arXiv · math/9812050
We proved that every rational curves in the primitive class of a general K3 surface of any genus is nodal.
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Xi Chen. 1999-07-20. Singularities of Rational Curves on K3 surfaces. https://arxiv.org/abs/math/9812050
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We prove that the moduli stack of index-one covers of semi-log-canonical surfaces of general type is isomorphic to the KSBA moduli stack of stable general type surfaces. Using the index-one covering Deligne-Mumford stack of a semi-log-canonical surface, we define the $\lci$ cover. The $\lci$ cover, as a Deligne-Mumford stack, has only locally complete intersection singularities. We then construct the moduli stack of $\lci$ covers so that it admits a proper map to the moduli stack of surfaces of general type. The smoothing of lci covering Deligne-Mumford stacks can not control all the non lci slc singularities. Thus, we construct the moduli stack of birational lci enhancement Deligne-Mumford stacks for all non lci slc singularities. This moduli stack is in general nonseparated. We construct a separated {\em algebraic space} from this moduli stack. Next, we construct a perfect obstruction theory on each of these moduli stacks and a virtual fundamental class in the Chow group. We then pushforward the virtual fundamental class to the KSBA moduli space. Thus, our construction proves Donaldson's conjecture on the existence of a virtual fundamental class for KSBA moduli spaces. A tautological invariant is defined by integrating a power of the first Chern class of the CM line bundle over the virtual fundamental class. This serves as a generalization of the tautological invariants defined by integrating tautological classes over the moduli space $\overline{M}_g$ of stable curves to the moduli space of stable surfaces.
Let $\mathcal{H}_2$ be the Lie algebra of polynomial Hamiltonian vector fields on the symplectic plane. Let $X$ be the moduli space of stable Higgs bundles of fixed relatively prime rank and degree, or more generally the moduli space of stable parabolic Higgs bundles of arbitrary rank and degree for a generic stability condition. Let $H^*(X)$ be the cohomology with rational coefficients. Using the operations of cup-product by tautological classes and Hecke correspondences we construct an action of $\mathcal{H}_2$ on $H^*(X)[x,y]$, where $x$ and $y$ are formal variables. We show that the perverse filtration on $H^*(X)$ coincides with the filtration canonically associated to $\mathfrak{sl}_2\subset \mathcal{H}_2$ and deduce the $P=W$ conjecture of de Cataldo-Hausel-Migliorini.
In this article, we study how the absolute coregularity of a projective log pair reflects on its fundamental group. More precisely, we conjecture that for a projective klt log pair $(X,D)$ of absolute coregularity $c$ (and arbitrary dimension) the fundamental group $π_1^{\rm reg}(X,D)$ admits a normal abelian subgroup of finite index and rank at most $2c$. We prove this conjecture in the cases $0 \leq c \leq 3$, building on the almost abelianity of the fundamental groups of klt Calabi-Yau pairs of dimension $\leq 3$. In the cases $c \in \{0,1,2\}$ and fixed dimension, we can furthermore bound the index of a solvable normal subgroup. In dimension three, we are able to prove almost abelianity of the fundamental group of the regular locus for projective klt Calabi-Yau pairs.