not generally hold for $g(S)>0$.

§.§ The calculation of winding numbers via graded quivers

In this subsection, we calculate the winding number of any simple closed curve in a surface model of a graded gentle algebra, in terms of the grading of the algebra.  As a result, we show in Proposition <ref> that any graded gentle algebra admits a unique surface model, which is essentially due to [23, 27].

Let $A$ be a graded gentle algebra. Let $(S, M, \eta,\zD)$ be any surface model of $A$. Then the winding number $w_{\eta}(\gamma)$ of any simple closed curve $\gamma$ (not necessarily non-separating) may be expressed using the gradings of the arrows in $A$.

Recall that the graded  admissible dissection $\Delta$ cuts the surface into polygons

\begin{align} \label{align:cuttingpolygons}
S\setminus \Delta = \cup_{f} P_f
\end{align}

where (the closure) of each $P_f$ is a polygon which has exactly one boundary segment, see Figure <ref>.  Note that the polygons appearing in (<ref>) bijectively correspond to the forbidden threats in the sense of <cit.>.

Let $\gamma$ be a simple closed curve in $S$. We choose a representative of $\gamma$ so that it intersects with the admissible arcs in $\zD$ in minimal position. Let $\ell_1, \dotsb, \ell_k$ be those arcs in $\Delta$ crossed by $\gamma$ in order. Then they cut $\gamma$ into segments $\gamma_1, \dotsb, \gamma_k$, where $\gamma_i$ lies between $\ell_i$ and $\ell_{i+1}$. Here, the indices are considered $\mod k$. We have (see e.g. <cit.>)

\[
w_\eta(\gamma) = \sum_{i=1}^k w_\eta(\gamma_i).
\]

Each segment $\gamma_i$ lies in a polygon $P_f$ and cuts $P_f$ into two subpolygons. Denote by $\alpha_1, \dotsb, \alpha_m$ all the oriented angles in the subpolygon which does not contain the boundary segment.  Then we have

\begin{align}\label{align:windingnumbersegments}
w_\eta(\gamma_i) = \begin{cases}
1-m +\sum_{j=1}^m |\alpha_j|  & \text{if the boundary segment in $P_f$ lies on the left of $\gamma$}\\
-1+m - \sum_{j=1}^{m}|\alpha_j| &  \text{otherwise}
\end{cases}
\end{align}

Here, the assumption for the first (resp. second) equality means that the direction of $\gamma$ agrees with (resp. differs from) the orientation induced by the subpolygon which does not contain the boundary segment. In particular, we have $w_\eta(\gamma^{-1}) = - w_\eta(\gamma)$ where $\gamma^{-1}$ is the inverse of $\gamma$ given by $\gamma^{-1}(t) = \gamma(1-t)$, for $t \in [0,1]$.

Let us look at the example in Figure <ref>. According to  (<ref>) the winding numbers of the segments $\gamma_{i-1}, \gamma_i, \gamma_{i+1}$ may be expressed as follows

\begin{align*}
w_{\eta}(\gamma_{i-1}) &=2-|\za_1'|-|\za_2'|-|\za_3'|,\\
w_{\eta}(\gamma_i) &= -3 + |\za_1|+\dotsb + |\za_4|, \\
w_{\eta}(\gamma_{i+1}) &= 3 - |\za_1''| - \dotsb- |\za_4''|.
\end{align*}

in 0,1,2,3
[green, thick] (-9+6*, -1.5) to (-9+6*,1);

[green, thick] (-9,1) to (-6,3) to (-3,1) to (-1,3);
[green, thick] (1,3) to (3,1) to (5,3) to (7, 3) to (9, 1);
[green, thick] (-5,-3) to (-3,-1.5) to (-1,-3) to (1,-3) to (3,-1.5) to (5,-3);
[green, thick] (7, -3) to (9, -1.5);
[green, thick] (-9,-1.5) to (-7,-3);

[pattern=north west lines] (-1.3,3.1) to [curve through = (-1, 3).. (0, 2.8)..(1, 3)] (1.3,3.1);
[pattern=north west lines] (-7.3,-3.1) to [curve through = (-7, -3).. (-6,-2.8)..(-5, -3)] (-4.7,-3.1);
[pattern=north west lines] (4.7,-3.1) to [curve through = (5, -3).. (6,-2.8)..(7, -3)] (7.3,-3.1);

[thick, ] (-9.5,0) to (9.5,0);
[thick, ->] (-10.5,0) to (-9.45,0);
[thick, >-] (9.35,0) to (10.5,0);
at (-6,.3) $\gamma_{i-1}$;
at (0,.3) $\gamma_{i}$;
at (6,.3) $\gamma_{i+1}$;

[bend left, ->] (-8.5, 4/3) to (-9, .5);
at (-8.2,.7) $\alpha'_3$;
[bend left, ->] (-5.5, 8/3) to (-6.5, 8/3);
at (-6,2.2) $\alpha'_2$;
[bend left, ->] (-3,.5) to (-3.5, 4/3);
at (-3.7,.7) $\alpha'_1$;
[bend left, ->] (-3, -1) to (-2.5, -1.875);
at (-2.15,-1.35) $\alpha_1$;
[bend left, ->] (-1.5,-2.62 ) to (-.5,-3);
at (-.88,-2.4) $\alpha_2$;
[bend left, ->] (.5,-3) to (1.5,-2.625);
at (.75,-2.35) $\alpha_3$;
[bend left, ->] (2.5,-1.875) to (3,-1);
at (2.3,-1.3) $\alpha_4$;
[bend left, ->] (3.5, 1.5) to (3, .5);
at (3.7,.8) $\alpha''_4$;
[bend left, ->] (5.5, 3) to (4.5, 2.5);
at (5.3,2.4) $\alpha''_3$;
[bend left, ->] (7.5, 2.5) to (6.5, 3);
at (6.5,2.4) $\alpha''_2$;
[bend left, ->] (9,.5) to (8.5, 1.5);
at (8.3,.8) $\alpha''_1$;

at (-6,-2.8) ;
at (0,2.8);
at (6,-2.8) ;

at (-8.3,-.5) $\ell_{i-1}$;
at (-2.6,-.5) $\ell_{i}$;
at (3.7,-.5) $\ell_{i+1}$;
at (9.7,-.5) $\ell_{i+2}$;

A local picture of a  simple closed curve $\gamma$ crossing through polygons of a graded admissible dissection $\zD$.

Let $A=\Bbbk Q/I$ be a graded gentle algebra. Let $(S_{1}, M_{1}, \eta_{1},  \zD_{1})$ and $(S_{2}, M_{2}, \eta_{2}, \zD_{2})$ be two surface models of $A$. Then there exists an orientation preserving homeomorphism $\varphi: S_{1}\xrightarrow{\simeq} S_{2}$ such that $\varphi(M_{1})=M_{2}, \ \varphi(\zD_{1})=\zD_{2}$,  and $\varphi_{*}(\eta_{1})$ is homotopic to $\eta_{2}$.

Since by definition $A(\zD_1) \simeq A \simeq A(\zD_2)$ as graded algebras, it follows that the ribbon graphs $\Gamma_1$ and $\Gamma_2$ associated to $\Delta_1$ and $\Delta_2$ are isomorphic and  thus it induces an orientation preserving homeomorphism $\varphi: S_1 \to S_2$ such that $\varphi(M_1) = M_2$ and $\varphi(\Delta_1) = \Delta_2$;  see e.g. <cit.>. (Note that the vertices in $\Gamma_i$, for $i =1,2$ are given by marked points and the edges are given by the admissible arcs in $\zD_i$.)

It remains to show that $\varphi_*(\eta_1)$ is homotopic to $\eta_2$. For this, by Remark <ref> it suffices to show that for any simple closed curve $\gamma$ in $S_2$ we have  $w_{\eta_2}(\gamma) = w_{\varphi_*(\eta_1)} ( \gamma)$. Note that $w_{\varphi_*(\eta_1)} (\gamma) = w_{\eta_1}(\varphi^{-1}(\gamma))$ as winding numbers are preserved under homeomorphisms, see e.g. [30]. Therefore, we need to show that for any simple closed curve $\gamma$ in $S_2$ we have

\begin{align}\label{align:toshow}
w_{\eta_2}(\gamma) = w_{\eta_1}(\varphi^{-1}(\gamma)).
\end{align}

To show this we may assume that $\gamma$ (after choosing a representative) intersects with the arcs in $\Delta$ in minimal position. Assume that $\gamma$ crosses  the polygons $P_1, \dotsc, P_k$ and is cut into segments $\gamma_1, \dotsc, \gamma_k$. Then $w_{\eta_2}(\gamma) = \sum_{i=1}^k w_{\eta_2}(\gamma_i)$. Since $\varphi$ is a homeomorphism and compatible with the dissections, it follows that $\varphi^{-1}(P_1), \dotsc, \varphi^{-1}(P_k)$ are the polygons of the dissection $\zD_1$ which intersect $\varphi^{-1}(\gamma)$. Note that, for each $1\leq i \leq k$,  the corresponding oriented angles in $P_i$ and $\varphi^{-1}(P_i)$ give rises to  the same arrows in the algebra $A$. Then by (<ref>) we obtain that $w_{\eta_2}(\gamma_i)$ and $w_{\eta_1}(\varphi^{-1}(\gamma_i))$ may be expressed by the same formula as in (<ref>) so they are equal, which shows (<ref>).

By Proposition <ref>, any graded gentle algebra admits a unique surface model, up to isomorphism.  We will see in Theorem <ref> below that this statement still holds if one replaces up to isomorphism' by up to derived equivalence'.

§ THE GEOMETRIC INVARIANTS OF THE GRADED GENTLE ALGEBRAS $A^{(N)}$

In this section, we study the surface model of a special class of graded gentle algebras of the form $A^{(n)}$ for each $n \geq 1$, which was introduced in [10] and plays an important role in the study of the existence of silting objects.

Let $Q^{(n)}$ be the following quiver

\begin{equation}
\label{equ:An}
\xymatrix{1 \ar@<1pc>[r]^-{\za_{1}} \ar@<-1pc>[r]^-{\zg_{1}}& 2 \ar[l]_-{\zb_{1}} \ar[r]^-{\zd_{1}}& 3 \ar@<1pc>[r]^-{\za_{2}} \ar@<-1pc>[r]^-{\zg_{2}} & 4 \ar[l]_-{\zb_{2}} \ar[r]^-{\zd_{2}} & \cdots \ar[r]^-{\zd_{n-1}} & 2n-1 \ar@<1pc>[r]^-{\za_{n}}  \ar@<-1pc>[r]^-{\zg_{n}}
& 2n \ar[l]_-{\zb_{n}}}
\end{equation}

and let $I^{(n)}$ be the ideal generated by   $
\{\za_{i}\zb_{i}, \zb_{i}\zg_{i}, \zg_{j}\zd_{j}, \zd_{j}\za_{j+1} \mid 1\le i\le n, 1\le j\le n-1\}.
We say that a graded gentle algebra $A = \Bbbk Q/I$ is of the form $A^{(n)}$ if $Q=Q^{(n)}$ and $I=I^{(n)}$. In particular, the difference between two algebras of the same form lies in their gradings.

§.§ The canonical surface model of $A^{(n)}$

In this subsection we calculate the geometric invariants considered in Theorem <ref> for any graded gentle algebra of the form $A^{(n)}$.

Let $A$ be a graded gentle algebra of the form $A^{(n)}$. Then by the general construction of surface models in [27, 23], the algebra  $A$ admits a surface model $(S, M, \eta, \Delta)$, where

* the surface $S$ is of genus $g(S) =n$ with exactly one boundary component (i.e. $b=1$) and with one marked point on the boundary;

* The graded admissible dissection $\Delta$ is illustrated in Figure <ref>;

* The line field $\eta$ is determined by the grading of $A$. (Recall that $\eta$ may be chosen so that it is transversal to the arcs in $\Delta$.)

We refer to the surface model above as the canonical surface model of $A^{(n)}$. The gentle algebras of the form $A^{(n)}$ may be viewed as the minimal' gentle algebras with genus $n$. Let $s_i, t_i$ be the natural non-separating simple closed curves around each genus hole for $1 \le i \le n$ (see Figure <ref>), which induce a symplectic basis of $\mathrm{H}_1(\overline S)$ as in Subsection <ref>.

Note that the dissection $\Delta$ cuts the surface $S$ into only one polygon. Then by (<ref>) (also see  <cit.>) the winding numbers of the natural non-separating simple closed curves $s_i$ and $t_i$ are given by

\begin{equation} \label{eq:windings_i}w_\eta(s_i) = |\za_i| + |\zb_i| -1\\
\mbox{ \, and \,
} \\
w_\eta(t_i) = |\zb_i| + |\zg_i| -1.
\end{equation}

By the Poincaré-Hopf index theorem, the winding number of the boundary equals $w_{\eta}(\partial S) = 2-4n$.

We now give the detailed calculation for the case $n=2$, see Figure <ref>.
From the figure on the left, by (<ref>) we have    $w_{\eta}(s_i) =|\alpha_i|+|\beta_i|-1$ and $w_{\eta}(t_i) = |\beta_i|+|\gamma_i|-1$ for $i=1,2$. From the figure on the right we obtain $w_\eta(\partial S) = -6$ since a simple closed curve $c$ homotopic to the boundary (in the neighborhood of the boundary) is cut by the dissection into segments $c_1, c_2, \dotsb, c_8$ in order. By (<ref>) we have
$$w_\eta(c_8) = -6+|\alpha_1|+|\beta_1|+|\gamma_1|+|\delta_1|+|\alpha_2|+|\beta_2|+|\gamma_2|.$$
Note that the inverse $c_1^{-1}$ of $c_1$ is an oriented angle from $l_1$ to $l_2$ and thus $w_\eta(c_1) =-w_{\eta}(c_1^{-1}) = -|\gamma_1|$. Similarly, we have $w_\eta(c_2) = -|\beta_1|, \dotsc, w_\eta(c_7)=-|\alpha_2|$. So $w_\eta(c) = \sum_{i=1}^8 w_\eta(c_i) = -6$, which satisfies the Poincaré–Hopf index theorem.

§.§ The geometric invariants of $A^{(n)}$

Inspired by (<ref>) we  denote
\begin{align}\label{align:numbersofthesumgradingsAn}
a_{i}:=|\za_{i}|+|\zb_{i}| \quad \text{and} \quad b_{i}:=|\zb_{i}|+|\zg_{i}| \quad \text{for each $1\le i\le n$.}
\end{align}

Then the invariant $\w{\mathcal A}(\eta)$ and the Arf invariant $\mathcal A(\eta)$ may be calculated using $a_i, b_i$. Precisely, in the case of  genus $g(S) =1$, we have

\begin{align}\label{align:wAA1}
\w{\mathcal A}(\eta)=\gcd(\{w_{\eta}(s_1), w_{\eta}(t_1)\})=\gcd\{a_{1}-1, b_{1}-1\}.
\end{align}

In the case of $g(S)>1$.  Note that since $w_\eta(\partial S)=2-4g(S)$, the equality $w_\eta(\partial S) = 2 \ \mod 4$ always holds. If there exists $1\leq i \leq n$ such that $a_ib_i$ is even then $\sigma(\eta) = 1$. Otherwise, $\sigma(\eta) = 0$ and
the Arf invariant $\mathcal A(\eta)$ is given by

\[\mathcal A(\eta) :=
\sum_{i=1}^n (\frac{1}{2}w_\eta(s_i) +1) (\frac{1}{2}w_\eta(t_i) +1) = \sum_{i=1}^n \frac{1}{4} (a_i
+1) (b_i +1)\ \mod 2
\]

[top color= blue!15] (-30:9.5) arc(-30:210:9.5);
[pattern=north west lines] (0,-.5) circle (.5);

[fill,circle, green, inner sep=0pt, minimum size=5pt] at (0,0);
[thick] (210:9.5) ..controls (-6.6,-7).. (-4, -4.75);
[dashed] (210:9.5) ..controls (-6.5,-5).. (-4, -4.75);
[dashed] (210:9.5) ..controls (-6.5,-4.5).. (-4, -4.75);
[thick] (-30:9.5) ..controls (6.6,-7).. (4, -4.75);
[dashed] (-30:9.5) ..controls (6.5,-5).. (4, -4.75);
[dashed] (-30:9.5) ..controls (6.5,-4.5).. (4, -4.75);
[thick] (-4, -4.75) ..controls (0,-1.8).. (4, -4.75);

[thick,green] (0,0) ..controls (-2.5, 8.5) .. (0,8.5);
[thick,green] (0,0) ..controls (2.5, 8.5) .. (0,8.5);
[thick,gray] (0,6.5) ..controls (-.5,6.4) and (-.5, 9.6) .. (0,9.5);
at (94:9)  $s_1$;
[thick,dashed,gray] (0,6.5) ..controls (.5,6.4) and (.5, 9.6).. (0,9.5);
[fill=white] (-.8,6.3) ..controls (0,6) .. (.8,6.3);
[fill=white] (-.8,6.3) ..controls (0,6.6) .. (.8,6.3);
[fill=white] (-.8,6.3) – (-.9,6.25);
[fill=white] (.8,6.3) – (.9,6.25);
[thick,gray]  (0, 6.25) circle (1.2);
at (83:4.8)  $t_1$;
[thick,green]  (0,0) ..controls (-.3, 6) .. (0,6.07);
[thick,green]  (0,0) ..controls (60:5) and (60:9) .. (65:9.5);
[dashed,green,thick]  (65:9.5) ..controls (70:9) and (3,5.5) .. (.4,6);
[dashed,green,thick]  (.4, 6) ..controls (.25, 6) .. (0, 6.07);
[bend left, ->] (106.4:1.5) to (92.7:1.5);
at (100:2)  $\gamma_1$;
[bend left, ->] (92.7:1.5) to (73.5:1.5);
at (83:2)  $\beta_1$;
[bend left, ->] (73.5:1.5) to (60:1.5);
at (67:2)  $\alpha_1$;
[bend left, ->] (60:1.5) to (46.5:1.5);
at (53:2)  $\delta_1$;

[thick,green] (0,0) ..controls (-2.5, 8.5) .. (0,8.5);
[thick,green] (0,0) ..controls (2.5, 8.5) .. (0,8.5);
[thick,gray] (0,6.5) ..controls (-.5,6.4) and (-.5, 9.6) .. (0,9.5);
at (94:9)  $s_2$;
[thick,dashed,gray] (0,6.5) ..controls (.5,6.4) and (.5, 9.6).. (0,9.5);
[fill=white] (-.8,6.3) ..controls (0,6) .. (.8,6.3);
[fill=white] (-.8,6.3) ..controls (0,6.6) .. (.8,6.3);
[fill=white] (-.8,6.3) – (-.9,6.25);
[fill=white] (.8,6.3) – (.9,6.25);
[thick,gray]  (0, 6.25) circle (1.2);
at (83:4.8)  $t_2$;
[thick,green]  (0,0) ..controls (-.3, 6) .. (0,6.07);
[thick,green]  (0,0) ..controls (60:5) and (60:9) .. (65:9.5);
[dashed,green,thick]  (65:9.5) ..controls (70:9) and (3,5.5) .. (.4,6);
[dashed,green,thick]  (.4, 6) ..controls (.25, 6) .. (0, 6.07);
[bend left, ->] (106.4:1.5) to (92.7:1.5);
at (100:2)  $\gamma_2$;
[bend left, ->] (92.7:1.5) to (73.5:1.5);
at (83:2)  $\beta_2$;
[bend left, ->] (73.5:1.5) to (60:1.5);
at (67:2)  $\alpha_2$;

[thick,green] (0,0) ..controls (-2.5, 8.5) .. (0,8.5);
[thick,green] (0,0) ..controls (2.5, 8.5) .. (0,8.5);
[thick,gray] (0,6.5) ..controls (-.5,6.4) and (-.5, 9.6) .. (0,9.5);
at (94:9)  $s_n$;
[thick,dashed,gray] (0,6.5) ..controls (.5,6.4) and (.5, 9.6).. (0,9.5);
[fill=white] (-.8,6.3) ..controls (0,6) .. (.8,6.3);
[fill=white] (-.8,6.3) ..controls (0,6.6) .. (.8,6.3);
[fill=white] (-.8,6.3) – (-.9,6.25);
[fill=white] (.8,6.3) – (.9,6.25);
[thick,gray]  (0, 6.25) circle (1.2);
at (83:4.9)  $t_n$;
[thick,green]  (0,0) ..controls (-.3, 6) .. (0,6.07);
[thick,green]  (0,0) ..controls (60:5) and (60:9) .. (65:9.5);
[dashed,green,thick]  (65:9.5) ..controls (70:9) and (3,5.5) .. (.4,6);
[dashed,green,thick]  (.4, 6) ..controls (.25, 6) .. (0, 6.07);
[bend left, ->] (106.4:1.5) to (92.7:1.5);
at (100:2)  $\gamma_n$;
[bend left, ->] (92.7:1.5) to (73.5:1.5);
at (83:2)  $\beta_n$;
[bend left, ->] (73.5:1.5) to (60:1.5);
at (67:2)  $\alpha_n$;

in 0, 2, 4
at (74-2.5*:2) [circle, fill, inner sep=.5pt];
at (74-2.5*:7) [circle, fill, inner sep=1pt];

Canonical surface model of an algebra of the form $A^{(n)}$, where the arcs in green form an admissible dissection.

[scale=0.4, transform shape]
[decoration=markings, mark=at position 0.5 with >]
in 0,1,...,7

[thick, green] (*40-70:5) to (*40-30:5);

in 1,..., 7
[shift=(*40-70:5), rotate=*40+20]
[bend left, ->] (160:.5) to (20:.5);

in 1,..., 8
[shift=(*40-70:5), rotate=*40+20]
(a) at (160:1.5);

[pattern=north west lines] (-70:5) to [curve through = .. (-90:4)..] (-110:5);

[fill,circle, green, inner sep=0pt, minimum size=6pt] at (-70:5);
[fill,circle, green, inner sep=0pt, minimum size=6pt] at (-110:5);

[bend left, postaction=decorate] (a1) to (a3);
[bend left, postaction=decorate] (a2) to (a4);
[bend left, postaction=decorate] (a5) to (a7);
[bend left, postaction=decorate] (a6) to (a8);
at (2.3,-1)  $s_1$;
at (1.6,2)  $t_1$;
at (-2.2,-1)  $t_2$;
at (-1.7,2)  $s_2$;

at (-30:4.3)  $\alpha_1$;
at (10:4.3)  $\beta_1$;
at (50:4.3)  $\gamma_1$;
at (90:4.3)  $\delta_1$;
at (130:4.3)  $\alpha_2$;
at (170:4.3)  $\beta_2$;
at (210:4.3)  $\gamma_2$;

at (-50:5)  $\ell_1$;
at (-10:5)  $\ell_2$;
at (30:5)  $\ell_1$;
at (70:5)  $\ell_2$;
at (110:5)  $\ell_3$;
at (150:5)  $\ell_4$;
at (190:5)  $\ell_3$;
at (230:5)  $\ell_4$;

[shift=(15,0), decoration=markings, mark=at position 0.5 with >]
in 0,1,...,7

[thick, green] (*40-70:5) to (*40-30:5);

in 1,..., 7
[shift=(*40-70:5), rotate=*40+20]
[bend left, ->] (160:.4) to (20:.4);
[postaction=decorate] (20:1.5) ..controls(90:2).. (160:1.5);
(c) at (90:2.1);

in 0, 8
[shift=(*40-70:5), rotate=*40+20]
(a) at (160:1.5);
(b) at (20:1.5);

[postaction=decorate] (b0) ..controls(90:-2).. (a8);
at (c1) $c_3$;
at (c2) $c_2$;
at (c3) $c_1$;
at (c4) $c_4$;
at (c5) $c_7$;
at (c6) $c_6$;
at (c7) $c_5$;
at (90:-2) $c_8$;

[pattern=north west lines] (-70:5) to [curve through = .. (-90:4)..] (-110:5);

[fill,circle, green, inner sep=0pt, minimum size=6pt] at (-70:5);
[fill,circle, green, inner sep=0pt, minimum size=6pt] at (-110:5);

at (-30:4.3)  $\alpha_1$;
at (10:4.3)  $\beta_1$;
at (50:4.3)  $\gamma_1$;
at (90:4.3)  $\delta_1$;
at (130:4.3)  $\alpha_2$;
at (170:4.3)  $\beta_2$;
at (210:4.3)  $\gamma_2$;

at (-50:5)  $\ell_1$;
at (-10:5)  $\ell_2$;
at (30:5)  $\ell_1$;
at (70:5)  $\ell_2$;
at (110:5)  $\ell_3$;
at (150:5)  $\ell_4$;
at (190:5)  $\ell_3$;
at (230:5)  $\ell_4$;

The polygon cut out by the admissible dissection for $A^{(2)}$. In the left polygon there are four closed curves $s_i, t_i$, for $i=1,2$; in the right one there is only one closed curve $c$ cut by the green arcs into $8$ segments $c_i$. Note that $c$ is homotopic to the boundary.

Then the following result is directly translated from Theorem <ref>.

Let  $A$ and $A'$ be two graded gentle algebras of the form $A^{(n)}$ with the numbers $a_i, b_i$ and $a_i',b_i'$ respectively as in (<ref>).

* Assume that $n=1$. Then $A$ and $A'$ are derived equivalent if

$$\gcd(a_1-1, b_1-1)=\gcd(a_1'-1, b_1'-1).$$

* Assume that $n>1$. Then $A$ and $A'$ are derived equivalent if one of the following two conditions holds

* there exist $1\le i, j\le n$ such that both $a_{i}b_{i}$ and $a'_{j}b'_{j}$ are even.

* All $a_{i},b_{i}, a'_{i}$ and $b'_{i}$ are odd for $1\le i\le n$, and

\begin{equation}\label{Arf} \sum_{i=1}^n \frac{1}{4} (a_i
+1) (b_i +1) = \sum_{i=1}^n \frac{1}{4} (a'_i
+1) (b'_i +1)\  \mod 2.\end{equation}

Consider the canonical models $A= (S,M, \zD, \eta)$ and $A'= (S, M, \zD, \eta')$. The first assertion is clear by (<ref>)  and [14].  Let us prove the second one.
In case there exist $1\le i, j\le n$ such that $a_{i}b_{i}$ and $a'_{j}b'_{j}$ are even then $\sigma(\eta) = 1 = \sigma(\eta')$.
For the case that all $a_i$, $b_i$, $a'_i$ and $b'_i$ are odd, then $\sigma(\eta) = 0 = \sigma(\eta')$ and by equation (<ref>), we have  $\mathcal A(\eta) =
\mathcal A(\eta')$. Now by  Theorem <ref>, the line fields $\eta$ and $\eta'$ are homotopic.  Hence by [14], the categories $\D(A)$ and $\D(A')$ are equivalent.

Let $A$ and $A'$  be two graded gentle algebras of the form $A^{(n)}$ with  corresponding sums $a_{i}, b_{i}$ and $a_{i}', b_{i}'$ respectively. Assume that $a_{i}=a_{i}'$ and $b_{i}=b_{i}'$ for all $i$. Then $A$ and $A'$ have the isomorphic surface models and thus are derived equivalent.

This directly follows  from Proposition <ref>, as under the assumptions all the geometric invariants for $A$ and $A'$ coincide.

Based on Corollary <ref>, we will simply write
\begin{align}\label{align:notationforAn}
A=(a_{1},b_{1};a_{2},b_{2};\dots; a_{n},b_{n})
\end{align}
for any of the derived equivalent algebras of the form $A^{(n)}$ with   numbers $a_i, b_i$.

We will show in Section <ref> that the conditions in Proposition <ref> are also necessary.

The following lemma will be used in the proof of Proposition <ref> below.

Let $(S, M)$ be the torus with exactly one boundary component and with one marked point on the boundary. Then any  admissible dissection $\zD$ on $S$ gives a graded gentle algebra of the form $A^{(1)}$.

Let $A=\Bbbk Q/I$ be the graded gentle algebra given by $\zD$. Recall from <cit.> that $|\Delta|=|M|+b+2g-2$, it follows that $|Q_0|=|\zD|=2$.  By <cit.>, we have that $|M| = 2 |Q_0| - |Q_1|$ which yields $|Q_1|=3$. Therefore, $A$ must be of the form $A^{(1)}$ (since there is only one ungraded gentle algebra with two vertices and three arrows between the two vertices).

Note that Lemma <ref> does not hold for $A^{(n)}$ with $n>1$. For instance, on the surface model of the graded gentle algebra of the form $A^{(2)}$ there is an admissible dissection $\Delta$ such that the associated gentle algebra $A(\Delta)$ is given by the following quiver
\[
\xymatrix{1<EMAIL_ADDRESS><EMAIL_ADDRESS>& 2\ar<EMAIL_ADDRESS>& 3<EMAIL_ADDRESS><EMAIL_ADDRESS>& 4 \ar<EMAIL_ADDRESS>\]

with relations $\{\alpha_1\beta_{2}, \beta_1\alpha_2, \alpha_2\beta_3, \beta_2\alpha_3, \delta \alpha_1, \beta_3\delta\}.$

§ EXISTENCE OF SILTING OBJECTS IN $\PER(A)$

The aim of this section is to provide a complete answer to the existence of silting objects in $\per(A)$ for a graded gentle algebra $A$ which is homologically smooth and proper.

Recall that an object $P\in \per(A)$ is called pre-silting if
$$\Hom_{\per(A)}(P, P[i])=0,\quad \text{for each $i> 0$},$$

and it is called silting if, moreover, $\per(A)=\thick(P)$.  We say that a (pre-)silting object $P$ is basic if its indecomposable direct summands are  pairwise non-isomorphic.

In general, it is not easy to check whether $\per(A)$ has a  silting object or not, but in case $A$ is a homologically smooth and proper graded gentle algebra, we give a complete answer in this section. Recall from Definition <ref> the graded gentle algebras of the form $A^{(n)}$. The main  result in this section is the following.

Let $A=\Bbbk Q/I$ be a homologically smooth and proper graded gentle algebra. Then $\per(A)$ admits a silting object if and only if $A$ is not isomorphic, as a graded algebra,  to an algebra of the form $A^{(1)}$ with $a_{1}=1=b_{1}$.

The proof of Theorem <ref> is postponed to Subsection <ref>.

For an ungraded  (or more generally, a non-positively graded) algebra $A$, $A$ itself is clearly a silting object in $\per(A)$. But $\per(A)$ does not always admit a silting object for a general  graded algebra $A$. For example, consider the homologically smooth and non-proper graded gentle algebra $A=\Bbbk[x]$ with $|x|=1$. By Koszul duality (e.g. [19]), $\per(A)$ is triangle equivalent to  $\D^b(A^!)$ where $A^!=\Bbbk[y]/(y^{2})$ with $|y|=0$, but  $\D^b(A^!)$ does not admit silting objects (otherwise we would have that $\Ext_{A^!}^{\gg0}(\Bbbk,\Bbbk)=0$, which contradicts with the fact that  $\Bbbk$ is of infinite projective dimension, see <cit.>).

We have seen in Proposition <ref> that all the algebras of the form $A^{(1)}$ with $a_1=1=b_1$ are necessarily derived equivalent. Since the existence of silting objects is invariant under derived equivalence, Theorem <ref> yields that among the derived equivalent classes of homologically smooth and proper graded gentle algebras, there is only one class corresponding to the form $A^{(1)}$ with $a_1=1=b_1$ whose perfect derived category does not admit silting objects.

§.§ Silting reduction of graded gentle algebras with respect to idempotents
In this subsection, we recall some results on the silting reduction of graded gentle algebras with respect to idempotents in [10], which play an important role in the sequel.
Silting reduction, introduced in [4, 16], is a useful tool to study the class of silting objects in a triangulated category $\T$ and it gives a
bijection between silting objects in some quotient category of $\T$ (which can be also realized as certain subfactor category of $\T$) and the silting objects in $\T$ with some properties.

Let $A=kQ/I$ be a graded gentle algebra.  Let $e$ be the sum of $m$ vertex idempotents. Without loss  of generality, we may re-label the vertices so that $e=e_{1}+\dots+e_{m}$.  We first recall the following definition.

The graded gentle algebra $A_{e}:=kQ_{e}/ I_{e}$ is defined as follows

* The set of vertices of $Q_{e}$ is $Q_{0}\backslash \{1,2,\dots, m\}$.

The arrows of $Q_{e}$ are of the form $[\za_{1}\cdots\za_{k}]:s(\za_{1})\ra t(\za_{k})$, where $\za_{1}\cdots\za_{k}$ is a path of $Q$ with $\za_i \za_{i+1} \in I$ for $1\leq i \leq k-1$ and $s(\za_{1}), t(\za_{k})\not\in \{1,2,\cdots,m \}$.

The grading of the arrow $[\za_{1}\cdots\za_{k}] $ is $|[\za_{1}\cdots\za_{k}]|:=\sum_{i}^{k}|\za_{i}|-k+1$.

* $I_{e}$ is generated by the set $\{[\za_{1}\cdots\za_{k}][\zb_{1}\cdots\zb_{t}]\mid \za_{k}\zb_{1}\in I  \}$.

The following result is to due to <cit.>. We refer to [8] for the definition of recollements.

* There is a recollement

\begin{equation} \label{mainrecoll}
\xymatrixcolsep{4pc}\xymatrix{
\D(A_{e}) \ar[r]^{i_{*}=i_{!}}
&\D(A<EMAIL_ADDRESS>\ar@/^1.5pc/[l]^{i^{!}}
\ar[r]^{j^{*}=?\ot_{A}^{\bf L}Ae}
&\D(eAe) \ar@/^-1.5pc/[l]_-{j_{!}=?\ot^{\bf L}_{eAe}eA} \ar@/^1.5pc/[l]^-{j_{*}=\RshHom_{eAe}(Ae, ?)}
}, \end{equation}

such that $i^{*}$ sends $e_{j}A$ to $e_{j}A_{e}$ and $i_{*}$ sends $\Top(e_{j}A_{e})$ to $\Top (e_{j}A)$ for any $j\not=1,2,\dots, m$.

* Assume $eA$ is  pre-silting in $\per(A)$. Then the silting reduction $\per(A)/\thick(eA)$ is triangle equivalent to $\per(A_{e})$ and
there is a bijection

\[ \{\mbox{silting objects in $\per(A_{e})$}\} \overset{1:1}{\longleftrightarrow} \{ \mbox{silting objects in $\per(A)$ containing $eA$ as a direct summand}\}.\]

§.§ The behavior of silting objects in the surface model

In this subsection, we show a remarkable behavior of silting objects in the surface model: Silting objects give rise to admissible dissections, see Proposition <ref> below. This is one of our motivations to study silting objects for graded gentle algebras and will play an essential role in the proof of Theorem <ref> below. We remark that in the case of an ungraded gentle algebra this was shown in [7].

Let $A$ be a homologically smooth and proper graded gentle algebra with a surface model $(S, M, \eta, \zD)$. Then the following holds

* Let $P$ be a basic pre-silting object in $\per(A)$. Then the graded arcs corresponding to $P$ form a graded admissible collection in the graded marked surface $(S, M, \eta)$.

* If, furthermore, $P$ is  silting then the admissible collection is a graded admissible dissection.

Let  $P=\bigoplus_{i=1}^N P_i$ be a basic pre-silting object in $\per(A)$, where each $P_i$ is indecomposable.  By the classification of indecomposable objects in $\per(A)$, see [14, 27], each $P_i$ corresponds to a curve $\ell_i$ in $S$ such that $\ell_i$ is either an admissible arc (with endpoints in $M$) or a primitive closed curve in $S\backslash \partial S$.

First we claim that $\ell_i$ cannot be a primitive closed curve for $1\leq i \leq n$. Indeed, if $\ell_i$ is a primitive closed curve then $\Hom_{\per(A)}(P_i, P_i[1]) \neq 0$, see <cit.> and Remark <ref> below. So $P_i$ is not a pre-silting object. It remains to show that $\ell_i$ has no self-intersections or does not intersect $\ell_j$ in the interior of $S$.  For contradiction suppose that there is a (self-)intersection.  Then by <cit.> and also <cit.>, we obtain that

\Hom_{\per(A)}(P_i, P_j[k_{i,j}]) \neq 0 \neq \Hom_{\per(A)}(P_i, P_j[1-k_{i,j}]) \quad \text{for some $k_{i,j} \in \mathbb Z$},

where $k_{i,j}$ is the grading induced by the corresponding (self)-intersection.
We infer that $P_i \oplus P_j$  or $P_i$ if $i=j$ is not a pre-silting object. As a result, we obtain that $\{\ell_i\}_{i=1}^n$ forms a graded admissible collection. This proves the first assertion.

One can complete $\{\ell_i\}_{i=1}^n$ to a graded admissible dissection corresponding to a graded gentle algebra $A'$ whose grading is induced by $\eta$.
Then by [14], there is a triangle equivalence $F:\per(A)\xra{\simeq} \per(A')$, and moreover, $F(P)\cong e'A'$ for some idempotent $e'\in A'$ where $e'$ also corresponds to $\{\ell_i\}_{i=1}^n$. Note that $\{\ell_i\}_{i=1}^n$ is an admissible dissection if and only if $e'=1$.
By Theorem <ref>, $\per(A)/\thick(P)\simeq \per(A')/\thick(e'A')\simeq \per(A'_{e'})$, which is zero if and only if $e'=1$, or equivalently, $\per(A)=\thick(P)$. Hence the second assertion follows.

Let $A$ be a graded gentle algebra, which is homologically smooth and proper. Let $X$ be an indecomposable object in $\per(A)$ corresponding to a primitive closed curve.  The fact that $\Hom_{\per(A)}(X, X[1]) \neq 0$, which is used in the proof of Proposition <ref>, is proved in <cit.> if $A$ is ungraded.

The graded case can be proved in a similar way. More precisely, by <cit.>, the indecomposable object $X$ can be read as

\[
X=\left(\bigoplus_{i=1}^n P_i[k_i] \otimes V, \alpha^*_0 \otimes T + \sum_{i=1}^{n-1} \alpha_i^* \otimes \mathrm{Id}_V\right)
\]

where $P_i$ is the projective indecomposable at vertex $i$, $\alpha_0$ is a path of degree $k_n-k_1+1$ from vertex $n$ to vertex $1$, which induces a dg $A$-module morphism of degree one from $P_1[k_1]$ to $P_n[k_n]$ and $T$ is an automorphism of  the $\Bbbk$-vector space $V$. Similarly, for $0<i <n$, $\alpha_i$ is a path either from $i$ to ${i+1}$ of degree $k_i-k_{i+1}+1$ or from ${i+1}$ to $i$ of degree $k_{i+1}-k_i+1$. The closed curve and the arcs in the admissible dissection are supposed to be in minimal position, see e.g. <cit.>. This particularly implies that if $\alpha_i^*$ and $\alpha^*_{i+1}$ have the same target (resp. source) then their first (resp. last) arrows must be distinct.

Note that $\alpha_0^* \otimes T$ induces a cocycle of degree one when viewed as an element in the Hom complex  $\shHom_A(X,X)$. Then by a similar argument to <cit.> we may show that $\alpha_0^* \otimes T$ cannot be a coboundary so that it gives a nonzero element in $\Hom_{\per(A)}(X, X[1])$.

Let us illustrate how Remark <ref> works in the following example.  Consider the  family of indecomposable modules $X_\lambda$ with $\lambda \in \Bbbk\setminus \{0\}$, for some graded gentle algebra $A$

\[ X_\lambda: \;
\xymatrix{
P_1[k_1] \ar[r]^-{\alpha_1^*} \ar@/_2pc/[rrrr]^-{\lambda\alpha_0^*} &  P_2 [k_2] \ar[r]^-{\alpha_2^*}  & P_3[k_3] & \ar[l]_-{\alpha_3^*}  P_4[k_4] \ar[r]^-{\alpha_4^*} &  P_5[k_5]
\]

where $V = \Bbbk$ and $k_1 = 0$. Note that we have

k_2     =|α_1|-1     k_3     = |α_1|+|α_2|-2

k_4     = |α_1| + |α_2| - |α_3| -1     k_5     = |α_1| + |α_2| - |α_3| + |α_4| -2 = |α_0|-1

In particular,  in order for  $X_\lambda$ to be  a well-defined dg-module we have to have

\[
|\alpha_0| - |\alpha_1| - |\alpha_2| + |\alpha_3| - |\alpha_4|=-1
\]

which  implies that  the winding number of the corresponding closed curve is zero. Note that $X_\lambda$ cannot be a band object of an ungraded gentle algebra $A$ (since the degrees of the $\alpha_i$ cannot all be zero).

Clearly $\lambda \alpha_0^*$, viewed as an element in the Hom complex $\shHom_A(X_\lambda,X_\lambda)$, is a cocycle of degree one.

\begin{align*}
\xymatrix@R=3pc{
X_\lambda:\ar[d]  && P_1 \ar[r]^-{\alpha_1^*}<EMAIL_ADDRESS>\alpha_0^*}<EMAIL_ADDRESS>&  P_2 [k_2] \ar[r]^-{\alpha_2^*}  & P_3[k_3] & \ar[l]_-{\alpha_3^*}  P_4[k_4] \ar[r]^-{\alpha_4^*} &  P_5[k_5]&\\
X_\lambda:&P_1 \ar[r]^-{-\alpha_1^*} \ar@/_2pc/[rrrr]^-{-\lambda\alpha_0^*} &  P_2 [k_2] \ar[r]^-{-\alpha_2^*}  & P_3[k_3] & \ar[l]_-{-\alpha_3^*}  P_4[k_4] \ar[r]^-{-\alpha_4^*} &  P_5[k_5] &
\end{align*}

We claim that $\lambda \alpha_0^*$ is not a coboundary. Indeed, assume that it is a coboundary then there exists
$$h:=\sum_{i, j} h_{i, j}\in \shHom^0_A(X_\lambda,X_\lambda)$$
where  $h_{i,j} \in \shHom^0_A(P_i[k_i], P_j[k_j])$ such that $\delta(h) = \lambda \alpha_0^*$. Here, $\delta$ is the differential of $\shHom_A(X_\lambda,X_\lambda)$. Restricting $\delta(h) = \lambda \alpha_0^*$ to the component $\Hom^1_A(P_1[k_1], P_5[k_5])$ we obtain

\begin{align}\label{align:homotopy}
h_{25}  \alpha_1^*  + h_{55} \lambda \alpha_0^* - \alpha_4^* h_{14} - \lambda \alpha_0^* h_{11} = \lambda \alpha_0^*.
\end{align}

Since $\alpha_0^*$ and $
\alpha_1^*$ (resp. $\alpha_4^*$) have the same source (resp. target), it follows that the last (resp. first) arrows of $\alpha_0$ and $\alpha_1$ (resp. $\alpha_4$) are distinct. It follows that $\lambda \alpha_0^*$ and  $h_{25} \alpha_1^*$ (resp. $\alpha_4^* h_{14}$)  are linearly independent, since the last arrow of the path corresponding to $h_{25}\alpha_1^*$ is the same as the one in $\alpha_1$.  So (<ref>) yields

\begin{align}\label{align:homotopy1}
(h_{55} \lambda  - \lambda  h_{11}) \alpha_0^* = \lambda \alpha_0^*
\end{align}

where $h_{55}, h_{11}$ are just multiplies of the corresponding idempotents. Then by abuse of notation we may just view them as elements in $\Bbbk$.

Note that at least one of $h_{55}, h_{11}$ is nonzero. Without loss of generality let us assume that $h_{55} \neq 0$. Then we restrict $\delta(h) = \lambda \alpha_0^*$ to $\Hom^1_A(P_4[k_4], P_5[k_5])$ and obtain that

\begin{align}\label{align:homotopy2}
h_{55} \alpha_4^* +  h_{35} \alpha_3^* - \alpha_4^* h_{44} - \lambda \alpha_0^* h_{41} = 0
\end{align}

Since $h_{55}\in \Bbbk$ by (<ref>),  we see that $h_{55}\alpha_4^*$ is a multiple of $\alpha_4^*$. By the same reason as above the last (resp. first) arrows of $\alpha_4$ and $\alpha_3$ (resp. $\alpha_0$) are distinct,  it follows that $h_{55} \alpha_4^*$  and $h_{35} \alpha_3^*$ (resp. $\lambda \alpha_0^* h_{41}$)  are linearly independent. So the equality (<ref>) yields that

\[
h_{55} \alpha_4^*- \alpha_4^* h_{44} = 0
\]

where $h_{44}$ is a multiple of the idempotent $e_4$ and will be viewed as an element in $\Bbbk$.
Similarly, by restricting $\delta(h) = \lambda \alpha_0^*$ to other components we obtain that

h_33 α_3^*- α_3^* h_44 = 0         h_22 α_2^* - α_2^* h_22 = 0
h_22 α_1^* - α_1^* h_11 = 0

As a result, we obtain that $
h_{11} = h_{22} = h_{33} = h_{44} = h_{55} \neq 0
and thus $\lambda = \lambda h_{55} - \lambda h_{11}=0$, which contradicts the assumption $\lambda \neq 0$.

§.§ The proof of Theorem <ref>

Our proof of Theorem <ref> makes use of a sufficient condition for the existence of silting objects given in [10] (we refer again to Definition <ref> for the definition of the graded gentle algebras of the form $A^{(n)}$).

Let $A$ be a homologically smooth and proper graded gentle algebra. Then
$\per(A)$ has a silting object if

* $A$ is not derived equivalent to $A^{(n)}$, or

* $A$ is derived equivalent to $A^{(n)}$ and $a_{i}\not=1$ or $b_{i}\not=1$ holds for each $1\le i\le n$.

Theorem <ref> shows that  in order to give a complete answer for the existence of silting objects, it thus remains to consider the graded gentle algebras of the form $A^{(n)}$. This is done in the following two propositions.

For any graded gentle algebra $A$ of the form $A^{(n)}$ with $n>1$, $\per(A)$ admits a silting object.

Let $A=(a_{1},b_{1};\cdots; a_{n},b_{n})$ be a graded gentle algebra of the form $A^{(n)}$ and let $N$ be an integer such that $a_{i}-4N\leq 0 $ and $b_{i}-4N\leq 0$ for each $1\le i\le n$.
Let  $A'=(a_{1}',b'_{1};\cdots; a'_{n},b'_{n})$ be  of the form $A^{(n)}$ with $a'_{i}:=a_{i}-4N$ and $b'_{i}:=b_{i}-4N$, for all $1\le i\le n$. Then $A$ is derived equivalent to $A'$ by Proposition <ref> (2). Since neither $a_i'$ nor $b_i'$ is $1$ for each $1\leq i \leq n$, it follows from Theorem <ref> that $\per(A')$ admits a silting object and hence so does $\per(A)$.

We note that the above proof relies on the description of the  geometric invariants in Proposition <ref>. In  Subsection <ref> below, we provide a  purely algebraic proof, which is based on  explicit algebraic computations and the recollements between graded gentle algebras studied in [10].

Let $A=(1,1)$ be of the form $A^{(1)}$. Then $\per(A)$ has no pre-silting objects and thus no silting objects.

Let $(S, M, \zD, \eta)$ be the canonical  surface model associated to $A$ and let $s_1$ and $t_1$ be the natural non-separating curves in $S$ forming a symplectic basis of the homology as in Figure <ref>. Since $a_1=1=b_1$,  by (<ref>) we have

\begin{equation} \label{equ:gcd=0}
\w{\mathcal A}(\eta)=\gcd\{a_{1}-1, b_{1}-1\}=0.
\end{equation}

Assume that $\per(A)$ admits a basic  pre-silting object $P$.
Then by Proposition <ref>,  $P$ gives rise to a graded admissible collection which we extend to a graded admissible dissection $\zD_{P}$ such that $A':=\h^{*}(\shEnd(P))$ is exactly the graded gentle algebra given by $(S, M, \zD_{P}, \eta)$.
By Lemma <ref>, $A' = (a', b')$ is also of the form  $A^{(1)}$.
Note that there is a triangle equivalence $F:\per(A)\xra{\simeq} \per(A')$ such that $F(P)\cong eA'$ for $e= e_1$ or $e= e_2$ by [14]. Since $P$ is pre-silting it follows that $eA'e$ is concentrated in non-positive degrees. In particular, one of  $a'$ and $b'$ is non-positive since the non-zero two-cycles in $A'$ have gradings $a'$ and $b'$, respectively.

Let $(S', M', \zD', \eta')$ be the canonical model of $A'$ with the canonical non-separating simple closed curves $s'$ and $t'$ as in Figure <ref>.
Since both $(S', M', \zD', \eta')$ and $(S, M, \zD_{P}, \eta)$  are surface models of $A'$, it follows from Proposition <ref> that there exists an orientation preserving homeomorphism $\varphi: S'\ra S$ such that $\varphi_{*}(\eta')$ is homotopic to $\eta$.
Thus we have

\begin{align}\w{\mathcal A}(\eta) &= \gcd(\{w_{\eta} (\varphi(s')), w_{\eta}(\varphi(t'))  \}) \label{equ:gcd1}\\ &=\gcd(\{w_{\eta'}(s'), w_{\eta'}(t')  \}) \label{equ:gcd2}\\ & =\gcd(\{a'-1, b'-1\}) \label{equ:gcd3}\\ &\neq 0,    \label{equ:gcd4}\end{align}

where  the equality in (<ref>)  follows from Remark <ref> which states that the invariant $\w{\mathcal A}(\eta)$ does not depend on the choice of the symplectic basis of $\mathrm{H}_1(\overline S)$.
We note that $\varphi(s'), \varphi(t')$ from a basis since a symplectic basis is preserved by homeomorphisms.
The equality in line (<ref>) follows since $\varphi_*(\eta')$ is homotopic to $\eta$ and thus for any simple closed curve $\gamma'$ in $S'$ we have
$$w_\eta(\varphi(\gamma'))=w_{\varphi_*(\eta')}(\varphi(\gamma')) = w_{\eta'}(\gamma').$$

The equality in line (<ref>)  follows from (<ref>) and  line (<ref>)  follows since  one of  $a'$
and $b'$ is non-positive.
This contradicts (<ref>) stating that $\w{\mathcal A}(\eta)  \neq 0$. Thus $\per(A)$ does not admit a pre-silting object and hence also no silting object.

We note that a direct  calculation  as stated in [10], also  shows  Proposition <ref> by showing that there always is a positive self-extension for any graded homotopy string or band complex in $\per(A^{(1)})$. A recent preprint  [24] also announces a  proof using  yet a different approach.

We are now ready to give a proof of Theorem <ref>.

The `only if' part follows from Proposition <ref>.

For the `if' part, let $(S, M, \zD, \eta)$ be a surface model of $A$.   Then if $|M| >1$ or $|M|=1$ and $g(S) >1$, by Theorem <ref> and Proposition <ref>, respectively, $\per(A)$ has a silting object. Now if $|M|=1$ and $g(S) =1$, then by Lemma <ref>,  $A = (a,b)$ is of the form $A^{(1)}$ with $(a,b) \neq (1,1)$ by assumption and by Theorem <ref>, $\per(A)$ has a silting object.

§.§ Pre-silting $\&$ partial silting in $\per(A)$

In this subsection, we give a family of examples of finite dimensional algebras  having pre-silting objects which are not partial silting. We note that in [24] a different example of a pre-silting object for a finite dimensional algebra which is not partial silting  was given.

Let $A$ be a finite dimensional (ungraded)  gentle algebra of the form $A^{(n)}$, for $n >1$. Then there exists a pre-silting object which is not a partial silting object.

Since $A$ is ungraded it follows that  $A=(0,0;\dotsb; 0,0)$. By Proposition <ref>, there is a triangle equivalence $F\colon \per(A) \simeq \per(A')$ where $A' = (1,1; 0,0;\dotsb;0,0)$. Let $e' = e_3 + \cdots +e_{2n} \in A'$ and let $P = e'A'$. Then $P$ is a pre-silting object in $\per(A')$ since $\Hom_{\per(A')}(P, P[i])=e'A'e'[i] =0$ for $i >0$ (Note that $e'A'e'$ is concentrated in degree zero).  Note that $A'_{e'}=(1,1)$ is of the form $A^{(1)}$ (see Definition <ref>).
Then by Proposition
<ref>, there is a bijection

\[ \{\mbox{silting objects in $\per(A'_{e'})$}\} \overset{1:1}{\longleftrightarrow} \{ \mbox{silting objects in $\per(A')$ containing $P$ as a direct summand}\}.\]

Note that the set on the left is empty by Theorem <ref>, it follows that $P$ is not partial silting.

We point out that in the graded case, by  Theorem <ref> and  Proposition
<ref>, one can easily construct more examples. In fact we have the following result.

Let $A=\Bbbk Q/I$ be a homologically smooth and proper graded gentle algebra and let $e$ be the sum of some vertex idempotents. Assume $eA$ is  pre-silting in $\per(A)$ and $eAe$ is homologically smooth. Then $eA$ is not partial silting if and only if there is a connected component of $A_{e}$ which is $(1,1)$ of the form $A^{(1)}$.

Note that $eAe$  and $A$ are homologically smooth and proper by our assumption and thus  so is $A_{e}$ (see <cit.>).
By Proposition
<ref>, $eA$ is partial silting if and only if $\per(A_{e})$ has silting objects.  The assertion follows by Theorem <ref>.

Let $A$ be a graded gentle algebra given by the graded quiver

\[\xymatrix{ & 1 \ar@<1pc>[r]^-{\za} \ar@<-1pc>[r]^-{\zg}& 2 \ar[l]_-{\zb} \ar[r]^{\zd} & 3 \ar[r]^{\eta}& 4}\]

with relation set $\{\za\zb,\zb\zg, \zg\zd\}$. Assume $|\za|+|\zb|=1=|\zb|+|\zg|$ and $|\eta|=0$. Then $(e_{3}+e_{4})A$ is  pre-silting in $\per(A)$ but it  is not partial silting.

On the other hand, we show that  almost complete pre-silting objects are always partial silting. Namely, we show the following.

Suppose that  $A=\Bbbk Q/I$ is a homologically smooth and proper graded gentle algebra with $|Q_0|=n$. Let $P$ be a basic pre-silting object with $n-1$ indecomposable direct  summands. Then $P$ is partial silting.

By Proposition <ref> we may assume $P = eA$ for $e = 1-e_i$ for some $i \in Q_0$. Without loss of generality we may assume that $i=1$.  Thanks to Proposition <ref>,  we only need to show that $\per(A_{e})$ has silting objects. Since by <cit.> $A_{e}$ is a homologically smooth graded gentle algebra  with only one vertex, we have $A_{e}\cong \Bbbk$ or $A_{e}\cong \Bbbk[x]$. The assertion clearly holds for $A_{e}\cong\Bbbk$. Now consider the case that $A\cong \Bbbk[x]$.
By Definition <ref>, $x$ is given by $[\alpha_1 \ldots \za_k]$ and $|x| = \sum_{i=1}^k \za_i - k+1$ where $\za_i \za_{i+1} \in I$, for $1 \le i \le k-1$ and $t(\za_k) = i$ and $\za_k \za_1 \notin I$.
Furthermore, since $eA$ is pre-silting, $eAe$ is non-positively graded and in particular, the arrows $\za_2, \ldots, \za_{t-1}, \za_k\za_1$ in $eAe$ are non-positively graded. Then $|x| = \sum_{i=1}^k \za_i - k+1$ is non-positive and $k[x]$ is silting in $\per(k[x])$.

§.§ An algebraic proof of Proposition <ref>

In this  subsection we give a  more algebraic proof of Proposition <ref> without the use of the geometric invariants in [23].

We start with the case of $n = 2$.

Given  any algebra  $A=(a_{1}, b_{1}; a_{2},b_{2})$ of the form $A^{(2)}$, there exists a graded gentle algebra $B=(\w{a}_{1},\w{b}_{1}; \w{a}_{2}, \w{b}_{2})$ of the form $A^{(2)}$ which is derived equivalent to  $A$ satisfying  $(\w{a}_{1}, \w{b}_{1})\not=(1,1)$ and  $(\w{a}_{2}, \w{b}_{2})\not=(1,1)$.
As a result, $\per(A)$ admits a silting object.

Denote by $P_i=e_iA$ the indecomposable projective $A$-module at the vertex $i$,  where the quiver of $A$ is  as in (<ref>). Clearly, $\bigoplus_{i=1}^4 P_i$ is a generator of $\per(A)$. Let us construct another generator $P'=\bigoplus_{i=1}^4 P_i'$ of $\per(A)$, where $P_1' = P_1, \ P_4'=P_4,$ the dg $A$-module $P'_{2}$ is given by the total complex  of the bi-complex

<EMAIL_ADDRESS>P_{3}\ar[r]^-{\tbinom{\zd_{1}}{\zb_{2}}} & P_{2}[|\zd_{1}|]\op P_{4}[|\zb_{2}|]  \ar[r]^-{(0 \  \za_{2})}  & P_{3}[a_{2}] \ar[r]^-{\zd_{1}} & P_{2}[a_{2}+|\zd_{1}|] \ar[r]^-{\zg_{1}} & P_{1}[a_{2}+|\zd_{1}|+|\zg_{1}|] \ar[r]^-{\zb_{1}}& P_{2}[a_{2}+b_{1}+|\zd_{1}|]},\]

and  $P'_{3}$ is given by the total complex of

\[ \xymatrix{ P_{3}\ar[r]^-{\zd_{1}} & P_{2}[|\zd_{1}|] \ar[r]^-{|\zg_{1}|} & P_{1}[|\zd_{1}|+|\zg_{1}|] \ar[r]^-{\zb_{1}} & P_{2}[|\zd_{1}|+b_{1}].

This is indeed a generator since one can check that $P_i \in \mathrm{thick}(P')$ for each $1\leq i \leq 4$. Note that as a graded module (ignoring the differential) $P_2'$ equals

\begin{equation}\label{align:p2'}\small
P_3 \oplus P_{2}[|\zd_{1}|-1]\op P_{4}[|\zb_{2}|-1] \oplus P_{3}[a_{2}-2] \op P_{2}[a_{2}+|\zd_{1}|-3]  \oplus P_{1}[a_{2}+|\zd_{1}|+|\zg_{1}|-4] \oplus P_{2}[a_{2}+b_{1}+|\zd_{1}|-5]
\end{equation}

and $P_3'$ equals

\begin{align}\label{align:p3'}
P_3 \oplus P_{2}[|\zd_{1}|-1] \oplus P_{1}[|\zd_{1}|+|\zg_{1}|-2]  \oplus P_{2}[|\zd_{1}|+b_{1}-3].
\end{align}

Let $(S,M, \zD, \eta)$ be the canonical model of $A$ as in Figure <ref>, where the arcs $\ell_i$  correspond to projective modules $P_i$.
The generator $P'$ gives a new admissible dissection  of the graded marked surface $(S,M, \eta)$ illustrated in Figure <ref>, where the blue (resp. red) arc corresponds to $P_2'$ (resp. $P_3'$). The graded algebra $A':=\h^{*}\shEnd_{A}(P') $, given by the dissection in Figure <ref>  is also   of the form $A^{(2)}$  with quiver

\begin{align}\label{align:newquivera2}
\xymatrix{1 \ar@<1pc>[r]^-{\za'_{1}} \ar@<-1pc>[r]^-{\zg'_{1}}& 2 \ar[l]_-{\zb'_{1}} \ar[r]^{\zd'_{1}}& 3 \ar@<1pc>[r]^-{\za'_{2}} \ar@<-1pc>[r]^-{\zg'_{2}} & 4. \ar[l]_-{\zb'_{2}} %\ar[r]^{\zd_{2}} & \cdots \ar[r]^{\zd_{n-1}} & 2n-1 \ar@<1pc>[r]^{\za_{n}}  \ar@<-1pc>[r]^{\zg_{n}}& 2n \ar[l]_{\zb_{n}}

Note that the arrow  $\za_{1}'$ in the quiver (<ref>) may be represented by the graded map
$$\alpha_1\in \Hom_A(P_{2}[a_{2}+b_{1}+|\zd_{1}|-5], P_1)\subset \Hom_A(P'_2, P'_1)$$
induced by the arrow $\alpha_1$, and thus its grading is $a_{2}+b_{1}+|\zd_{1}|+|\za_{1}|-5$. Here we use the explicit decomposition (<ref>) of $P_2'$ as a graded module. It is easy to check that this map is compatible with the differential.
Similarly,  the arrow $\beta_1'$ is represented by

$$\beta_1 \in \Hom_A(P_1, P_{2}[|\zd_{1}|-1]) \subset \Hom_A(P_1', P_2')$$

whose grading is $|\zg_{1}|+|\zd_{1}|-1$, and
$\gamma_1'$ is represented by

\[
\gamma_1\in \Hom_A(P_2[|\delta_1|-1], P_1) \subset \Hom_A(P_2', P_1')
\]

whose grading is $|\zg_{1}|+|\zd_{1}|-1$;

In the same manner, we may write down the explicit maps at the level of complexes which induce $\zd_{1}'$, $\za'_{2}$, $\zb_{2}'$ and $\zg_{2}'$. We leave the details as an exercise to the reader. As a result, we obtain that  $|\zd_{1}'|=2-a_{2}$, $|\za_{2}'|=|\za_{2}|$, $|\zb_{2}'|=|\zb_{2}|$ and $|\zg_{2}'|=3-b_{1}+|\zg_{2}|$. Then $A' = (a'_1, b'_1; a'_2, b'_2) = (a_{1}+a_{2}+b_{1}-4, b_{1}; a_{2}, 3-b_{1}+b_{2})$.

Now starting from $A=(a_{1},b_{1}; a_{2},b_{2})$, we have two ways to construct algebras derived equivalent to $A$: using the construction above, we obtain $A'=(a_{1}+a_{2}+b_{1}-4, b_{1}; a_{2}, b_{2}-b_{1}+3)$ and using Koszul duality, we obtain $A^{!}=(2-a_{2},2-b_{2}; 2-a_{1}, 2-b_{1})$, see for example <cit.>.  Combining and repeating these two constructions, we  obtain an algebra  $B=(\w{a}_{1},\w{b}_{1}; \w{a}_{2}, \w{b}_{2})$ of the form $A^{(2)}$ with $(\w{a}_{1}, \w{b}_{1})\not=(1,1)$ and  $(\w{a}_{2}, \w{b}_{2})\not=(1,1)$.

Then it follows from Theorem <ref> that $B$ admits a silting object and thus so does $A$.

Let $A=(a_{1}, b_{1}; a_{2},b_{2}; a_{3}, b_{3})$ be of the form $A^{(3)}$. Then $\per(A)$ has silting objects.

If $(a_{i},b_{i})\not=(1,1)$ for some $1\le i\le 3$, we have that $\thick(eA)=\per(eAe)$ has silting objects for $e=e_{2i-1}+e_{2i}$ by Theorem <ref>.
Note that $A_{e}$ is of the form $A^{(2)}$ (see Definition <ref>) and $\per(A_{e})$ has silting objects by Lemma <ref>. Then $\per(A)$ has silting objects by Proposition <ref>.

If $(a_{i},b_{i})=(1,1)$ for all $1\le i\le 3$.
Let $P_{i}'$ be as in the proof of Lemma <ref> for $i=1,\dots, 4$. Let $P_{5}'=P_{5}$ and $P_{6}'=P_{6}$. Then $\bop_{i=1}^{6}P_{i}'$ gives another admissible dissection in the canonical model of $A$ which is similar to Figure <ref>. Let $A':=\h^{*}\shEnd(\bop_{i=1}^{6}P_{i}')$. Then

\[A'=(a_{1}+a_{2}+b_{1}-4, b_{1}; a_{2}, 3-b_{1}+b_{2}; a_{3}, b_{3})=(-1,1;1,3;1,1).\]

By our discussion above, $\per(A)\simeq \per(A')$ has silting objects.

[scale=0.4, transform shape]

[top color= blue!15](-30:9.5) arc(-30:210:9.5);
[ pattern=north west lines] (0,-.5) circle (.5);
[thick] (210:9.5) ..controls (-6.6,-7).. (-4, -4.75);
[dashed] (210:9.5) ..controls (-6.5,-5).. (-4, -4.75);
[dashed] (210:9.5) ..controls (-6.5,-4.5).. (-4, -4.75);
[thick] (-30:9.5) ..controls (6.6,-7).. (4, -4.75);
[dashed] (-30:9.5) ..controls (6.5,-5).. (4, -4.75);
[dashed] (-30:9.5) ..controls (6.5,-4.5).. (4, -4.75);
[thick] (-4, -4.75) ..controls (0,-1.77).. (4, -4.75);

[thick,green] (0,0) ..controls (-2.5, 7.5) .. (0,7.5);
[thick,green] (0,0) ..controls (2.5, 7.5) .. (0,7.5);
[fill=white] (-.8,6.3) ..controls (0,6) .. (.8,6.3);
[fill=white] (-.8,6.3) ..controls (0,6.6) .. (.8,6.3);
[fill=white] (-.8,6.3) – (-.9,6.25);
[fill=white] (.8,6.3) – (.9,6.25);
[thick,red]  (0,0) ..controls (-.3, 6) .. (0,6.07);
[dashed,red] plot [smooth] coordinates(45:9.5) (55:9) (1.1, 5.9) (0, 6.07) ;
[bend left, ->] (108.5:1.5) to (93:1.5);
at (101:2)  $\gamma_1'$;
[bend left, ->] (93:1.5) to (71.5:1.5);
at (83:2)  $\beta_1'$;
[bend left, ->] (71.5:1.5) to (50:1.5);
at (62:2)  $\alpha_1'$;
[bend left, ->] (50:1.5) to (26:1.5);

at (107:2)  $\delta_1'$;
[fill=white] (-.8,6.3) ..controls (0,6) .. (.8,6.3);
[fill=white] (-.8,6.3) ..controls (0,6.6) .. (.8,6.3);
[fill=white] (-.8,6.3) – (-.9,6.25);
[fill=white] (.8,6.3) – (.9,6.25);
[thick,green]  (0,0) ..controls (-.3, 6) .. (0,6.07);
[thick,green]  (0,0) ..controls (60:5) and (60:9) .. (65:9.5);
[dashed,green]  (65:9.5) ..controls (70:9) and (3,5.5) .. (.4,6);
[dashed,green]  (.4, 6) ..controls (.25, 6) .. (0, 6.07);
[bend left, ->] (118:1.5) to (92.7:1.5);
at (106:2)  $\gamma_2'$;
[bend left, ->] (92.7:1.5) to (75:1.5);
at (84:2)  $\beta_2'$;
[bend left, ->] (75:1.5) to (60:1.5);
at (67:2)  $\alpha_2'$;

[thick,red] plot [smooth] coordinates (0, 0) (125:7.5)  (160:8)  (190:7) (0, -1.5) (3, -4.02);
[dashed,red] plot [smooth] coordinates(3, -4.02)  (3.5, -3.55) (35:4)  (70:5.7) (75:6.2);
[thick, red] plot [smooth] coordinates(75:6.2)  (80:6.2)  (100:8.6) (115:9.5);

[thick,blue] plot [smooth] coordinates (0, 0) (100:5) (120:8) (140:8.5) (170:8.5)  (200:8) (0, -2.1) (1.5, -3.02);
[thick,blue] plot [smooth] coordinates (0, 0) (61:8) (100:9.5);
[dashed, blue] plot [smooth] coordinates (1.5, -3.02)   (2, -2.5)  (10:2.5) (80:5) (101:8.6) (100:9.5);

[top color= blue!15](-30:9.5) arc(-30:210:9.5);
[ pattern=north west lines] (0,-.5) circle (.5);
[thick] (210:9.5) ..controls (-6.6,-7).. (-4, -4.75);
[dashed] (210:9.5) ..controls (-6.5,-5).. (-4, -4.75);
[dashed] (210:9.5) ..controls (-6.5,-4.5).. (-4, -4.75);
[thick] (-30:9.5) ..controls (6.6,-7).. (4, -4.75);
[dashed] (-30:9.5) ..controls (6.5,-5).. (4, -4.75);
[dashed] (-30:9.5) ..controls (6.5,-4.5).. (4, -4.75);
[thick] (-4, -4.75) ..controls (0,-1.77).. (4, -4.75);

[thick,green] (0,0) ..controls (-2.5, 7.5) .. (0,7.5);
[thick,green] (0,0) ..controls (2.5, 7.5) .. (0,7.5);
[fill=white] (-.8,6.3) ..controls (0,6) .. (.8,6.3);
[fill=white] (-.8,6.3) ..controls (0,6.6) .. (.8,6.3);
[fill=white] (-.8,6.3) – (-.9,6.25);
[fill=white] (.8,6.3) – (.9,6.25);
[thick,red]  (0,0) ..controls (-.3, 6) .. (0,6.07);
[dashed,red] plot [smooth] coordinates(45:9.5) (55:9) (1.1, 5.9) (0, 6.07) ;
[bend left, ->] (108.5:1.5) to (93:1.5);
at (101:2)  $\gamma_1'$;
[bend left, ->] (93:1.5) to (71.5:1.5);
at (83:2)  $\beta_1'$;
[bend left, ->] (71.5:1.5) to (50:1.5);
at (62:2)  $\alpha_1'$;
[bend left, ->] (50:1.5) to (26:1.5);

at (107:2)  $\delta_1'$;
[fill=white] (-.8,6.3) ..controls (0,6) .. (.8,6.3);
[fill=white] (-.8,6.3) ..controls (0,6.6) .. (.8,6.3);
[fill=white] (-.8,6.3) – (-.9,6.25);
[fill=white] (.8,6.3) – (.9,6.25);

[thick,green]  (0,0) ..controls (-.3, 6) .. (0,6.07);
[thick,green]  (0,0) ..controls (60:5) and (60:9) .. (65:9.5);
[dashed,green]  (65:9.5) ..controls (70:9) and (3,5.5) .. (.4,6);
[dashed,green]  (.4, 6) ..controls (.25, 6) .. (0, 6.07);
[bend left, ->] (118:1.5) to (92.7:1.5);
at (106:2)  $\gamma_2'$;
[bend left, ->] (92.7:1.5) to (75:1.5);
at (84:2)  $\beta_2'$;
[bend left, ->] (75:1.5) to (60:1.5);
at (67:2)  $\alpha_2'$;

[thick,red] plot [smooth] coordinates (0, 0) (125:7.5)  (160:8)  (190:7) (0, -1.5) (3, -4.02);
[dashed,red] plot [smooth] coordinates(3, -4.02)  (3.5, -3.55) (35:4)   (75:6.2);
[thick, red] plot [smooth] coordinates(75:6.2)  (80:6.2)  (100:8.6) (115:9.5);

[thick,blue] plot [smooth] coordinates (0, 0) (100:5) (120:8) (140:8.5) (170:8.5)  (200:8) (0, -2.1) (1.5, -3.02);
[thick,blue] plot [smooth] coordinates (0, 0) (61:8) (100:9.5);
[dashed, blue] plot [smooth] coordinates (1.5, -3.02)   (2, -2.5)  (10:2.5) (80:5) (101:8.6) (100:9.5);

[thick,green] (0,0) ..controls (-1.8, 7.5) .. (0,7.5);
[thick,green] (0,0) ..controls (1.8, 7.5) .. (0,7.5);

[fill=white] (-.8,6.3) ..controls (0,6) .. (.8,6.3);
[fill=white] (-.8,6.3) ..controls (0,6.6) .. (.8,6.3);
[fill=white] (-.8,6.3) – (-.9,6.25);
[fill=white] (.8,6.3) – (.9,6.25);
[thick,green]  (0,0) ..controls (0, 6) .. (.4,6.15);
[thick,green]  (0,0) ..controls (60:5) and (60:9) .. (75:9.5);

[dashed,green] plot [smooth] coordinates (.4, 6.15) (.55, 6) (80:6.3) (77:9) (75:9.5)  ;
[bend left, ->] (120:1.5) to (103:1.5);
at (112:2)  $\delta_3$;
[bend left, ->] (103:1.5) to (92.7:1.5);
at (98:2)  $\gamma_3$;
[bend left, ->] (92.6:1.5) to (77:1.5);
at (84:2)  $\beta_3$;
[bend left, ->] (77:1.5) to (60:1.5);
at (67:2)  $\alpha_3$;

Another admissible dissections for $A^{(2)}$ and $A^{(3)}$.

Now we are ready to give another proof of Proposition <ref>.

We apply induction on $n$. The assertion holds for $n=2$  by Lemma <ref> and $n = 3$ by Lemma <ref>.
Assume it is true for $n=2,3,\dotsc, m-1$ for some $m>3$.  Then we need to show that  $\per(A)$ has a silting object for any algebra $A=(a_{1},b_{1};a_{2},b_{2};\dots; a_{m},b_{m})$ of the form $A^{(m)}$.

Let $e=e_{1}+e_{2}+e_{3}+e_{4}$ be the sum of the idempotents corresponding to the vertices $1, \dotsc, 4$ in $Q^{(m)}$, see Definition <ref>. Then $eAe\simeq (a_{1},b_{1};a_{2},b_{2})$ and  $A_e = (a_{3},b_{3};\dots; a_{m},b_{m})$. Note that $\per(eAe)$ has silting objects by Lemma <ref> and  $\thick(eA)\xra{\simeq} \per(eAe)$. Then $
\thick(eA)$ has a silting object $P$, and moreover, the silting reduction $\per(A)/\thick(P)$ is equivalent to $\per(A_e)$ by Theorem <ref>. Then there is a bijection between silting objects in $\per(A_e)$ and silting objects in $\per(A)$ containing $P$ as a direct summand. Since $\per(A_e)$ has silting objects by our assumption, $\per(A)$ has silting objects.

§ A COMPLETE DERIVED INVARIANT FOR GRADED GENTLE ALGEBRAS

In this section we give a   second application of Theorem <ref>. Namely we  give  a complete derived invariant for graded gentle algebras.
This invariant is given in terms of their geometric surface models. By the equivalences in [14] and [23] this  then also gives a derived invariant for partially wrapped Fukaya categories of graded surfaces with stops.

More precisely, we show the following result which was conjectured in

Let $A$ and $B$ be two graded gentle algebras which are homologically smooth and proper with associated surfaces models  $(S_{A}, M_{A}, \zD_{A}, \eta_{A})$ and
$(S_{B}, M_{B}, \zD_{B}, \eta_{B})$,  respectively. Then $\D(A)$ is triangle equivalent to $\D(B)$ if and only if there exists an orientation preserving homeomorphism $\varphi: S_{A}\ra S_{B}$ such that $\varphi(M_{A})=M_{B}$  and $\varphi_{*}(\eta_{A})$ is homotopic to $\eta_{B}$.

The `if' part is due to [14, 23].

Assume that $F:\D(A) \ra \D(B)$ is a triangle equivalence.  We prove the `only if' part by considering the following two cases according to the existence of silting objects.

Case 1
Assume that $\per(A)$ admits a silting object $P$. We may assume that $P$ is basic. Then $F(P)$ is a basic silting object of $\per(B)$. Moreover, by Proposition <ref>,
$P$ and $F(P)$ give rise to admissible dissections $\zD_{P}$ of  $(S_{A}, M_{A})$ and  $\zD_{F(P)}$ of $(S_{B}, M_{B})$, respectively. Since

\[\h^{*}(\shEnd_{\per(A)}(P))\cong \bop_{i\in\Z}\Hom_{\per(A)}(P, P[i]) \cong \bop_{i\in\Z}\Hom_{\per(B)}(F(P), F(P)[i])= \h^{*}(\shEnd_{\per(B)}(F(P))),\]

both $(S_{A}, M_{A}, \zD_{P}, \eta_{A})$ and $(S_{B}, M_{B}, \zD_{F(P)}, \eta_{B})$ are surface models of $\h^{*}(\shEnd_{\per(A)}P))$. By Proposition <ref>, there exists an orientation preserving homeomorphism $\varphi: S_{A}\ra S_{B}$ such that $\varphi(M_{A})=M_{B}$, $\varphi(\zD_{P})=\zD_{F(P)}$,  and $\varphi_{*}(\eta_{A})$ is homotopic to $\eta_{B}$.

Case 2    Assume that neither $\per(A)$ nor $\per(B)$ admits silting objects. By Theorem <ref>, $A$ and $B$ must be of the form $A^{(1)}$, that is $A=(a_A,b_A)$ and $B = (a_B,b_B)$ such that $a_A, b_A, a_B$ and $b_B$  are all equal to $1$.
Let $(S, M, \zD, \eta)$ and $(S', M', \zD', \eta')$ be the canonical surface models of $A$ and $B$, respectively. Then $S'=S, M'=M, \zD' = \zD$. Since $\w{\mathcal A}(\eta)=0=\w{\mathcal A}(\eta')$, we have that $\eta$ is homotopic to $\eta'$ by Theorem <ref>. It follows that the identity map $\rm{id}\colon S\to S'$ gives a homeomorphism between these two surface models.

By Proposition <ref>, the surface model  (up to algebra isomorphism) is unique. So there exists an orientation preserving homeomorphisms $\varphi^{A}: S_{A}\ra S$ (resp.$\varphi^{B}: S'\ra S_{B}$) such that $\varphi^{A}(M_{A})=M$ (resp. $\varphi^{B}(M')=M_{B}$) and $\varphi^{A}_{*}(\eta_{A})$ is homotopic to $\eta$ (resp. $\varphi^{B}_{*}(\eta')$ is homotopic to $\eta_{B}$).
The  homeomorphism $\varphi=\varphi^{B}\circ \rm{id} \circ \varphi^{A}$ satisfies the condition we want.

As an application of Theorem <ref> we give a complete derived invariant for homologically smooth and proper graded gentle algebras. Recall  the invariants $\sigma(\eta), \w{\mathcal  A}(\eta)$ and $\mathcal A(\eta)$ from Subsection <ref>. Then we have the following.

Let $A$ and $B$ be two homologically smooth and proper graded gentle algebras with associated surfaces models  $(S_{A}, M_{A}, \zD_{A}, \eta_{A})$ and $(S_{B}, M_{B}, \zD_{B}, \eta_{B})$,  respectively. Then $A$ and $B$ are derived equivalent if and only if $S_A$ and $S_B$ have the same number $b$ of boundary components and there exists a numbering of these boundary components such that for each $i=1,\dots, b$, one has

\[\#(M_{A}\cap\partial_{i} S_{A})= \#(M_{B}\cap\partial_{i} S_{B}),\]

\[w_{\eta_{A}}(\partial_{i} S_{A})=w_{\eta_{B}}(\partial_{i} S_{B}),  \]

and in addition,

* if $g(S_{A})=g(S_{B})=1$, then $\w{\mathcal A}(\eta_{A})=\w{\mathcal A}(\eta_{B})$;

* if $g(S_{A})=g(S_{B}) >1$, then $\zs(\eta_{A})=\zs(\eta_{B})$ and $\mathcal A(\eta_{A})=\mathcal A (\eta_{B})$ whenever the latter two invariants are defined.

The proof follows from Theorems <ref> and <ref>.

As an application of the last corollary, we  give a complete classification of derived equivalence classes  for the graded gentle algebras of the form $A^{(n)}$. Recall that a sufficient condition is given in Proposition <ref>.

* Let $A=(a,b)$ be a graded gentle algebra of the form $A^{(1)}$. Let $A'$ be any graded gentle algebra. Then $A'$ is derived equivalent to $A$ if and only if $A' = (a',b')$ and
$$\gcd\{a-1, b-1\} = \gcd\{a'-1, b'-1\}.$$

* Let $A=(a_1,b_1; \dotsb; a_n,b_n)$ and $A'=(a_1',b_1'; \dotsb; a_n', b_n')$ be two graded gentle algebras of the form $A^{(n)}$ for $n>1$. Then $A$ and $A'$ are derived equivalent if and only if one of the following two conditions holds

* there exist $1\le i, j\le n$ such that both $a_{i}b_{i}$ and $a'_{j}b'_{j}$ are even.

* All $a_{i},b_{i}, a'_{i}$ and $b'_{i}$ are odd for $1\le i\le n$, and

\begin{equation*} \sum_{i=1}^n \frac{1}{4} (a_i
+1) (b_i +1) = \sum_{i=1}^n \frac{1}{4} (a'_i
+1) (b'_i +1)\  \mod 2.\end{equation*}

The first assertion follows from Lemma <ref> and Corollary <ref>, and the second one directly follows from Corollary <ref>.

In particular, $(a, b)$ is not derived equivalent to $(1,1)$ unless $(a, b) =(1,1)$. By Remark <ref> the class of the algebras of the form $A^{(n)}$ for each $n>1$ is not closed under derived equivalence.

§ ACKNOWLEDGEMENTS

We would like to thank Yu Zhou and  Alexandra Zvonareva for useful discussions. The first-named author also thanks the Institute of Algebra and Number Theory at the University of Stuttgart for the hospitality while working on this paper.

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