Mirror symmetry and stability conditions on \(K3\) surfaces (Q2863520)

The PhD thesis under review is concerned with some questions arising in the context of mirror symmetry for \(K3\) surfaces. It consists of two parts. The first one presents several results concerning Fourier-Mukai partners of, and stability conditions on, a smooth projective \(K3\) surface \(X\). In the second one the author provides an explicit example of mirror symmetry for \(K3\) surfaces.NEWLINENEWLINEWrite \(\mathcal{T}\) for \(\mathrm{D}^b(X)\), the bounded derived category of coherent sheaves on \(X\), and consider the numerical Grothendieck group \(N(X)=N(\mathcal{T})\), which can be identified with the extended Néron-Severi group \(H^0(X,\mathbb{Z})\oplus \mathrm{NS}(X)\oplus H^4(X,\mathbb{Z})\). This lattice is contained in the so-called Mukai lattice, which is the full integer cohomology \(H^*(X,\mathbb{Z})\) of \(X\), and the geometry of the latter lattice is described in detail in Section 2.NEWLINENEWLINEOne then considers the period domain of elements \([z]\) in \(\mathbb{P}(N(\mathcal{T})_\mathbb{C})\) satisfying \(z.z=0, z.\overline{z}>0\) and defines the Kähler moduli space \(\mathrm{KM}(\mathcal{T})\) to be the quotient of a connected component of this period domain by the group of autoequivalences of \(\mathcal{T}\). The space \(\mathrm{KM}(\mathcal{T})\) can be compactified to \(\overline{\mathrm{KM}}(\mathcal{T})\) using the Baily--Borel construction and the components of the boundary \(\overline{\mathrm{KM}}(\mathcal{T})\setminus \mathrm{KM}(\mathcal{T})\) are called cusps, of which there are three types. \textit{S. Ma} proved in [Int.\ J.\ Math.\ 20, No.\ 6, 727--750 (2009; Zbl 1216.14037)] that there is a bijection between so-called 0-dimensional standard cusps and Fourier--Mukai partners \(Y\) of \(X\), that is, \(K3\) surfaces \(Y\) such that \(\mathrm{D}^b(Y)\cong \mathcal{T}\). This result is recalled in Section 3, while Section 4 presents some information concerning Bridgeland stability conditions on \(K3\) surfaces. Recall that a stability condition \(\sigma\) consists of a homomorphism \(Z_\sigma: N(\mathcal{T})\rightarrow \mathbb{C}\) and a heart of a bounded t-structure \(\mathcal{A}(\sigma)\) subject to some conditions. The space of those stability conditions which satisfy some additional technical assumptions is known to be a finite-dimensional complex manifold. It is also known that a special open subset of \(\mathrm{KM}(\mathcal{T})\) can be identified with a quotient of the distinguished component \(\mathrm{Stab}^\dagger(\mathcal{T})\) of the space of stability conditions on \(\mathcal{T}\) described by \textit{T. Bridgeland} in [Duke Math.\ J.\ 141, No.\ 2, 241--291 (2008; Zbl 1138.14022)].NEWLINENEWLINEGiven an equivalence between \(\mathrm{D}^b(X)\) and \(\mathrm{D}^b(Y)\), there is an induced map \(\Phi_*\) on the associated stability manifolds and this map is said to respect the distinguished component if \(\Phi_*\mathrm{Stab}^\dagger(X)=\mathrm{Stab}^\dagger(Y)\). It was known that shifts, isomorphisms and line bundle twists have this property. The author proves in Section 5 that (a) Fourier--Mukai equivalences induced by the universal family of a fine compact two-dimensional moduli space of Gieseker-stable sheaves, (b) spherical twists along Gieseker-stable spherical vector bundles and (c) spherical twists along \(\mathcal{O}_C(k)\) for a \((-2)\)-curve \(C\) and \(k\in \mathbb{Z}\) all respect the distinguished component.NEWLINENEWLINEIn the next section the following theorem, which gives a more geometric explanation of Ma's result recalled above, is proved. Denoting the quotient map from \(\mathrm{Stab}^\dagger(\mathcal{T})\) to the special open subset of \(\mathrm{KM}(\mathcal{T})\) by \(\pi\), the author shows that, given a standard cusp \([v]\), there exists a path \(\sigma(t)\) in \(\mathrm{Stab}^\dagger(\mathcal{T})\) such that \(\pi(\sigma(t))\) converges to \([v]\). Furthermore, fixing an equivalence \(\Phi: \mathrm{D}^b(Y)\cong \mathcal{T}\), which exists by the result quoted above, the heart \(\Phi(\text{Coh}(Y))\) can be described as the limit of the hearts \(\mathcal{A}(\sigma(t))\). In addition, the author describes paths of stability conditions mapping to special paths, called linear degenerations to a cusp, in the Kähler moduli space. It turns out that the stability conditions of such a path can be explicitly described for \(t\gg 0\) via data on the \(K3\) surface associated to the cusp.NEWLINENEWLINEThe next result, proved in Section 7, concerns moduli spaces of \(\sigma\)-stable objects in \(\mathcal{T}\). Namely, given a vector \(v\in N(X)\) satisfying some properties, the author considers the moduli space of \(\sigma\)-stable, where \(\sigma\) is a \(v\)-general stability condition (this roughly means that stable=semistable), objects \(E\) with Mukai vector \(v\) modulo some equivalence relation and proves that this moduli space is represented by a \(K3\) surface, which is derived equivalent to \(X\).NEWLINENEWLINEIn the following section the author recalls mirror symmetry for \(K3\) surfaces. Roughly, the usual weight \(2\) Hodge structure on \(H^2(X,\mathbb{Z})\) is determined by a nowhere vanishing holomorphic two form \(\Omega\) and it also induces a weight \(2\) Hodge strucutre \(H_B\) on the Mukai lattice \(H^*(X,\mathbb{Z})\). On the other hand, given a Kähler form \(\omega\) on a \(K3\) surface \(Y\), which makes \(Y\) into a symplectic manifold, and a so-called B-field \(\beta\in H^2(Y,\mathbb{R})\), the element \(z=(1,i\omega+\beta,\frac{(i\omega+\beta)^2}{2})\) defines a weight \(2\) Hodge structure \(H_A\) on \(H^*(Y,\mathbb{Z})\). Now, a complex \(K3\) surface \(X\) and a symplectic \(K3\) surface \(Y\) are called mirror dual if \(H_B(X)\cong H_A(Y)\). It also makes sense to talk about the period map of a family of \(K3\) surfaces and of mirror symmetry for families.NEWLINENEWLINEIf \(X\) is a quartic \(K3\) surface, then the Fubini--Study metric makes it into a symplectic manifold and the author explicitly computes the period map for the natural family of quartics (roughly given by scaling the symplectic form and varying a B-field) in Section 9. The last section presents the computation of the period map for the so-called Dwork family, which is constructed from the Fermat pencil of quartics by taking the quotient by a finite group and resolving singularities.